GIFT OF MICHAEL REESE MATHEMATICAL PEOBLEMS THE FIRST AND SECOND DIVISIONS OF THE SCHEDULE OF SUBJECTS CAMBRIDGE MATHEMATICAL TRIPOS EXAMINATION. DEVISED AND ARRANGED JOSEPH WOLSTENHOLME, M.A. LATE FELLOW AND TUTOR OF CHRIST'S COLLEGE; SOMETIME FELLOW OF ST JOHN'S COLLEGE; PROFESSOR OF MATHEMATICS IN THE ROYAL INDIAN ENGINEERING COLLEGi:. Tricks to shew the stretch of human brain, Mere curious pleasure, or ingenious pain." Pope, Essay on Man. TJNIVKT^'STTY^I SECOND EDrflON^!&^ATjL^'^NLA RO£D. Ilontion : MACMILLAN AND CO. 1878 [The lii'jht r,f Tmvi'h'tlon is rcscrrcJ.] v^" Cambrttigc : PRINTED BY C. J. CLAY, M.A. AX THE UNIVEBSITY PRESS. ~5& l^y^ PREFACE TO THE FIRST EDITION. This " Book of Mathematical Problems" consists, mainly, of ques- tions either proposed by myself at various University and College Examinations during the past fourteen years, or communicated to my friends for that purpose. It contains also a certain number, (between three and four hundred), which, as I have been in the habit of devoting considerable time to the manufacture of pro- blems, have accumulated on my hands in that period. In each subject I have followed the order of the Text-books in general use in the University of Cambridge; and I have endeavoured also, to some extent, to arrange the questions in order of difi&culty. I had not sufficient boldness to seek to impose on any of my friends the task of verifying my results, and have had therefore to trust to my own resources. I have however done my best, by solving anew every question from the proof sheets, to ensure that few serious errors shall be discovered. I shall be much obliged to any one who will give me information as to those which still remain. I have, in some cases, where I thought I had anything ser- viceable to communicate, prefixed to certain classes of problems fragmentary notes on the mathematical subjects to which they relate. These are few in number, and I hope will be found not altogether superfluous. This collection will be found to be unusually copious in problems in the earlier subjects, by which I designed to make it useful to mathematical students, not only in- the Universities, but in the higher classes of public schools. 3 t> t+o VI rUKFACK. 1 have to fXpR'ss my bust tlianks to Mr K. Morton, Fellow of Christ's College, ibr his great kindness in reading over the proof sheets of this work, and correcting such errors as were thereby discoverable. NOTICE TO THE SECOND EDITION. The present edition has been enlarged by the addition of such other problems from my accumulated store as seemed to myself worthy of preservation. About one hundred of these, and pro- bably the most interesting, have appeared in the mathematical columns of the " Educational Times," and many of the others have already been used for Examination purposes. The "Fragmentary Notes" have been increased, and I hope improved. Answers are given in the great majority of cases and sometimes hints for the solution. I have taken much pains to avoid mistakes, and although, from the nature of the case, I dare not venture to expect the errors to be few .in number, I hope they will not often be found of much importance. The greater number of the proof sheets have been read over by my colleague, Professor Minchin, and many improvements are due to his suggestion. I am deeply grateful for his kind and efficient help. I shall be thankful for information of misprints or other mistakes which are not in the list of Errata. R. I. E. College, Nov. 3, 1878. CONTENTS. PBOBLEUS 1—117. 118—1^3. 124 (1—74). 125—142. 143—153. 154—199. 200—223. 224—233. 284—251. 252—261. 262—281. 282—296. 297—307. 308—317. 318—347. 348—361. 362—375. 376—401. PAGE GEOMETRY (Euclid) 1 ALGEBRA. I. Highest Common Divisor 12 II. Equations 13 III. Theory of Quadratic Equations 17 IV. Theory of Divisors 21 V. Identities and Equalities 22 VI. Inequalities 31 VII. Proportion, Variation, Scales of Notation ... 35 VIII. Progressions 36 IX. Permutations and Combinations 38 X. Binomial Theorem 39 XI. Exponential and Logarithmic Series .... 42 Xn. Summation of Series 46 XIU. Recurring Series 50 XIV. Convergent Fractions 53 XV. Poristic Systems of Equations 61 XVI. Properties of Numbers 64 XVII. Probabilities 65 PLANE TRIGONOMETRY. 402—428. I. 429—484. U. 485—510. III. 511—524. IV. 525—584. V. 585—598. VI. 599—629. VII. Equations .... Identities and Equalities Poristic Systems of Equations Inequalities .... Properties of Triangles Heights and Distances. Polygons Expansions of Trigonometrical Functions, Functions f)30— 635. Vlll. Scries Inverse 70 76 85 91 94 102 104 109 CONTENTS. CONIC SECTIONS, GEOMETRICAL. rnOBLEMS PAGE 636— 712. I. rnrabolii 115 713— 8-43. II. Central Conies 122 844— 882. III. Rectangular Hyperbola 134 CONIC SECTIONS, ANALYTICAL. 883- - 913. s5 r I. 914- 976- - 975. -1088. •is II. III. 1089- -1121. ^<^ IV. 1122- -1152. V. 1153- -1235. VI. 1236- -1287. VII. 1288- -1400. VIII 1401- -1455. IX. 1456- -1540. X. 1541- -1571. XI. straight Line, Linear Transformation, Circle Parabola referred to its axis Elliijse referred to its axes .... Hyperbola, referred to its axes or asymptotes Polar Co-ordinates General Equation of tlie Second Degree Envelopes (of the second class) Areal Co-ordinates Anharmonic Ratio, nomographic Pencils and Ranges. Involution .... Reciprocal Polars and Projections Invariant Relations bet\Yeeu Conies. Covariants 138 145 153 178 183 188 204 215 236 244 258 1572—1641. THEORY OF EQUATIONS 266 1642—1819. DIFFERENTIAL CALCULUS 281 1820—1856. HIGHER PLANE CURVES 810 1857—1993. INTEGRAL CALCULUS 319 1994- -2030. I. 2031- -2071. II. 2072- -2134. HI. 2135- -2156. IV. 2157- -2173. V. 2174- -2179. VI. 2180- -2187. vn. 2188- -2207. vin SOLID GEOMETRY. Straight Line and Plane 347 Linear Transformations. General Equation . . 354 Couicoids referred to their axes 361 Tetrahedral Co-ortlinates ...... 371 Focal Curves : Reciprocal Polars 374 General Functional and Diiiferential Equations . . 377 Envelopes 378 Curvatuie 379 CONTENTS. IX STATICS. PROBLKMS PAaE 2208—2243. I. Composition and Resolutiou of Forces . . 883 2244—2257. II. Centre of Inertia . 388 2258—2279. III. Equilibrium of Smooth Bodies . 390 2280—2295. IV. Friction . 393 2296—2304. V. Elastic Strings 396 2305—2329. VI. Catenaries, Attractions, &c 397 DYNAMICS, ELEMENTARY. 2330—2356. I. Rectilinear Motion : Impulses 2357—2379, 11. Parabolic Motion under Gravity . 2380 — 2391. III. Motion on a smooth Curve under Gravity 402 406 409 2392—2436. NEWTON. (Sections I. II. III.) . 412 DYNAMICS OF A PARTICLE. 2437—2477. I. Rectilinear Motion, Kinematics 418 2478—2512. II. Central Forces 424 2513 — 2559. III. Constrained Motion on Curves or Surfaces: Particles joined by Strings 429 2560—2573. IV. Motion of Strings on Curves or Sm-facea . . . 437 2574—2596. V. Resisting Media. Hodographs 440 2597—2613. I. 2614—2632. U. 2633—2657. III. 2658—2692. IV. 2693—2764. 2765—2784. DYNAMICS OF A RIGID BODY. Moments of Inertia, Principal Axes .... 444 Motion about a fixed Axis 446 Motion in two Dimensions 449 Miscellaneous 454 HYDROSTATICS 463 GEOMETRICAL OPTICS 474 2785—2815. SPHERICAL TRIGONOMETRY AND ASTRONOJIY 477 EEEATA. Page 73, question 415 is wrong. — — 107, line 6 from the bottom, for "all values of 6" read "values of ^ between - - and - ." 108, line 10, for +336?j + lG2, read -336n + 164. 109, line 5, dele the second "tbat." 211, question 1267, insert "Prove that." 273, line 9, for x + a^^, x + a.^ read x + b-^^, x + h.2. 289, question 1686, for u}-^" read ?t-»-i; and for x^'^", read x^"''^. 327, line 6 from the bottom, for in (m - 1), read m {n - 1). „ line 3 from the bottom, for +... + dx„, read dx.2,,. 430, question 2519, for tt- , read w + ip. 444, line 3 from the bottom, for 23 read 32. 446, line 2 from the bottom, for 31 read N. GEOMETEY. 1. A point is taken witliin a polygon ABC ...KL; prove that OA, OB,...OL are together greater than half the perimeter of the polygon. 2. Two triangles are on the same base and between the same parallels; through the point of intersection of their sides is drawn a straight line parallel to the base and terminated by the sides which do not intersect: prove that the segments of this straight line are equal. 3. The sides AB, AC of a triangle are bisected in D, E, and CD, BE intersect in F : prove that the triangle BFC is equal in area to the quadrangle ADFE. 4. AB, CD are two parallel straight lines, E the middle point of CD, and F, G the respective points of intersection of AC, BE, and of AE, BD: prove that FG is parallel to AB. 5. Through the angular points of a triangle are drawn three parallel straight lines terminated by the opposite sides ' : prove that the triangle foz'med by joining the ends of these lines will be double of the original triangle. 6. If «, b, c be the middle points of the sides of a triangle ABC, and if through A, B, C be drawn three parallels to meet be, ca, ab re- spectively in A', B', C, the sides of the triangle A'B'C will pass through A, B, C respectively, and the triangle ABC will be double of the triangle A'B'C. 7. In a right-angled triangle the straight line joining the i-ight angle to the centre of the square on the hypotenuse will bisect the right angle. 8. Through the vertex of an equilateral triangle is drawn a straight line terminated by the two straight lines drawn through the ends of the base at right angles to tlie base, and on this straight line as base is described another equilateral ti'iangle : prove that the vei-tex will lie either on the base of the former or on a fixed straight line parallel to that base. ^ All straight lines are supposed to be produced if necessary. W. P. 1 _ GEOMETRY. '■••' ^.■- '■'Fh.VOiigih' th«- 'angle C of a pm-iillelogram ABCD is Jiawn a stniiglit line inoeting the two sides AJi, AD in P, Q: prove that the rectangle uuder Bl\ DQ is of constant area, 10. In any quadrangle the squares on the sides together exceed the squares on the diagonals by the square on twice the line joining the middle points of the diagonals. 11. If a straight line he divided in extreme and mean ratio and produced so that the part produced is equal to the smaller of the seg- ments, the x'ectangle contained liy the whole line thus produced, and the part produced together -with the square on the given line will be equal to four times the square on the larger segment. 12. Two equal circles touch at A, a circle of double the radius is drawn having internal contact with one of them at B and cutting the othei- in two points: prove that the straight line AB will pass through one of the points of section. 1 3. Two straight lines inclined at a given angle are drawn touching respectively two given concentric circles : their point of intersection will lie on one of two fixed cii-cles concentric with the given cii'cles. 14. A chord CD is drawn at right angles to a fixed diameter AB of a given cii'cle, and DP is any other chord meeting AB in Q : prove that the angle PCQ is bisected by either CA or CB. 15. AB is the diameter of a circle, P a point on the circle, PM perpendicular on AB ; on AM, MB as diameters are described two cii-cles meeting AP, BP in Q, R respectively : prove that QR will touch both circles. 16. Given two straight lines in position and a point equidistant from them, prove that any cu'cle through the given point and the point of intersection of the two given lines will intercept on the lines segments whose sum or whose difference will be equal to a given length. 17. A triangle cii'cumscribes a circle and from each point of contact is di'awn a perpendicular to_the straight line joining the other two : prove that the straight lines joining the feet of these perpendiculars will be parallel to the sides of the original triangle. 18. From a fixed point of a given circle are di-awn two chords OP, OQ equally inclined to a fixed chord : prove thai PQ will be fixed in dii-ection. 19. Through the ends of a fixed chord of a given circle are drawn two other chords parallel to each other : prove that the straight line joining the other ends of these chords will touch a fixed circle, 20. Two circles with centres A, B cut each other at right angles and their common chord meets AB in C ; DE is a chord of the first circle passing through B : prove that A, D, E, C lie on a circle. 21. Four fixed points lie on a circle, and two other cii'cles are drawn touching each other, one passing through two fixed points of the four and the other through the other two : prove that their point of contact lies on a fixed circle. GEOMETRY. 3 22. A cii-cle A passes through tlie centre of a cii-cle B : prove tliat their common tangents will touch A in points lying on a tangent to B ; and convei'sely, 23. On the same side of a straight line ^17? ai-e described two seg- ments of circles, AP, AQ are chords of the two segments including an angle equal to that between the tangents to the two circles at A : prove that F, Q, B are in one straight line. 24. The centre ^ of a circle lies on another circle which cuts the former in ^, C ; ylZ) is a chord of the latter circle meeting BG in E and from D are drawn DF^ DG to touch the former circle : prove that G, E, F lie on one straight line. 25. If the opposite sides of a quadrangle inscribed in a circle be produced to meet in P, Q, and if about two of the triangles so formed cii'cles be described meeting again in E : F, E, Q will be in one straight line. 26. Two cii-cles intersect in A and through A any two straight lines BAG, B'AG' ai-e drawn terminated by the circles : prove that the chords BB\ CG' of the two circles are inclined at a constant angle. 27. If two circles touch at A and FQ be any chord of one circle touching the other, the sum or the difference of the chords AF, AQ will bear to the chord FQ a constant ratio. 28. Four points A, B, G, F are taken on a cii'cle and chords PA', FB', FG' drawn parallel respectively to BG, GA, AB : prove that the angles AFA', BFB', GFG' have common internal and external bisectors. 29. Two cii'cles are di-awn such that theii" two common points and the centre of either are corners of an equilateral triangle, F is one common point and FQ, FQ' tangents at F terminated each by the other circle : prove that QQ' will be a common tangent. 30. On a fixed diameter AB of a given circle is taken a fixed point G from which perpendiculars are let fall on the straight lines joining A, B to any point of the circle : prov^ that the straight line joining the feet of these perpendiculars will pass through a fixed point. [If I) be this fixed point and the centre, the rectangle under OG, OD will be half the sum of the squares on OG, 0A.'\ 31. Four points are taken on a cii'cle and the three paii'S of straight lines which can be drawn through the four points intersect respectively in E, F, G: prove that the three pairs of straight lines which bisect the angles at E, F, G respectively will be in the same dii'cctions. 32. Through one point of intersection of two circles is cb-awn a straight line at right angles to their common chord and terminated by the circles, and through the other point is drawn a straight line equally inclined to the straight lines joining that point to the extremities of the former straight line : prove that the tangents to the two circles at the points on this latter straight line will intei-sect in a point on the common chord. 1—2 4 GEOMETRY. 33. Two circles cut each other at A and a straight linc^^lC is di-awn terminated by the circles ; with B, C as centre are described two circles each cutting at right angles one of tlie former ciicles : prove that the.se two circles and the circle of which JJC is a diameter will have a common chord. 3i. Cii-cles are described on the sides of a triangle as diameters and each meets the perpendicular from the opposite angular point on its diameter in two points : prove that these six points lie on a circle. 35. The tangents from a point to a cii-cle are bisected by a straight line which meets a chord 2^Q of the cii-cle in B : prove that the angles JiOP, OQR are equal. 36. A straight line PQ of given length is intercepted between two straight lines OP, OQ given in position ; through P, Q are drawn straight lines in given directions intersecting in a point R, and the angles POQ, PRQ are equal and on the same side of PQ (or supplementary and on opposite sides) : prove that R lies on a fixed circle. 37. From the point of intersection of the diagonals of a quadrangle inscribed in a circle perpendiculars are let fall on the sides : prove that the sum of two opposite angles formed by the straight lines joining the feet of these perpendiculars is double of one of the angles between the two diagonals. 38. If OP, OQ be tangents to a circle, PR any chord tlu-ough P, then will QR bisect the chord drawn through parallel to PR. 39. Two chords AB, AC of a circle are drawn and the perpendicu- lar from the centre on AB meets AC m. D : prove that the straight line joining D to the pole of BC will be parallel to AB. 40. A cii"cle is drawn subtending given angles at two given points : prove that its centi-e lies on a fixed circle with respect to which the two given points are reciprocal ; and conversely that if a circle be drawn with its centre on a given circle and subtending a given angle at a fixed point it will also subtend a fixed angle at the recipx'ocal point. 41. Prove the following construction for finding a point P in the base BC of a triangle ABC such that the ratio of the square on AP to the rectangle under the segments BP, PC may be equal to a given ratio : — Take the centre of the circle ABC and divide AO in 0' so that the ratio of AO' to O'O may be equal to the given ratio, the circle whose centre is 0' and radius O'A will meet BC in two points each satisfying the required condition. If P, Q be the two points AP, AQ will be equally inclined to the bisector of the angle A and will coincide with this bisector when the given ratio has its least possible value, which is when O'O is to ^0 in the duplicate ratio of BC to the sum of the other two sides. Also the construction holds if 0' lie in OA produced, AP, AQ being then equally inclined to the external bisector of A and coinciding with it when the given ratio has its least possible value outside the triangle, which is when 00' has to O.'l the duplicate ratio of BC to the difference of the other two sides. GEOMETRY. 5 42. If a circle touch each of two other circles the straight line passing thi'oxigh the points of contact will cut off similar segments from the two circles. 43. Two circles have internal contact at A, a straight line touches one circle at F and cuts the other in Q, Q': prove that QF, FQ' subtend equal angles at A. If the contact be external, FA bisects the external angle between QA, Q'A. 44. A straight line touches one of two fixed circles which do not intersect in F and cuts the other in Q, Q': prove that there are two fixed points at either of which FQ, FQ' subtend angles equal or supple- mentary. 45. Any straight line is drawn through one corner ^ of a parallelo- gram to meet the diagonal and the two sides which do not pass through A in. F, Q, F : pi'ove that AF will be a mean proportional between FQ, FE. 46. In a triangle ABC are given the centres of the escribed circles opposite B, C and the length of the side BC : prove that (1) -4 lies on a fixed straight line ; (2) AB, AC are fixed in direction ; (3) the circle ABC is given in magnitude ; and (4) the centre of the circle ABC lies on a fixed equal circle. 47. Any three points are taken on a given cii'cle and from the middle point of the arc intercepted between two of the points per- pendiculars are let fall on the straight lines joining them to the third point : j)rove that the sixm of the squares on the distances of the feet of* these perpendiculars from the centre is double the square on the radius. 48. At two fixed points A, B are drawn AC, BD at right angles to AB and on the same side of it and of such magnitude that the rectangle A C, BD is equal to the square on AB : prove that the cii'cles whose diameters are AC, BD will touch each other and that their point of contact will lie on a fixed cir-cle. 49. ABC is an isosceles triangle right angled at C and the parallelo- gram ABCD is completed ; with centre D and radius DC a circle is described : prove that if P be any point on this cii'cle the squares on FA, FC will be together equal to the square on FB. 50. A circle is described about a triangle ABC and the tangents to the cii'cle at B, C meet in -4'; through A' is drawn a straight line meet- ing AC, AB in the points B', C : prove that BE, CC will intersect on the circle. 51. If Z) be the middle point of the side BC of a triangle ABC tixA the tangents at B, C to the cii'cimiscribed cii-cle meet in A' , the angles BAA', DAC will be equal. 52. The side BC of a triangle ABC is bisected in D, and on DA is taken a point F such that the rectangle DP, DA is equal to the rectangle BD, DC : prove that the angles BFC, BAC are together equal to two right angles. 6 GEOMETRY. ^ 53. If the circle inscribed in a circle ABC touch £C in i), the circles inscribed in the triangles AJW, VAC will touch each other. Also a similar property holds for the escribed circles. 54. Given the base and the vertical angle of a triangle : prove that the centres of the four cii'cles which touch the sides of the triangle will lie on two fixed cii'cles passing through the extremities of the base. 55. A circle is drawn through B, C and the centre of perpendicu- lars of a triangle ABC ; D is the middle point of BC and AD is produced to meet the circle in E : prove that AE is bisected in D. 5G. The straight lines joining the centres of the four circles which touch the sides of a triangle are bisected by the circumscribed circle ; also the middle jjoint of the line joining any two of the centres and that of the line joining the other two are extremities of a diameter of the cii-- cumscribed circle. 57. With three given points not lying in one straight line as centres describe thi-ee circles which shall have three common tangents. 58. From the angular points of a triangle straight lines are drawn perpendicular to the opposite sides and terminated by the circumscribed cii-cle : prove that the parts of these lines intercepted between their point of concourse and the cii-cle are bisected by the corresponding sides respec- tively. 59. The radii from the centre of the circumscribed circle of a tri- angle to the angular points are respectively perpendicular to the straight lines joining the feet of the perpendiculars. 60. Three circles are described each passing through the centre of perpendiculars of a given triangle and through two of the angular points : prove that their centres are the angular points of a triangle equal in all respects to the given triangle and similarly situated : and that the rela- tion between the two triangles is reciprocal. Gl. If the centres of two of the cii'cles which touch the sides of a triangle be joined, and also the centres of the other two, the squares on the joining lines are together equal to the square on a diameter of the cii'cumscribed circle. 62. The centre of perpendiculars of a triangle is joined to the middle point of a side and the joining line produced to meet the cii-cumscribed circle : pi-ove that it will meet it in the same point as the diameter through the angular point opposite to the bisected side. 63. From any point of a given circle two chords are draA\ai touch- ing another given circle whose centre is on the cii'cumference of the former : prove that the straight line joining the ends of these chords is fixed in direction, 64. ABC is a triangle and the centre of its circumscribed circle ; A'B'C another triangle whose sides are parallel to OA, OB, OC ; and through A', B', C are drawn straight lines resjiectively parallel to the corresponding sides of the former triangle : prove that they will meet in a point which is the centre of one of the circles touching the sides of the triangle A' B'C, GEOMETRY. 7 G5. A triangle is dra\\ni having its sides parallel to the straight lines joining the angular points of a given triangle to the middle points of the opposite sides : prove that the relation between the two triangles is reciprocal. 6G. Two triangles ai'e so related that straight lines dra^vn through the angular points of one parallel respectively to the sides of the other meet in a point : prove that straight lines dra\vn through the correspond- ing angular points of the second parallel to the sides of the first will also meet in a point ; aad that each triangle will be divided into three tri- angles which are each to each in the same i-atio. G7. The diameter AB of a circle is produced to C so that BC = AB, the tangent at A and. a parallel to it through G are drawn and any point P being taken on the latter the two tangents from F are drawn foi-ming a triangle with the tangent at A : prove that tliis triangle will have a fixed centroid. 68. A common tangent AB is drawn to two circles, CD is their common chord and tangents are drawn fi-om A to any other circle through C, D : prove that the chord of contact will pass through B. 69. Four straight lines in a plane form four finite triangles : prove that the centres of the four circumscribed circles lie on a circle which also glasses thi-ough the common point of the four circumscribed circles. 70. A triangle ABO is inscribed in a circle and A A', BB', CO' are chords of the circle bisecting the angles of the triangle (or one internal and two external angles) and meeting in B : prove that B'C", C'A', A'B' respectively bisect BA, BB, EC at right angles : also the circles BBC, EB'C will touch each other at E, EA being the common tangent. 71. Two of the sides of a triangle are given in position and the area is given; through the middle point of the third side is drawn a straight line in a given direction and terminated by the two sides : pi'ove that the rectangle under the segments of this straight line is constant. 72. In the hexagon AB'CA'BC the three sides AB', CA', BC arc parallel, as are also the three BA', CB', AC : prove that AA', BB', CC will meet in a point. 73. Two pai'allelogi'ams ABCD, A' BCD' have a common angle B : prove that AC, A'C, DD' will meet in a point; or, if the parallelograms be equal, will be parallel. 74. On two straight lines not in the same plane are taken points A, B, C ; A', B', C respectively : prove that the three sti-aight lines each of which bisects two coi'responding segments on the two straight lines will meet in a point. 75. Four planes can be drawn each of which cuts six edges of a gi^'en cube in the corners of a regular hexagon, and the other six pro- duced in the corners of another regular hexagon, whose area is three times that of the first, and whose sides are respectively perpentlicular to the central radii drawn to the corners of the firat. 8 GEOMETRY. 76. Given the circumscnbed circle and the centre of perpendicuhirs of a triangle, prove that the feet of the i>erpendiculai-s lie on a fixed, circle, and the straight lines joining the feet of the iKjrpendiculars touch another fixed circle. 77. Given the circumscribed circle of a triangle and one of the circles which touch the sides, prove that the centres of the other three circles -which touch the sides will lie on a fixed circle. 78. If 0, K he the centres of the circumscribed and inscribed circles of a triangle, L the centre of perpendiculars, and OK be pro- duced to // so that OH is bisected in K, then will HL ^R - 2r, where li, r are the radii of the two cii'cles. 79. In any triangle ABC, 0, 0' are the centres of the inscribed circle and of the escribed cii-cle opposite A ; 00' meets BG in Z), any straight line through D meets AB, AG respectively in h, c, Ob, O'c in- tei-sect iu P, O'b, Oc in Q: prove that P, 2, ^ lie in one straight line perpendicular to 00'. 80. The centre of the cii'cumscribed circle of a triangle and the centre of perpendiculars are joined : prove that the joining line is divided into segments in the ratio of 1 : 2 by each of the straight lines joining an angular point to the middle point of the opposite side. 81. The side BG of a triangle ABC is bisected in i>, a straight line parallel to BG meeting AB, AG produced in F, F respectively is divided in Q, so that P(?, BD, QP' are in continued proportion, and through Q is drawn a sti'aight line EQR terminated by AB, AG and bisected in Q: prove that the triangles ABC, ABE' are equal. 82. On AB, AG two sides of a triangle are taken two points D, E ; AB, AC are produced to P, G so that BF is equal to AD and GG to AE; BG, CF, EG are joined, the two former meeting in H : prove that the triangle FUG is equal to the two triangles BUG, ADE together'. 83. If two sides of a triangle be given in position, and their sum be also given, and if the third side be divided in a given ratio, the point of division will lie on one of two fixed straight lines. 84. Two circles intersect in ^, jB, PQ is a straight line through A terminated by the two circles : prove that BP has to BQ a constant ratio. 85. Throiigh the centre of perpendiculars of a triangle is drawn a straight line at light angles to the plane of the triangle : prove that any tetrahedron of Avhich the triangle is one face and whose opposite vertex lies on this straight lino will be such that each edge is perpendicular to the direction of the opposite edge. 86. A, B, G, D are four points not in one plane, and AB, AG respectively lie in planes perpendicular to CD, BD : prove that AD lies in a plane i^erpendicular to BG ; and that the middle points of these six edges lie on one sphere which also passes through the feet of the shortest distances between the opposite edges. GEOMETRY. 9 87. In a certain tetrahedron each edge is pei'pendiciilar to the direc- tion of the opposite edge : prove that the straight line joining the centre of the cii-cumscribed sphere to the middle point of any edge will be equal and parallel to the straight line joining the centre of perpendiculars of the tetraheth-on to the middle point of the opposite edge. 88. Each edge of a tetrahedron is equal to the opposite edge : prove that the straight line joining the middle points of two opposite edges is at right angles to both : also in such a tetrahedron the centres of the in- scribed and cii'cumscribed spheres and the centres of gravity of the volume and of the surface of the tetrahedron coincide. 89. If from any point be let fall perpendiculars Oa, Oh, Oc, Od on the faces of a tetrahedron ABC D, the perpendiculars from A, B, C, D on the corresponding faces of the tetrahedron abed will meet in a point 0', and the relation between and 0' is reciprocal. 90. The gi'eatest possible number of tetrahedrons which can be constructed having their six edges of lengths equal to six given straight lines all unequal is thirty; and when they are all possible the one of greatest volume is that in which the three shoi'test edges meet in a point, and to them are opposite the other three in opposite oixler of magnitude. 91. Two tetrahedrons A BCD, abed are so I'elated that straight lines drawn from «, h, c, d pei'pendicular to the corresponding faces of A BCD meet in a point : pi-ove that straight lines drawn from A, B, C, D per- pendiciilar to the corresponding faces of abed will meet in a point o, and that vol. OBCD : vol. ABCD :: vol. obcd : vol. abed. 92. A solid angle is contained by three plane angles : prove that any straight line throiigh the vertex makes with the edges angles whose sum is greater than half the sum of the containing angles, and extend the proposition to any number of containing angles. 93. Two circles are drawn, one lying altogether within the other; 0, 0' are the two points which are recipi-ocals with respect to either circle, and FQ is a chord of the outer cii-cle touching the inner : prove that if PP', QQ' be chords of the outer circle passing through or 0', P'Q' will also touch the inner circle. 94. The circles described on the diagonals of a complete quadri- latei'al as diameters cut orthogonally the cii-cle circumscribing the triangle formed by the diagonals. 95. Four points ai-e taken on the circumference of a circle, and through them are drawn three pairs of straight lines, each intersecting in a point: prove that the straight line joining any one of these points to the centre will be perpendicular to the straight line joining the other two. 96. A sphere is described touching three given spheres: prove that the plane passing through the points of contact contains one of four tixed straight lines. 97. Four straight lines are given in position : prove that an infinite number of systems of three circles can be found such that the pomts of 10 GEOMETRY. iiiteraectiou of the four straight lines shall be the centres of similarity of the circles taken two and two. 98. In two fixed circles are drawn two parallel chords PP', QQ' ; PQ, P'Q' ai-e joined meeting the circles again in 7?, S ; li\ S', respec- tively : prove that the points of intersection of QQ', liR' and of PP', SS' lie on a fixed straight line, the radical axis of the two circles. 99. The six i*adical axes of the four circles taken two and two which touch the sides of a triangle are the straight lines bisecting internally and externally the angles of a triangle formed by joining the middle points of the sides of the former triangle. 100. If two circles have four common tangents the circles de- scribed on these tangents as diameters will have a common radical axis. 101. Four points are taken on a circle and from the middle point of the chord joining any two a straight line is drawn perpendicular to the chord joining the other two : prove that the six lines so drawn will meet in a point, which is also common to the four nine points' circles of the triangles each having three of the points for its angular points. 102. Given in position two sides of a triangle including an angle equal to that of an equilateral triangle ; prove that the centre of the nine points' circle of the triangle lies on a fixed sti-aight line. 103. Given in position two sides of a triangle and given the sum of those sides, prove that the centre of the nine points' cii-cle lies on a fixed straight line. lO-t. The perpendiculars let fall from the centres of the escribed, circles of a triangle on the corresponding sides meet in a point. 105. The straight lines bisecting each a paii' of opposite edges of a tetrahedron A BCD meet in and through A, P, C, D respectively are drawn planes at right angles to OA, OB, OC, OD : prove that the faces of the teti-ahedron bounded by these planes will be to one another as OA :0B :0C : OP. 106. A straight line meets the produced sides of a triangle ABC in A', B', C respectively : prove that the triangles ABB', ACC, A'CC, A'BB' A\dll be proportionals. 107. A point is taken within a triangle ABC, and through A, B, C are drawn straight lines parallel to those bisecting the angles BOC, CO A, AOB : proAe that these lines will meet in a point. 108. Straight lines A A', BB', CC are drawn through a point to meet the ojiposite sides of a triangle ABC : prove that the straight lines drawn from A, B, C to bisect B'C, C'A', A'B' will meet in one point ; and that straight lines drawn from A, B, C parallel to B'C, C'A', A'h' will meet the respectively opposite sides in three points lying on one straight line. 109. If two circles lie entirely without each other and any straight line meet them in P, P' ; Q, Q' respectively, there are two jioints such that the straight lines bisecting the angles POF , QOQ' shall be always at right angles to each other. ^ GEOMETRY. ^S;^^*^/^^ 11 110. Given two circles which do not intersect, a tangent to one at any point P meets the polar of P with respect to the other in P' : prove that the cii'cle whose diameter is PP' will pass through two fixed points. 111. A point has the same polar with respect to each of two circles : pro^'e that any common tangent will subtend a right angle at that point. 112. Given two points A, P, a straight line PA Q is drawn through A so that the angle PBQ is equal to a given angle and that BP has to BQ a given ratio : prove that P, Q will lie on two fixed circles which pass through A and B. 113. If be a fixed point, P any point on a fixed circle and the rectangle be constructed of which OP is a side and the tangent at P a diagonal, the angular point opposite will lie on the polar of 0. 114. If OA', OB', OC be perpendiculars from a jioint on the sides of a triangle ABC, then will AB' . BC . CA' + B'C . G'A .A'B=2a A'B'C x diameter of the circle ABC. 115. From a fixed point are let fall perpendiculars on two con- jugate rays of a pencil in involution : prove that the straight line join- ing the feet of these perpendiculars passes through a fixed point. 116. If be a fixed point, P, P' conjugate points of a range in involution and PQ, P'Q be ch-awn at right angles to OB, OP' ; Q ■will lie on a fixed straight line. 117. In any complete quadrilateral the common radical axis of the three circles whose diameters are the three diagonals will pass through the centres of perpendiculars of the four triangles formed by the four straight lines. ALGEBRA. I. Highest Common Divisor. 118. Reduce to their lowest terms tlie fractions ll.x-^+24a:^+125 55a:* + 24a;''+ 1 ^> a;'+24a;+55 ' ^^' 125a:*+24a; + 1 ' 9a;'+ 5a;- 2 . , 9a;^ + lla;''-2 ^^ 27a;='-45a;*-16' ^^ 81a;'+lla;+4' 2a;»-lla;'-9 . . a:^+ 11x^-54 ^•^ 4^?Tm^r81' ^' a;^+lla;+12' a^- 209.x +56 , . 8a;^-377a;^4-21 16x"-a;°+ 16x^32 a;«+ 2rf^+ Sx"- 2.r'+ 1 ^' 32x'-'+16x*-x'+16' ^ ' Qx'+x'+llx'-lx^-V ^'^^^ (a + /ix)^ + (A + 6a;)^ ^ ^ a + '2Jix + hx^ "*" a{h + bxy-2h{a'+ luc) {h + ^bx) + b{a + hxy ' 119. Simplify tlie expressions a:(l-f)(l-z') + y{l-z''){l-x') + z{l-x'){l-7f)-4xyz x + y + z — xyz ' a(6+c-a)- + &(c+a-Z')' + c(«.+6-c)-+(& + c-a)(c+«-6)(a+6-c) rt^(6 + c-a) + 6'(c+a-6)+c^(a+6-c)-(6+c-a)(c+a-6)(a+6-c) ' (1) (2) (3) (4) a" (6 - c) + 6" (c - a) + c^ (a - 6) 1 1 (a - i<) (a - c) (a - c^ (6 - c) (6 - c^) (6 - a) 1 1 + (c-fZ)(c-a)(c-6)"^(f^-a)(rf-6)(c^-c)' ALGEBRA. 13 ,_, bed cda . .. ^ (5) / rri \7~~~:7\ + 71 \n — jwl \ + *^^o simiLor terms, ^ ' {a-h){a-c){a-d) (b -c){b -d){b-a) ' (a\ n' (« + ^)(«+c) , 7,2 {h + c){b + a) (c+a)(c + b) ^ ^ {a-b){a-c) (b-c){b-a) {c-a){c-b)' {a + b){a + c) (b +c){b + a) (c + a) (c+b) ^^ {a-b){a-c) {b-c){b-a) {c-a){c-b)' /R\ n* (^ + ^)(« + g) . 7,4 {b + c){b + a) {c + a){c + b ) ^^ {a-b){a-c)'^ {b-c){b-a) (c-a){c-b)' (Q) 8aW + ^(6- +V- a')(c'+a'-b' ) ( a'+b'-c' ) ^ ' {a + b + c){~a + b+c){a-b + c){a + b-c) 120. Prove thcat (ab - cd) (a' -b' + c'- d') + (ac - bd) (a' + ¥-c'- d') {a'-b- + c'- if) (a- + 6-- c' - cf) + 4 {ab - cd) {ac - bd) _ {b + c) {a + (/) 121. Prove that {b- c){l + ¥){\ + c') + {c-a){l + c'){\ + or) + {a-b){\ +a'){l+¥) a{b-c){\+b'){\+c')+b{c-a){\+d'){\+a') + c{a-b){\+a'){l+lr) _\—bc — ca — ab ~ a + b + c — abc 122. Prove that {{a+b){a + c)+2a{b + c)Y-- {a - by{a - c)" __ {{b + c){b + a) + 2b(c + a)Y -{b- cY{b - a)' b ^ {{c + a){c + b) + 2c{a + b)Y-{c - ny{c-by c = 8(6 + c){c + a){a + b). 123. Prove that {(6 - cy + {c- ay + {a - by} {«-" {b - cy + b'{c- ay + c' (« - by} = 2{b-cy{c-ay{a- by + {a {b -cy + b{c- ay + c{a- 6)'}". II. Equations. 124. Solve the equations (1) (a;+l)(a;+2)(a; + 3) = (a;-3)(a; + 4)(a; + 5), (2) (05+ \){x + 2)(a; + 3) = (x- \){x -2){x- 3) + 3(4u;- l)(a; + 1), 14 ALGEBRA. (3) (x +a){x + a + b) = {x + h) {x + 3rt), 1 _1___L _L ^> x+l'^ x-^5~ x + Z'^ x + T (6) 20 10 a:+20 15 5 a;+10 rK4 15' 1 x-\- 6a 2 + 5- + x — oa 3 x + 2a 6 ic + a ' X b - 1 X + b - ■a ' x + b- -c ^' a b - + «- -c 6 + (c + J — c x + a — c x + b x + a ^^ («-6)x' + a-/8 (6-c)a; + /3-y {c-d)x + y-h {d-af \d — a)x + ^ — a (10) 4(a;-a)' = 9(a:-6)(a-6), (11) 2{x-2ay=={^x-2b){Za-b), (12) a;(a;-5)(a;-9) = (x-6)(a;^-27), (13) (a; + 7) (a;'-4)= (a;+ 1) (a;' + Ua;+ 22), a;'' + 3 x^-x+\_^x''-2x+\ (1*) "^ITi -^ aj_2 --^ x-Z ' /- ^x «* — a;4-l 03^ — Sx'.+ l _ 1 (15) -r-T- + — r-^=2^-: = 0. x—\ a — 3 4x — 8' riG^ ^ + 1 a^ + 2_ 110^ + 18 ^^^^ x-l^a:-2-''llx-18' n-x 111 1 (17) -+r+- = 7 , ^ ' a X a+ b + X (18) a- — -7 + 5 5 = a;, ^ ^ a-b b—a (19) (a? - 9) (a: - 7) {x - 5) (cc - 1) = (a; - 2) (a: - 4) (cc - 6) (a; - 10), (20) (a + a;)- + (6 + tB)--(a-6)^ (21) (a + cc)^ + (6+a;)^:=(a-5)^, /2o^ (6-c)' , (o-< ■ (^-5)^ _o ^ '' (6-cf-(a;-a)^ (c-a)*-(a;-6)^ (a-6)^-(a;-c)* ' (23) {.r(a + 6-a:)}- + {a(6 + r«-a)}^ + {6(« + £«-6)}^ = 0, (24) ALGEBRA. 15 (x + a) (x + h) {x - a) (x-b) _(x + c) (x + d) (x - c) {x - d) {x-a)(x-h) {x + a){x + b) {x-c){x-d) {x+c){x->fdy /o^x 5 5 18 1 (25) ^ + - — , = T + —^ + ? , ^ ' x + 2 x + 4: x + l x + 3 x + o (26) {x' - 18a; - 27)' = (x + 1) (x + 9)^ (27) (x^-27)' = (.T-5)(.'K-9)'', /oQ\ /a;'-lla;+19\" ^x-2 ,. ('') [ x^^x-n )'-'^2 = ^' (29) (x + 3)- (,x- - 9x + 9) {2x' -6x+9) + (x' + 3x- 9)' = 0, (30) «;(.. + 4) + 1(^4) ^10, (31) x''+l + (x + iy = 2{x' + x + lY, 1 2 9 14 7 ^^ 10a;-50 x-6 x-7 x-8 x-d ' m) 13x^-10 ^^"^^ 12 ^^^-^^ ^ ' a;" - 4a; + 5 *^ a;^ - 6x + 6 ' /Sn _i L+_L_JL_1 ^ ^ £C-1 a;-2 x-3 x-4: 30' ^^^^ a;'-7a; + 3~ar' + 7a; + 2"^' nn 2 3 ^a; ^ ^ o;^ + 2x - 2 a;^ - 2x- + 3 2 ' (37) 45 2^^^ +15 ^,^ ^ ' a;^ + 3a; + 3 a;" + a; + 1 ' ,„„, 7a; + 10 2a; + 4 , (38) - — - - — -_ = x% ^ ' a;- - 4a; + 5 a; - 2a; + 2 7a; -4 72a; -32 G5x' ^^^^ :?TT~a;^-4a; + 8^~y~~"' 6 2 2 (40) 3 (a;- + 2) + ^ + ^ tts = ^ T' ^ ' ^ ' x-\ (x-l) a;^ + a;+l' (41) (2a;' - 7x' + 9a; - 6)'' = 4 (ic" - a; + 1) {x^ - 3a; + 3)', . (42) (x - 2) (a; + 1)=' {x^ + 2a; + 4) (a;' - a; + 1)' + 15a;' + 8 = 0, (43) x'' + l+(a;+l)» = 2(a* + a; + l)^ (44) a;'" + 1 + (a; + 1)'" = 2 (a;» + a; + 1) V 15a;' (x= + a; + 1)', (45) 1 6a; (a; +^1) (a; + 2) (x + 3) = 9, (46) a:* + 2x'-lla;' + 4a;+4-0, 16 ALGEBRA. (47 (48 (19 (50 (51 (52 (53 (54 (55 (56 (57 (58 (59 (60 (61 (62 (63 (64 (65 (66 (67 ,2 9 ,**."" "^ ^ a;*- 7^+10 .-c"- 13a: + 40 40 20 + = a;' -10a; + 19, 8 12 a:* + 2a:-48 a;' + 9a; + 8 a;" + lOo; a;' + 5a:-50 + 1=0, — . ^,-T + „. + xix^V) " (a;-l)(a;-3) " (x - 1) (a; + 2) " a;* - 4 (a;' + l/ = 4(2a;-l), 2x _ 7 3 3 x" + 3a; — 7 a;" + a; - 3 ' 4a; +69 9.r+23 + 1-0, 8a;' a;^ + 2a; + 3 a;^ + a; + 1 ' a;' = 6a; + 6, 4a;' - 6a; + 3, a;' + 6a;- =36, xi/{x + y) = l2x+ 3y, xy (4.^ + y - a;^) = 12 (a; + i/ - 3) ; a^\/l -y^-yJ^ -x' = xy-Jl -x^Jl-y-^l; x^ + Qxy^ + 2?/' = 90, y (a;' + xy + 7/) = 21 ; x{y + z — x) = a', y (z + x — y) = h^, z (x + y — z) = c^; a;V2a;/ + 2y^ = l = i-A + |; x'' + 2yz = a, y' + 2zx = b, z^ + 2xy = e; u^ + v* + 2xy = a, x^ + y^ +2uv = h, ux + vy=c, vx + uy = d; xyz = a (y^ + z') — h[z'+ x") — c {x' + y^) ; , 11 1 1 7 11 cy + oz— , az + cx= ■:: , bx + ay= : ax by ex h^-c^ c'-a^ a^-b' 3, x+ = 2/ + ^^ = * + _„, = s/'^y^; yz zx xy X + y + z = mxyz, yz + zx + xy = n, {l+x'){l+f-)(l + z') = {l-ny', (68) 10/ + 13s='-6y^ =242, 5s'' + 10a;'-2sa; =98, 13x'' + 5?/'' -16a;y = 2; proving that an infinite number of solutions exist j ALGEBRA. 17 (G9) x; + 2x.^x^ + 2x^x^r=a^, x^'+2x^x^ + 2x^x. = a^, x,' + 2x^x^ + 2x^x^^a^, x^ + 1x^x^ + 2x^x, = a^, x^^+2x^x^ + 2x^x^ = a^; (70) x^^+ x^ + 2x^x, + Ix^x^ = «, , x^x, + x,x^ + xx^ =. 6, , .T,^ + x^ + 2it' a:-, + Ixx, - «,, aj,.r, + xx. + ai^.i; = ?> , a S ill 4" — '— i J" 40 '1 ' x^ + a:/ + Ix^x^ + 2a!^.'r;, - a^, x.^x.^ + x^x^ + x^x, = b^ ; (71) a'' + 2/" + «' + Gxijz = - 4 or «, a;?/* + y^^ + zx^ =5 or 5, xz^ + ya;' + «?/" = — 1 or c ; (72) £Cj^ + 3x^x.f + Sx^ {x^ + cc/) + Cx'^-r^r^ = 17 or a,, fljg'' + Zxjr.^ + Sjc^ (ajg^ + x^) + Gaj^ajga;, = 13 or a^, x^ + 'ix.jv^ + 3.^1 (x^ + .r^^) + Cajg.T^aj^ =15 or a^, x^ + Zxx} + 3a;, (a;,^ + cc/) + 6a:^a;,aj, = 19 or a^ ; and shew liow to solve systems of equations like (69) and (71) with any odd number of nnkno-wTi quantities, and systems like (70) and (72) with any even number. [From (69) may be obtained (a;, + a)a?2 + ul'x^ + ta^x^ + oi*x^" = a-^ + ?>.] III. 125. In the equation a b c rf - + + J- + = 0, X — ma X — mc x + mo x + ma prove that, if a + 6 + c + (? = 0, the only finite value of x will be m [ac + bd) a + b W. P. 18 ALGEBRA. 12G. In tlic equation X + />, X + b^ x + b^ x + b^ ' ]trovc that, if ctj + a^ + a^ + a^ r^ 0, and aJJ^ + ajb.^ + a.p^ + aj)^ = 0, the only finite value of x will be ?A;±^A;_±f .j<-±". V - (6, + J. + i,, + 6.). aj)' + ajbj' + a J) J' + aj}' \ i - 3 4/ 127. The equation {x + Jx" - be) {y + J if - ca) {z + Jz' - ab) = abc is equivalent to ax- + by- + cz- = ahc + 2x?/2;. 128. Find limits to the real values of x and y Avhich can satisfy the equation x" + \2xy + 4/ + 4a; + 8y + 20 = 0. [a; cannot lie between — 2 and 1, nor y between — 1 and |.] 129. If the roots of the equation ao(^ + 2hx + b-0 be possible and diflferent, the roots of the equation (a + b) {ax' + 2Aa; + 6) = 2 {ab - h') {x' 4 1 ) will be impossible : and vice versd. 130. Prove that the equations x+y+z = a + b + c, X 11 z ^ abc 1- — 4- - = () a' b' c' ' are equivalent to only two independent equations, if be + ca+ ab = 0. 131. Obtain the several equations for determining a, /3, y so that the equations X* + ijJX^ + Gqx^ + irx + s = 0, {x" + 2px + a)" = {^x + yf, may coincide : and in this manner solve the eqtiation {x' + 3x - G)= + 3x- = 72. ALGEBRA. 19 132. Shew bow to solve any biquadratic of the form x' + ^ax" + ^ — , , .-^ =0, 46 [by putting it in the form \x'+ax + —^—j =^bx + a ^ ^^ ' j j] and hence solve the equations (1) a* -8a;' -108 = 0, (2) x*-lOx'-34:5Q^O. 133. Prove that the equation x^ + Sax- + 3hx -\ — = a can be solved directly, and that the complete cubic x^ + Zpx^ + 3qx +r-0 can be reduced to this form by the substitution x = y + h. Prove that the roots of the auxiliary quadratic are a(^-yy + P(y-ay + y (a-l3y^ J -Z3(/3-y){y-a)(a-(3) 0-yY + {y-ay + {a-l3y a, (3, y being the roots of the original cubic. 134. The roots of the equation (x + a-c){x + b + c)(x+ a-d){x + b + d) = e will all be real if IGe-c {a -b-2cy {a -b-2ciy and > - 4:{c-dy (b + c + d-ay. 135. Detemiine X so that the equation in x 2A X 2B ^ + -+ = x + a X X — a may have equal roots ; and if X,, X^ be the two values of X, JC,, x,^ the corresponding values of x, prove that x^x^^a', \,X^=:{A-By. 136. Prove that, if two relations be satisfied, the expression {x' + ax + m) (x' + bx + 7ii) {x' + cx + in) will contain no power of x except those whose index is a multiple of 3. Resolve a;''-20x-^+ 343 and a^H- 3Ga;''+ 1000 into their real quadratic factors and identify the roots found fx'om the expression in x'' with those of the several quadratic factors. 2 2 20 ' ALGEBRA. 137. Tlie equations x + y z xu — z x + j/ x + y _z _ xy a b h ah — h" li:ivo the unique solutioii X y ^ _ -{n + bf "TdTH? " , 7"«&-/r " A ~ (a-bV + ih' ' rt - 2 ,- b-2 — -i- a + b (i + b 1 38. Tlie four equations a: + y-2z_ X- + y- - 2s^ ^ xy-z^ ^ xy(x + y) - 2^" ^ xY - ^* a + b a- + b^ ~ ah ab{a + h) d'U' ai-e consistent and equivalent to tlio three x + y _ X1J _ fa - /^Y lab a+ b ~ ah~ \a + b) ^ ^~ a + b' 1 39. The system of equations .r, -x^-a (x'3 - x^), x^-Xi = a(x^-x.), Xi-Xs = a{x^-x^), Xr, — x^ = a(x^ — x^), ■will be equivalent to only two independent equations ii a{a- 1) = 1. [This may also be proved from Statical considerations.] 1 40. The six equations 3 (cy + bz) {by -^cz) 3 _ {by + cz) (be + yz) bc + yz cy + bz , „ (az + ex) (cz + ax) . (cz + ax) (ca + zx) b' - — , y = -^ , ca -\rzx «« + ex 2 (hx + ay)(ax + by) ^3 (ax + by) (ah + xy) ~ ab + xy ' ^ bx + ay ' ai'o equivalent only to the two independent equations ax + by + cz=0, ayz 4- bzx + cxy + abc - 0. [Geometrically these equations expi-ess relations between the six joining lines of a quadrangle inscribed in a cii'cle.] f. c'y + h'z a'z + c'x h'x + a'y ax + hu + fs = 0, — J = '— -x+ 1/ + z, •^ ' a b G J ^ prove tliat a{bh' 4- cc — aa!) + h'icd + act! — hb') + c(aa + bh' — cc) = 2a'b'c\ a\bb' + cc' — aa) + b {cc + aa — bb') + c {cia + bb' — cc) - 2a' be, a (bb' + cc' — aa) + b'{cG' + aa' — bb') + c (aa + bb' — cc) = 2ab'c, a (bb' + cc' — aa) + b {cc' + aa' — bb') + c'{aa' + bb' — cc') = 2abc', wliicli are equivalent to tlie two-fold relation corresponding terms in the two being taken with the same sign. 142. From the equations x^ + 2yz =a, y'+ 2zx -b, s" + 2xy = c, obtain the result 3 {yz + ZX + xy) -a + b + c- J a' + b' + c^-bc- ca - ab. IV. Theory of Divisors. 143. Determine the condition necessary in order that x^ +px + q and x^ + p'x + q may have a common divisor x + c, and prove that such a divisor will also be a divisor of px' + (S' — p')'^ — S"'* 144. The expression x" + 3ax' + Zbj-" + c.^ + 3dx' + 3cx +/ will be a complete cube if T,^ /c d c-a^ , 145. The expression x^ - bx^ ■\- cx' ■\- dx - e will be the product of a complete square and a complete cube if Ub_U_rH__il 5 ~ b " c ~ c' ' 146. Prove that ax"" +bx + c and a + bx* + cx^ will have a common quadratic factor if b'-c^ = (c- - a" f b-){G^ - a* + ab) ; 22 ALGEBRA. and that ax" + tx* + c and a + &.«^ + cxi' avHI liavc a common quadratic factor if {it'-c'){a'-c' + hc)=-a'b\ 147. Prove that a^x* + a,x' + a^x^ + OgX + a^ and a^ + a^x + «j,x' + OgO;'' + a^x'* will have a common qiiadratic factor if 148. Prove that ox" + 6x' + c and « + &x* + cx^ will have a common quadratic factor if (a" + ah — c") (a" — ah - c°)* - h'c-(a^ — c°). 149. The expression x' +px- + qx + r will be divisible by x" + ax + b if a^ — 2pa^ + {p- + q) a ■¥ r - pq = 0, and W — qh' + rph - r" = 0. 150. The expression x* +2}x + q will be di"\T.sible hj x^ + ax +b if a" - iqa" = p^ and (6^ + q) {b" - qf = p%^. 151. The highest common divisor of p{x^ —V) —q{x^ —V) and (y — p)x' — gx'"'' +/) is {x— 1)", p^ q being numbers whose greatest com- mon measure is 1 and q being greater than p. 152. If w be any positive whole number not a multiple of 3, the expression x^" + 1 + (x + 1)'" will be divisible by x^ + x + 1 j and, if n be of the form 3r — 1, by (x^ + x + l)^ 153. Prove that {ah - 7r) {x (X - X) + 1/ (y- Y)^- h (x -X)^ + 27. (x - X) {y-Y)-a{y- Yy will be divisible by (x - Xy + (y - Y)" if X^-Y' _XY _ 1 a — h li A* — ah ' V. Identities and Equalities. 154. Prove that . (1) {a-¥h + cY=a^ + h^ + c' + Z{h+c){c + a){a + h), (2) J-+_J , 2 (6-cr + (c-ar+(c.-6r ^ 6-c c-a a-6^ (6-c)(c-a)(a- 6) (3) (,S'- 6^)(^'- c'') + (^-c^)(^- a') + {S-a'){S- h') = is{s — a) {s — h)(s — c), where 2S=a^ + h^ + c\ and 2s=a + 6 + c; ALGEBRA. 23 (i) {b-c){l+ab){l+ac) + {c-a){l + hc)(l +ba) + (a -b){l+ m)(l + cb) ={b-c){c- a){a - b), (5) a{b-c)(l + ab) (1 + ac) + the two similar terms E - abc (b — c)(c- a) (a — b), (6) (b-c){\ + ic'b){l + a*c) + the two similar terms = - abc (a + 6 + c) [b — c){c- o) [a — h), (7) (b^ — c") (1 — a'b) ( 1 — «"f) + the two similar tenns = (1 + abc) (a^ + b^ + c'+bc + ca + ab) {b -c) (c- a) {a - b), (8) 26"c*(c + a)- (a -i- by + the two similar tenns = a* (b + cy +b*{c + ay + c' {a + by + 1 Ga'bV {be + ca + ab), (9) (a' + 2bcy + {b' + 2cay+ {c'+2aby - 3 {a' + 2bc) (b' + 2ca) {c' + 2ab) = (a^ + b^ + c^ - Zabcy, (1 0) 8a'6'c* + {¥ + c- - a^) {c' + a' - ¥) {a' + b'- c') = (a- + b" + c''){a + b + c){- a + b + c) {a-b + c) {a + b + c), (11) (a - by (a - cy+ (b - cy{b- ay + {c-ay-{c-by = (a' + ¥ + c- — bc — ca — aby = l{{h-cy+{c-ay + {a-hy], (12) ft (6 - c) (5 + c — o) ' + the two similar terms E IGoic (6 — c) (a — Z*) (ft - c), (13) a{b — c){b + c — ay + the two similar terms = 1 Qabc (b - c) (a - b) {a - c) {(ft + & + c)" - 4 (a" + 6" + c% (i 4) {bed + cda + dab + obey - abed {a + b + e + dy E {be — ad) {ca — bd) {ab — cd), (15) {a + b + e + dy - 4: {a + b + c + d){bc + ad + ca + bd + ab +cd) + 8{bcd + cda + dab +abc) = - {b -i-c-a- d){c + a-h-d) {a+ b-c - d), {b + cy + (c 4- ay + (ft + by-3{b+ c){c + a){a + b) _ ^ ^^^^ a'+b^' + c^-Sabc (17) (x^-rB + l)(x*-i»/ + l)...(x2"-x2-'+l) _ a;2"^' + x"-" + 1 ~ af + x-¥ 1 * (18) {{b - cy + (c - ay + (ft - b)'] {ft* (6 - cy + b\c - ay + e'{a - 6)*} = 3{b-cy{e-ay{a-b)\ 24 ALGEJBIU. 155. Prove the followmg identities, where o, h, c, d arc the roots of the equation x' - Lj^ + Mx^ - xYx- + P - 0, and the product of their differences (J -c){c- a) {a - b) {a -d){b- d) (c - d) is denoted by A, ( 1 ) (6'c' '^d + a'd" h + c) (6 - c) (a - d) + (cV 6 + c/ + 1/d^ c + a) (c -a){b- d) + (a'i* ^T7+ c'd' a+h) {a -b){c- d) = 0, (2) {{b + c)' + {a + d)-} (b - c) {a - d) + the two similar terms E 0, (3) {b'c^ + a'd'){b-c){a-d)+ E -A, (4) {¥-c'){a'-d'){bc + ad) + = A, (5) {¥-c'){a'-d;-){b-¥c){a + d) + E-A, (6) {bc{b + cY + ad{a + df]{b-c){a-d)+ E-A, (7) ij/ + c'){a' + d'){b-c){a-d) + E A, (8) (6c"F+? + af?f7T^)(6-c)(rt-cZ)+ eXA, (9) {be b'T^+ ad ci+d) {b' - c') (a' - d') + E ZA, (10) {{b + cy + {a + dy}{b-c){a-d)+ E - 5ZA, (11) (bY + a\P){b-c){a-d) + E-J/A, (12) {bc + ad){b^-c^){a^-d')+ E J/A, (13) {bc + ad){b-cY{a-dy + E J/A, (U) {b + cy{a + dy{b-c){a-d) + = -2M\ (15) {b'c' + a'd'){b'-c'){a'-cr)+ = (Ly~F)A, (16) {b' + c'){a' + d'){b^-c^){a^-d'} + = {P-LN)A, (17) {bc + ady{b'-c'){a'-d')+ ^{iP-LN)\ (18) (h + c-a- dy {b- c){a-d) + E A. 156. The expression {ax^ + hf + cz^) -^ {be {y - zY + ca (5; - a;)- + cd) {x - ?/)-} will have a constant value for all values of x, y, « which satisfy the equation ax + b>/ + cz-0. 157. li xy + x + y = 2, then will a;* -8a; _y*-8y _xy (4 -a;y) ,1 + a;^ ~ T+]y ay - 1 ALGEBPtA, 25 158. If IJ,^ + iii^m^ = I J, + ?/y?ii = IJ.^ + ^^h'>h > then will <^7-')^;'",";')/"-',v''"- H- -^ = 0. 159. If 5:4 = 1 and?i'+^-=^,, a a a+b thon will + - ,— = - , ) . a b \a + bj 160. Havins: given =«, —^ — =6, =c, find tkc relation ^ ° y + z ' « + .<; x + y between a, b, c ; and prove that x' 9/ z' a — abc b — abc c — abc 101. Having given the equations X + y + z =1, ax +by +CZ =d, a^x + b^y + c'z = dP, prove that a^x + b^y + ch- d^ -{d-a) (d-b) (d- c), and a*x + b*y + c^z ~d^ — {d — a) (d-b) (d—c) {a + b +c + d). 162. If - + r + - = — 1 J then, for all integral values of «, a b c a+b+c ^ ' 111 1 163. If x + y + z^xyz, ox \i yz + zx + xy ^\, 2x 2i/ 2z 2x 2y 2^ l-x' 1-y' l-z' l-x'l-9/l-z''' , y + z z + x x + y^y + zz + xx + y \—yz \—zx \—xj l—yzl—zxl-xy' 16-1. 1£ yz + zx + xy = (yz)~^ + (zx)"^ + (.^■y)~' ^ m, then will (1 +7/z){l+zx ) (1 + xy) _ I +m (1 + x^) (1 + y^yij^i - {i^r^^ • 1 ijo. Having given the system of equations bz + cy cx + az ay + bx ax + by + cz b—c c—a a—b a+b+c ' prove that (b + c)x+(c + a) y + {a + b)z^O, bcx + cay + abz - 0, and that either a-^b + c^O or abc^(b- c) (c - a) (a - b). 26 ALGEBRA. IGG. If a, b, c bo real quantities satisfying tbe equation then will a^, i", c" be all less than 1, or all gi-eater than 1. 1G7. If X, y, z be finite quantities satisfying the equations ax (i/ + z — x) + by {z+ x — y) + cz {x + i/ — z) - 0, « V + b^7/- + c-z' = 2bcijz + 2cazx + lahxy, then will X y z xyz a(b — cy b{c- ay c (a — h)' abc (x" + y'^ + ^ - yz - zx — xy) 168. If yz + zx + xy-Q and (6 - c)" x + (c-a)-y + (a- b)" z ~ 0, then wall x{b — c) = y (c — a) = z{a — b). 1 G9. If x{b-c) +y {c-a) + z(a-h) = 0, then will bz — cy ex — az ay — bx b — c c — a a — b 1 70. If X, y, z, u be all finite and satisfy the equations x=by + CZ+ du, y -ax +CZ + du, z = ax + by + du, u = ax + by + cz, ., .,, a b c d ^ then will r + r — .4-^ +- .= 1. 1+a 1+6 1+c I +d 171. If 2^' = ^^,and?^^ = -, o — c x c —XI y then will x--'if ^xy a — b z ' z' - X- and if a + ^ = 6 + O-G then will each member of the equation be equal to c + h — G c — a'' X' — y" a — b 172. Having given the equations yz 0? + a;' zx a^ + y"^ , c* c* + a'x" c* c' + aY' . , . xy a* + z'^ „ prove that -^ = ^4^^,^, , yz^zx + xy^^ a\ and a* xyz = c* (x- + y + s). ALGEBRA. 27 173. Having given X 7/ + S y \ -x' m + ')ii/z ^ l — 'if m + nzx ' prove that, if x, y be unequal, Z X ^- U 1 /^ — - f yz-k-zx-\-xy + in-^\ = v, 1 — s^ m + nxy and (y^^)"' + {"'^)~' + (^"Z/)"' = 7i - 1- a;-— V i!C '2/ 174. If ^ = !, and rr, y ^^ unequal, then will eacli member ^ ~" 111 of this equation be equal to — — ^ , to a; + y + ;?, and to - H h 1 xy X y z 1 75. If 7 — ^^^ = 7 r-» = -„ , each member will be equal to {y-zf (z-x)- (x-yf ahc ^ («' + &' + c' - 26c - 2ca - 2ab). 176. 1£ X, y be unequal and if {2x-y-zy ^ (2y-z-xy X y ' each member will be equal to (22 - a: - 2/)' . , „ g , . ^^ — ^-'- , to 9 {x- + y +z- -yz-zx- xy), and to - 27 {x (y — z)- + y{z— x)^ + z[x — y)'] -^{x^ y + z). 177. Having given the equations alx + hmy + cnz = al'x + hm'y + cn'z = ax^ + by" + cz^ = 0, pi'ove that x (mri' — mn) + y (ul' — n'l) + z {Im' — I'm) = 0, and that (m — VIZ — ?t — n'yY {n - u'x -I- I'zf {l — l'y — in— mxf _ r ^ r • — V. a c 178. Having given the eqiiations Ix + my + ?«; = 0, (6 - c) y + (c - a) - 4 (a - 6) - = 0, 9 8 prove that ^'y^ (tji^ - 71^/) + wi^^x {^ix - Iz) + ii'xy {ly - vix) {r + m'^-n'f imn (h -c){c- a) (a - h). 28 ALGKlillA. ] 79. Having given a + b -{-c + (l = a' — b' -c +d' = ^ aa + bb' -f cc + del', ^. ^ aa'^ + bb"- + cc" -i-d a,6, + ajb,^ + ...+ ab_^ = 0, = 0, afi;-^ + a./);-^ +...+ ab,;-^ = 0, and the single value of x Avill then be equal to — + — 4- ... H T T r ^1 ^., b ' ^ "a,o," - + ... + « 6 " - 181. Having given the equations X y I {inb + nc — la) m {nc + la - mh) 71 {la + rab - nc) ' ,, , mz + m/ 7ix + lz ly + mx prove that = — ; — ^ ; a c and that x{py + cz — ax) y{cz + ax — by) z{ax + by-cz)' 182. If a, b, c, X, y, z be any six quantities, and a^ = bc — x^, 5j = ca — y', c^ = ab- z', x^=yz-ax, y^^zx-by, z^^xy-cz; and a„, b^, c„, x„, y„, z,^ be similarly formed from a^, b^, c,, x^, y^, and so on; then will a b c x y z = {ax" + 6?/- + cz' — aic — 2icy5;) 3 . ALGEBRA. 20 183. Prove the following equalities, Laving given that a + b + c = 0, ft' + 6' + c' _a^ + ¥ + c^ a- + h- + c^ 5 " 3 2 ' a' + ¥+c'' _a'' + b'+ c' a" + 6' + c' _ «" + ?>" + c' a' + ?>' + c" 7 ~ 5 2 ~ -6 2 ft" + ^," 4. c" _ g'' + 6=^ + c-^ «" + &" + c" _ (g'' + ^.■' + c"')^ ft" -f- &- + c" n ~ 3 'Z 9 2 " 184. Prove that (ft + 5 + C)' - ft' - &■' - C' , / „ ,2 , , ; X (ft + 6 + cf - aJ" -h^ - (? ■^ ^ ' 185. If ft + 6 + c + (Z = 0, prove that n^ + l^ + c' + (p _ a^ + h^ + c'+rr n' + Jr + c' + rP 5 " 3 2 18G. Having given « + 6 + c + ft'+5'+c'=0, ft' + 6' + c'+ft" + 6'='+c'^=0, prove that «'+ &''+ c'' + ft''' + h'' + c'^ _ ft'+ 6'+ C-+ «'-+ 6'-+ c'^ ft'' + 5' + c'+a"+6'''+c'^ 7 ~ 2 ^ 5 187. Prove that (2a-b-c + h^c J^d? ={2b-e-a + T^i J'^^f- 188. If X= ax + C7/ + bz, T-cx + by + az, Z = bx + ay + cz, then will X' + Y' + J^- 3XYZ^ (ft" + b' + c' - Zabc) (x' + y' + z^- 2xyz). Hence shew how to expi*es3 the product of any nnmber of factors of this form in a similar form. [By means of the identity a^ + ¥ + c' — 3a5c = (ft + 6 + c) (a + w5 + ojV) (a + oj'i + wc), where w* + w + 1 = 0.] The same equation will be true if X=ax + by + cz, Y=a7/ + bz + cx, Z=az + bx+ci/, and these two are the only essentially different arrangements. 189. 1{ x + I/ + Z-X1/Z and x'' = yz, then will y and z be ca|xd»lc of all values, but x^ cannot be less than 3. 30 ALGEBRA. 190. If a; + y + c - a' + y + ^' = 2, then will x{\-xY = y{\-yy = z{\-zy; also the gi-eatcst of the three x, y, z lies between ^ and 1, the next between 1 and J, and the least between ^- and 0; and the difierence between the gi-eatest and least cannot be less than 1 nor gi-eater than 2 ^/3• 191. Having given the equations {ij + zY = Wyz, {z + xf = ^¥zx, {x + yy= ic'xy; prove that «' + &' + c^ ± 2abc = 1 . 192. Having given the equations y z z X , X y z y X z y ^ prove that a" + 6' + c* = 2i'c- + 2cV + 2a%- + a-J/c^. 193. Having given the equations X y 2 . x {by' + cz' - ax') y' (c^ + ace' - hy) z {ax' + 6?/' - cz') ' prove that, if x, y, z be all finite, 194. Having given the equations jc" + 2/' + s' = (y + ^) {z +x){x + y), a{if + i? -x-) = h {z"" + X- -y-) ^c(x- + y^ - ^; prove that a^ + &^ + c^ = (6 + c) (c + a) (a + b). 195. If « (6 + c - a) {¥- + c^-a-) = b{c + a-b) {c" + a' - &') and a, 5 be unequal, then will each member be equal to c (a -f & - c) (a^ + 6^ - c^) and to 2abc (a + b + c); also 4a6c + (& + c - a) (c + a - 6) (a + 6 - c) = 0. [This relation is equivalent to ai'+¥ + c'={b + c) (c + a) (a + b).] 196. If a; = 6^ + c' - a", y = c^ + a^ - b% and z = a' + F- c\ prove that 2/ V + 2;V + a;y - xyz {y + z) {z + x) {x + y) is the product of four factors, one of which is iabc +{b + c-a) {c + a-b){a+b- c), and the other three are formed from this by changing the signs of a, h, c respectively. ALGEBRA. 81 197. If -+ r+ -•--+T'+ -7=0; a b c a be then will - {''■ + i' _ ^Y + 1 (i ^ "1 - f:y H- f (»■ + % - i)'= 0. a\o c a) b\c a b) c\a b cj 198, Simplify the fraction a (5-° - c^) + 6 (c' - ft ^ ) + c (g" - 6") ^ 6 (c - a)^ + c (a - hf -aib- c)' ' and thence the fraction whose numerator is or {h — cy — 2bc (a - h)" {a — c)' + the two similar expressions, and denominator «" (6" — c")- — 26c (a- — &") {a' - (f) + the two similar expressions. [The numerator and denominator ia the last case are each equivalent to (b-cy{c-ay-(a-by.] 199. If ¥ + bc + c'= Zif + 2uz + 3^", c' + ca + a- = Zz' +2zx+ 3x', and a' + a6 + 6" = 3a3" + 2xy + 3y", then will 3 {he + c« + aby = 32 {^/V + :^c(f + xry^ + xyz (x + 2/ + «)}. VI. Inequalities. [The symbols employed in the following questions are always sup- posed to denote real quantities. The fundamental proposition on which the solution generally de- pends is a* + 6* => 2ab. Limiting values of certain expressions invoh^ing an unknown quan- tity in the second degree only may be found from the condition that a quadratic equation shall have real roots: — e.g. "To find the greatest jy* 4a; 4- 7 and least values of -^ — ^ 2 •" Assuming the expression = y, we obtain the quadratic in x, x'{l-y)-2{2-y)x+7-4y = 0, and if X be a real quantity satisfying this equation we must have (2-2/r>(l-y)(7-4y), or 3y^ - 7y + 3 < 0, so that y must lie between "^ — and —- — , which are accord- o u ingly the least and greatest possible values of the expression.] S2 ALGEBRA. 200. If .T, y, z 1)0 llirce positive qiiantitics wliose sum i.s unity, then will (l-a;)(l-2/)(l-c)>8x?/c. 201. Prove that 4 (rt^ + // + c" + f^) > (a + 6 + c + d) {fC' W + e'+ cP) > (a= + Z*- + c" + r/')' ,- 1 Gahcd. 202. Trove that {Sa'bV + (6- + c" - a") (c= + a'- h"-) (a' + h'-c')}' > 3 {26V + 2cV + 2a'b' - a* - h* - c*Y except Avlicn a = h = c. 203. If a, h, c be positive and not all eqxial «■'' + Z>^ + c' + 3ahc >a-{b + c) + b" (c + a) +c^ (a + b). 204. If {a+b + cY <4:{b + c)(c + a) {a + b), then will ft^ + 6" + c" < 2bc + 2ca + 2ab. 205. If a, h, c he positive and not all eqnal, the expression a''(a-b) [a - c) A-b" {b - c) {b - a) + c''{c~a) (c-b) will be positive for all integi-al values of n, and for the values and - 1 . 206. Prove that, if oi be a positive whole number, 2 > n > 91 - ; and that A - 1) ^3 - 5) ... A - ^^zi) . 1. 207. If (7, &, c be the sides of a triangle, then will 1 1 11119 > -+ v+ -> 6+c— « c+a—b a+b—c a b c a+b+c' and {b + c- a)" {c + a- bf (« + 6 - c)" > {b" + c" - a") (c'f a" - 6") (rt^+ 6" -r) ; also X, y, z being any real quantities, "' (•'« - y) (^ - -) + ^'{y-~) {y-^) + c-(z-x) (z-y) cannot be negative. If x + y + z = 0, a^yz + b^zx + c'xy cannot be posi- tive. 208. If xyz = {l-x){l-y){l-z) the greatest value of either of these equals is J, x, y, z being each posi- tive and less than 1. 209. Prove that {axb + c + bye + a + cza + b)' > iabc {x + y+z) (ax + by + cz) ; a, b, c, X, y, z being all positive and a, b, c unequal. ALGEBRA. 33 210. Prove that, for real values of x, 2 (a- x) {x + Jx" + ¥) h, and c be positive, the greatest value which the expi'ession 16 (x - a) {x -b) {x-a- c) {x- b + c) can have for values of x between b — c and ci + c is {a — by(a — b + 2c)*. 213. If2J>m, x'' — 2mx + p^ p — m , p + m — — pr —„ > ^ and < . 03" + 2mx + p' p + 7n p - vi 214. The expression -^ — ^ '■ car + ox + a will be capable of all values whatever if 6" > (a + c)' ; there wtlII be two values between which it cannot lie if b^ < (a + c)' and > 4ac ; and two values between which it must lie if b" < iac. oir rr-i • (x-a)(x-b) 215. The expression ) ^7 j-; (x — c){x — a) can have any real value whatever if one and only one of the two a, b lie between c and d : otherwise there will be two values between which it cannot lie. 216. The expression x' +bx + c will always lie between two fixed limits if ¥ < 4c* ; there will be two limits between which it cannot lie if a^ + c' > ab and i* > 4c- ; and the expression will be capable of all values if a" ■¥0^ < ab. w. p. 3 34 ALGEBRA. ,.,- ,™ . nx° + 2hx+b 217. The expression , ^, — -,, — ,, '■ ax + 2/1X+ will be capable of all values, provided that {ah' - a'bf < 4 (a'h - ah') {lib - hh') {or, wLich is equivalent, (2M' - ab' - a'bf < 4 (li^ - ab) (h'' - a'b')}. Prove that this inequality involves the two h' > ab, h'" > a'b' ; and investigate the condition (1) that two limits exist between w^hich the value of the expression cannot lie, (2) that two limits exist be- tween which the value of the expression must lie. [ (1) {ab' - a'bf > 4 {a'h - ah') {bh' - b'h), 7r > ab, (2) {ab' - a'by > 4 {a'h - a'h) {bh' - b'h), 7r < ab.] 218. If cCj, x^, X.J, ... K, be real quantities such that so o "-In x^ + X,- + ... + xr- — x^x^ — xjc^ — ... - ic;^_,£C__ - x^^ + — = U, then will 0, x^, x„, x.^, ... x^^, 1 be in ascending order of magnitude. 219. If a;," + a;,' + ... + x^" + xjx^ + x^x^ + x.x„ + . . . = 1, then none of ' 2n the quantities x^, x^, ... x^ can be greater than l ; and tlieii- sum 2 must lie between 2 and ?i+ 1 220. If x^ + o^j" 4- ... x^ + 1m {x^x^ + x^x^ + x^x_, + ,..)= 1 , m being positive and < 1, then none of the quantities x^, «/, ... x^ can be ,,1 1 +mn — 2 ,,,. greater than . : and their sum must lie between (1 -m) (1 +mn- 1) :j and . 1-071 1 + m w - 1 221. If ccj- + £c/ + . . . + xj' — x^Co — x„x^ — ... - x^^_{c^^ - — — - , then will x^ and will have its least value when r is the intearer next below lo*?, n. VIT. Proportion, Variation, Scales of Notation. 224. If h-\-c + d, c+ d + a, d+a + b, a + h + c be proportionals, then will a—db—c 22.5. If 2/ vary as the sum of tln-ee quantities of which the first is constant, the second varies as x, and the third as .^•": and if {a, 0), (2a, a), (3a, 4a) be three pairs of simultaneous values of x and y, then when X = «a, y= (n— 1)'^ a. 226. A triangle has two sides given in position and a given peri- meter 2s : if c be the length of the side ojiposite to the given angle, the area of the triangle will vary as s — c. 227. The radix of the scale in which 49 denotes a square number must be of the form (r + 1) (;•+ 4), whei-e r is some whole number. 228. The radix of a scale being 4r + 2, prove that if the digit in the units' place of any number N be either 2r + 1 or 2r + 2, iV* will have the same digit in the units' place. 229. Find a number (1) of three digits, (2) of four digits, in the denary scale such that if the first and last digits be interchanged the result reiiresents the same numljcr in the nonary scale : and prove that there is only one solution in each case. [The numbers are 445, 55G7 respectively.] 230. If the radix of any scale have more than one prime factor there will exist two and Only two digits ditferent from unity such that if any number iV have one of these digits in the units' place, iV* will have tlie same digit in the units' place. 3—2 so ALGf:BRA. 231. Prove that the product of the niimbers denoted by 10, 11, 12, 13, increased by 1, will be the square of the number denoted by 131, whatever be the scale of notation. 232. Prove that \2m - I -^ \m\m - I is always an even number except when m is a power of 2, and the index of the power of 2 contained in it = q—]y, where q is the sum of the digits of 2m — 1 wlien expressed in the binaiy scale and 2'' is the highest power of 2 which is a divisor of m. 233. The index of the highest power of g which is a divisor of \pq-7- (!^>)' is the sum of the digits of ^^ when expressed in the scale whose radix is q. VIII. Arithmetical, GeometHcal, and Ilarmonical Progressions. 234. If the sum of m terms of an A. p. be to the sum of n terms as ni^ : n"; prove that the ni^^ term will be to the n^^ term as 2m -\ : 2n-\. 235. The series of natural numbers is divided into groups 1 ; 2, 3, 4 ; 5, 6, 7, 8, 9 ; and so on : prove that the sum of the numbers in the n^^ group is n^ + [n — iy. 236. The sum of the products of every two of n terms of an A. p., •whose first term is a and last term I, is n{n-2){Zn-l){a + Tf+in{n-\-\ )al 24 {n - 1) • 237. The sum of the ];)roducts of every three of n terms of an A. p. , whose first tenn is a and last tei-m I, is n(n-2){a-vl) , , „, , ^^'^ a / ^\ n ■ 48(,/_i) {'"' 0^ - 3) (« + 0- +^{n+l) al], ... - ^ n(n-l)(n-2) ^, ,, , nCn-\){n-2) and lies between — ^^ r4r al (a 4 I) and — ^^ j^ ^ (a + If ; 12 ^ ^ 48 \ / ^ and the sum of the products of every three of n consecutive whole numbers beginning with r is '^(''-'^H^-^) {(^ + 2r - 1)^ -(n + 1) (n + 2r - 1)}. 238. Having given that . , , and ^ are in A.P. : prove o — c c — a a — o that a^ + c''-2b^ _a + b + c a' + c'-2b' ~ ' 2 • ALGEBRA. 37 239. If a, b, c; h, c, a; or c, a, 6 be in A. p., then will § {a + h+cf= a' (6 + c) + h" (c + a) + c^ (a + &) ; and if in G. P. , 2-10. If a, I be the first and n^^ terms of an A. p. the continued product of all the n terms will be a + ^\" {alf and < ^-^ j . 241. The first term of a G. p. is a and the n^^ term I] prove that the r^^ term is 242. If a, 6, c be in A. p., a, p, y in H. p., and aa, b/3, cy in g. p.., then will 7 111 243. The first term of an ii. p. is a and the n^^ term I, prove that the r^^ term is {n — 1) at {n-r)l + {r—\)a' Prove that the sum of these n terms is < (« + ^) ^ ; and theii- con- tinued product < (aiy. 244. If a, b, c be in H. p. , then will 1 4 1 11 b — c c — a a — b c a' 245. If a, b, c, d be four positive quantities in h. p. , a + d>b + c. 246. Prove that b + c, c + a, a + b will be in h. p. , if a', b^, c" be in A. p. 247. If three numbers be in g. p. and the mean be added to each of the thi-ee, the three sums will be in H. p. 248. Prove that, for all values of x except — 1, 2-- '"-^>";3^»"-'^' 2- + ... E 91+ 1, (2) (m + 1 )"-(«- 1) m (m + 1)""^ + fr-2)(»-3) ^^, ^^^ ^ ^^,._, (?^-3)(^^-4)(;^-5) ,, _,„_« m"+' - 1 - ^ , ., '^ ' nf (m+iy ^+ . . . E -— , 1 3 ^ ' m — 1 (3) (?? + q)" - {n - 1) pq {p + g)"-^ + (^^- H^^~^) p-y (p + ?)"-* n being a positive integer. to ALGEBRA. 41 268. If p be nearly equal to q, then will ~ — J^ bo nearly equal N q' 269. If w be nearly equal to q, ; r-f^ — \ -4^ is a close ai)- proximation to (^ )"; and if diflfer from 1 only in the r + l"" decimal place, this approximation wUl be coiTect to 2r places. . 270. If a, denote the coefficient of a' in the expansion of ( — —-\ in a series of ascending powers of x, the following relation will hold among any three consecutive coefficients, (r + 1) a^^ , - 2na, - (r - 1) o,_, - 0. 271. If \j ^^3 be expanded in ascending powers of x, the coeffi- cient of a;""^' * is (?i + 2r) 2" ', n, r being positive integers (including zero). 272. If (1 + xf = a^ + a^x + . . . + ao! + . . ,, then \\\Vi a* + 2a/ + 3«3^ + . . . + na^ n a J + a'^ + «,- + . . . + a^^ 2 ' w being a positive integer. 273. Prove that «(..-!) , n(^^-l)(7^-2)(7^-3) .., ^+ p ^ + p^S ^ +---=2,^n' 274r. The sum of the first n coefficients of the expansion in ascend- • (1+a;)". ?i(» + 2){?i + 7)_„_^ , . ing powers ot x ot ' ' is — =^ -^ 2 , n bemg a positive integer. 275. Prove that, if ?i be a positive integer, 276. The coefficient of a;""^""" in the expansion of (1 +a;)" (1 — a:)~^ is 2""^ {(?i + 2r) (?i + 2r - 2) + «} ; and that of a;"*'"^ in the expansion of (1 + a;)" (1 - a-)-* is J {n + 2r - 2) 2"- {(n + 2>-) (?i + 2r - 4) + 3je}. 277. If the expansion of (1 + a;)" (1 + a:")" (1 + a;^)" be X-^a^x-i^a^ + ...+«x' + ... and /SjE a,+ff8 + «jj+...,AS'j,=:aj + aj + a"'+ ..., 'and *so on to S,, then will ^, = ^^^ ^;= ... ^s.^Xi^^"- 1). 42 ALGEJUIA. 278. If tlie expansion of (1 + a; + a;'' + . . . + xf')" bo 1 + a^x + a.p^ + . . . + aX+"- and /S^, = a, + «^+i + aj,p+, + ..., /Sj = a3 + ap^.a + «j^^.2+ ..., and so on to >S',,, then will S^- S^= ... = S^. 2 ( 9. Prove that )-pf — —^ — ^tt r — —, — Sa — a" {b-c) + Jr (c - a) + c" {a - b) is equal to tlic sum of the homogeneous products of n dimensions of a, h, c. 280. ProA-c that the coefficient of aj"" in the expansion of {\-ax)-^{\-bx)-"- in ascending powers of x is (r + 1) (ct-^^ - V*') - {r + 3) ab (a"*' - S"^') {a-br 281. If x = -i ) 5-/> n rs will be equal to the sum of the b + n{a — b) {I —xf ^ first n terms of its expansion in ascending powers of x; a, b being any unequal quantities. XI. Exponential and Logarithmic Series. [In the questions under this head, n always denotes a positive Avhole number.] 282. Prove the following identities : — (1) n"" -{n + \){n-lY + ^'^^^'>'\ n-^Y - ... towtermssl, (2) (,i_l)"-w(9i-2)-+'-?^^^^ (J^-3)''-... to7Z::Tterms = |7i-l, (3) (?i-2)"-7i(n-3)"+ ^ \ n-4:Y-... to7i-2termsE|^+ji-2", oi (n't ^ 1 I _ (4) r - n 2" + -^12— 3" - ... to n+ 1 terms = (- 1)"| w. [(1) is obtained by means of the expansion of e~' (e'- 1)"'^', (2) from that of €"'(£'-!)", (3) from that of £--^(e'- !)"+', and (4) from that of e-"(e--ir.] 283. Prove the following identities by the consideration of the coefficients of poAvers of x in the expansions of (c'-c"')" and of its 2x^ ( 2.' equivalent -(20; + -,^ 3 +....!:- ALGE15KA. 43 (1) n-''-n{H-2y-'' + ''^''~^\ }i-4:y-''-...=0, (2) ?i"-9i(H-2)" + '^*'^— \w-4)''-...EJ£i2"-', (3) n''^'-7i{n-2y^'+ "'^'^~^\ n-iy^'- ... = %\n + 2 2"-' : I '^ the number of terms in each series being - or — - — , and r a Avhole number < - . 284. If ^S*^ denote the series {2H + iy-{2n+l){2n-iy + ^-' |^ ^ (2m -3/ -...to n+\ terms, then will ^^ = ^^ = .9^ = . . . = S„_„_^ = 0, ^^„^ , = 2"-" \2n+\, _ 2'" {2n + 1) 1 27i + 3 and S^„^^ = 13 ' . 285. Prove that the sums of the infinite series m 1 1 1 ^' 1.2.3"^ 3.4.5"^5. 6. Z"^'"' /o\ 1 1 1 ^"^ 1. 2. 3. 4 "^3. 4. 5. 6 "^5. 6.7.8 ^■•* 1 4 9 ^^ 1.2.3.4'*'3.4.5.6"^5.6.7.8"^"" are respectively log 2 - ^, f log 2 - ^, and i log 2 - Jj- ; and that, if S^ denote the sum of the infinite series 1 1 1 1.2.3...(w + 2)'^3.4.5...(w + 4)'^5.6.7...(w+6)"^'"' (n + l)S^ = 2S„_^-^, 286. The coefficient of re' in the expansion of (1 + a;)" being denoted by a^, prove that «o/'""' - «> - l)"-' + ct, (p - 2)-' -...+(- 1)"- V. = «o(^i-p)""t«. {n-p-iy-'+a^{n-p-2y-'-... + (- \y-'-\_,_„ 2) being a whole number < n. [From the expansion of c"'' (e' — 1 )" containing no lower power of x than ic", so that the coefficient of .-c""' is zero. This result might be used to prove (264).] 44 ALGEBRA. 287. By means of the identity- log (1 - x') = log (1 - a;) + log (1 + a; + a;*), prove that the sum of n terms of the series n{n+l) {n - V) n {n + \){n + 2) if- + \5 - 3(— 1)""' is if n be of the forms 3r or 3r - 1 ; and — ^ — ^j— if 9i be of the form 3r + 1 ; also that the sum of n terms of the series 1 h — ^ L — jg /_ ly- ' if ?i be of the forms 3r ± 1 and —^ — if n be of the form 3r. 288. By means of the identity log (1 - a? + af ) + log (1 + x + x^) = log (1 + a;^ + a;*), pi-ove that, if/OO denote the sum of ?i+l terms of the series f{2n) = (- \Yf{n) j and that, if F{n) denote the sum of ?i+ 1 terms of the series oiiii+V) {n-l)n{n + \){n+2) \f~ + [5 ••■' {2n + l)F{n) = {-lY or 2{-l)"-'. , a;^ - 3a3 + 2 , « + 2 ^ , a; + 1 289. By means of the identity prove that ;3 " \b ^ o«T— i^-^){n-S)...(n-i) _ 2- - 1 . ■'-^'^ 17 ■'••■=2^^Tl' the series being continued so long as the indices of the powers of 3 are positive. 290. Denoting by u^ the series V + 2" + ^ + ... + ^-^^^^ + ...10 inanity, ALGEBRA. 45 prove that nhi—l) n(n—l)(n-2) , ,.„ w„_, + w,= u„+i -»i^*„ + I iy — M„_, ^ rP w„_„+ ... + (- 1) ?*, 11 n(n—l and that ii^^j - ^c^^ = u^^ + ?«w^_i + ^-^ — w^_„ + • • • + n^^■^ + w„ ; and by means of either of these prove that w^ = 4140€. 291. If ?*^ denote the infinite series 3""' 4"-' |2 [3 then will \=u ^,+u +nit , + — ^^r — - 111 C- 1V+' 292. If t, =^_^ + j^_...+L_^. then will \=v +v _+ -^- + ...+ '-— + , - " "' 2 \n-l ^n + l 293. Having given prove that the limit of ,- — ^ when n is indefinitely increased is ic, - 2u. 294. If there be a series of terms u^, «,, ti^, ...u^..., of which any one is obtained from the preceding by the formula w_ = mt,_, + (— 1)" and if w^ = 1, then will n {n — 1) n{n — \) In = ?« + nn. , H ^-^^ — - it , + ...+ L_ n n-I 2 "-2 W, + WM,+W„. ?t 1 Prove also that ,— tends to become equal to - as n increases \n ^ c indefinitely. 295. Prove that 2"+'- 2=2"-— o.-» , (n-2)(n-3) ,,.^ n + 2 - 1 2 [4 4G ALGEBRA. and that ]>" + 9" E {p + ry) ' - npq (2> + 9) -+ -' |2 pq- {p + 5) ?i (n — 4) (w — 5) - . . , _ , — - — ^ ^pY(p + + 2 a&c . . . { 22 (a') + 32 {ah) }. Also if a be any other quantity and if S^ now denote (a + sJ-2(a + s„_,X + 2(a + vJ-... + (-l)'->2(a + s,r, thenwiU /S', =^^ = ^3=...=^_,eO, /S'„E nabc ..., 2,S„^,E '?^+ l«5c... (2a + « + 6 + c + ...), and 1 2,S'„^ ^ e i?^ + 2 ajc . . . { 22 (a") + 32 {ab) + Ca2 («) + Ga'}. [These results are deduced from the identities (1) (e«^ - 1) (e^^ - 1) ... E £««^ - 2 (e«'-'^) +2 (c^'-^^) - . . . (2) c«'(€aJ'-l)(e&^--l) {...) = e(<»+«").c_2£(»+«'-i)-^ + 2{€<"+«'-2)-^}-... by taking the expansions of every term of the form e"" and equating the coefficients of like powers of x up to a;""^".] XII. Sumviation of Series. [If zt_ denote a certain function of n and S_ = u, + ii,+ ... + u 'n» the summation of the series means expressing S^^ as a function of n involving only a fixed number (independent of n) of terms. The usual ALGEBRA. 47 artifice by which this is efTected consists in expressing ?t^ as the difference of two quantities, one of Avhich is the same function of n as the other Ls of w— 1, {U^— ^^„_,). This being effected we have at once Thus if u^ be the pi'oduct of r consecutive terms of a given a. p., beginning with the ?i*'^, we have it,_ E {« + ?i - 1 6} (rt + nh) ... {« + (?i + r - 2) i} _ (a + ?i - 1 6) (rt + ?i 6) . . . (rt + n + r - 1 6) - (« + ?i — 2 i) . . . (« + 9i + r - 2 6) "" (7+1)6 ~ ^ whence T = (^ +"^^1 &)(^ + »&) ••> + ^ "^ ^- 1 ft) (r + l)6 and „ _ 1 " =(r + l)6 U" + H - 1 6) (« + ?i6) ...{a + n-^r-\h) -{a-h)a{a+h) ... (« + r-16)}. The sums of many series can also be expressed in a finite form by equating the coefficients of x" in the expansions of the same function of X effected by two different methods of whicli examples have been already given in the Binomial, Exponential, and Logarithmic Series. In the examples under this head, n always means a positive whole number.] 297. Sum tlie series : — ^> 2. 4^2. 4. 6^2.4. 6. b^'--"*"--^ ^^^ -" 1.2 2.2^ w2" /n 3 2.3^ wS" 2.4... 2w(2?i+2)' (4 [5 ^•••^|n^tj_' /K\ '^ 2?-- wr" W) i — TT+l ^+--- + i ■ |r 4- 1 \r+ 2 7i + r' 1 1 (6) + ... + (l+:.)(l + 2..) (l + 2^)(l + 3a.)-^--^(i^,,^.)(i^— i^y ^'^ 1.3"^ 1.3.5"^"-"^1.3.5...(2«+l)' /ox 1 , 3 _ 5 2?i-l ^'''' 3'"3.7-'3T7Tri-'-+3T77Tl...(4n-l)' 48 ALGEBRA. ,f.. 1 5 n' + 71-1 ,,„, 4 9 211" + 371-1 ,,,,1 5 11 7J* + W-1 2.5 5.10 10.17 (l+n^)(l+^r:n^) 2\ > (l+iK)(l+ar) (l+a.-*)(l+a;=') (1 + a;") (1 +«;"+') ' /V< /V» /J»* /V» /V»* /V'^ (13) T-r;:.+ i—zi 1-7^4 + 1 + a- 1 + ojM + a;* 1 + a;M +a;* 1 +a;« + on > 1 + a;^ 1 + a;* ' " 1 + «■ ,, , , £c (1 — aic) ax (1 — «°a;) (14) ^ -I ^^ h ^ ' (1 + x) (1 + ax) (1 + a'x) (1 + ax) (1 + a^x) (1 + a^x) a"~^x (1 —a"x) (1 + a" '«) (1 + a"x) (1 + a""^ V) (15) ._L+ __ _^+l (r+l)(2r+l) 2) + r (p + ?') (p + 2r) (jw + r) (j() + 2r) ( j»? + 3r) (r+l)(2r+l) ^ .. (n-lr+1) (p + r) (p + 2r) ... (p + nr) 298. Prove tliat ^ w^w-l}_ ?z(w-l)(?^-2) ^ 7^(7^-l)(7^-2)(^^-3 ) |2 |3 "^ g • + 0^-i)(-i)". 299. Prove that (1 + r + r"" + ?-^) (1 + r-+r* + r^) ... (1 + r2""' + r^" + r3.2"->) _ l-r2"-r2"+' + r3-2'' 300. Prove that /ix 1 1-3 1-3.5 (^> 4-'4:6+4T6-8+-**'°° = ^' .nx n 3 3.5 3.5.7 <2) ^■*- 8+ sTTo-' 8^0712 + •••*°«^ = 2' ,-, , 11 11.13 11.13.15 (^> 1-^ 14-^14:16+ 14716.18 + - *°=^ = 12; ALGEBRA. 49 and generally that, if ^;, q, r and q-j) -r be positive, the sum of the infinite series 1 + i^ + ** + (^l*') (Z±_2**) + (^ + 1') {P + 2r) (;9 + 3r ) ^ (7 + r (g- + r) (^ + 2r) (3- + r) (g- + 2r) (^ + 3r) will be [The sum of n terms of the last series is g-;j-rl q{q + r) ...{q + n-\r) ) .301. Prove that 2 2jA 2.4:.e...2n _ 2. 4. 6. ..(2^+ 2 ) 3"*'3.5"^""^3.5.7...(2n+l)-3.5.7...(2ri+l) 302. Pi'ove that, m being not less than ?i, /i\ 1 ** w(«-l) w(?i-l) (?i- 2) , . , (1) 1 + - + ; / , + —} -f) ^ + ... to ?i + 1 terms 7n m {m - 1) m (m - 1) (m - 2) _ «i + 1 ~ 7)1- n+1' (2) 1 , o^'f 3''^''~^^ I i ^^(^^-l)(»-2) , m m{m-l) m(7n- l)(m-2) (m + l)(m + 2) ~ (m -n+1) (m -n + 2)^ (3) U3-^ + 6^^i^i^UlO n0i-l)(n-2 l^ m m (in - 1) m (w - 1) (m - 2) (m + 1 ) (m + 2 ) (m +^) ~ (m - 71 + 1) (ni - n + 2) (»i - w + 3) ' 303. Prove that \-n-+ ^(^~^) «(«-!) _ n{n-l){n-2) a{a-l){ a- 2) b \2 b{b-l} |3 6(6-1) (6-2) 304. Prove that, if x be less than 1, ^ _ {r-l){r -2)^^{r-l ){r-2 ) {r-3){r-i) ^ j3 15 (r + 1 ) ("r + 2) (r + 1 ) (r + 2)~{r + 3) (r + 4 ) , |3 ^"^ |5 •''■ "•■• = (\+xY. W. P. 4. 50 ALGEBRA. [Obtained by expanding numerator and denominator of tlic fraction 305. If .,.^i--ii!^-^^-(-^)(-;-^)(--^^-)-..., \'A li ^,,a r^n- ''0'-^)(''-^) ^ n(n-l){n-2) {n-3){n-A) _ " |3_ [5 ' tlien -will 2i„" + v/ = 2 (w,w„_i + "y..^',,-,)- 30G. If .Hl-!i(!ifl).^^"("-')(»-S)(''-").U..:, and v,E?w; ^^ ~ -x+..., tlien will w/ + v/ = (1 + x^)" = (1 + a^) 0^ w„_, + ^'„^'„_,). 307. Prove the identity , ?i+l?lW-l,- ,, 9i+ 1 7i9i-l ^i- 2 Vi- 3 ,„ n + l 12 (2m + 1)+ — (22t + l)'-... n— 2 n — 3 = 2"- n- 1 2"-' (u + 1) + ^ — ^ 2"-" (zt + 1)^ 'Jt — 3 n— 4: n — 5 ^ „ ~2"-^u + iy+ \1 XIII. Recurring Series. [Tlie series u^ + ?t, + it^ + • • • + ^^„ is a recuiTuig series if a fixed number (r) of consecutive terms are connected by a relation of the form »,. +2\^K-, +;'A-2 + ••• +i^,_,«,-,+, = 0, {A) in which n may have any integral value, but 2\y P--'* •-•IK-i are inde- pendent of n. It follows that the series a^ + a^x + aj)tf + . . . + a^x" + ... is the expansion in ascending powers of x of a function of x of "the form A+Ax+...+A_„x''~^ ,. . - . „ , T— , J- ~ ;^j (the generating function of the series) : and if the scale of relation {A) and the first r - 1 terms of the series be given this function wall be completely determined ; when by separating this ALGEBRA. 61 function mto its partial fractions :j '— + ' — + . . . and expanding each we obtain the ri}^ tei-m of the series and the sum of n terms. Thus the n^'' term of such a series is J^^a.^"'^ + -Sa^a""' + ...where a^, a^, ... are the roots of i\iQ auxillavi/ equation x"'^ +2'>^x''~'' +p„x^~^ + ... + J>t-\ ~^» and B^, J^^i--- constants which can be determined from the first »""' teiTDS of the series. If however two roots of the equation be equal (say a^^a^) we must write B^i instead of 7>„, if three be equal (a.^=a^^a^ the corresponding tenns will be (B^ 4- nB,^ + n'B^ a^""' and so on. If the scale of relation is not given we shall require 2 (r— 1) terms of the series to be known to determine completely the generating function ; thus if four terms are given we can find a recurring series with a scale of relation between any three consecutive terms and whose first four terms are the given terms.] "^ 308. Pi'ove that every a. p. is a recurring series and that its generating function is — r^ rj^ , « being the first term and h the ( 1 — xy common difference. 309. Find the generating functions of the following series : — (1) l + 3x + 5x' + 7a;='+... (2) 2 + 5a;+13x'+35a;='+... (3) 2 + 4a;+Ua)-+52aj='+... (4) 4 4-5a; + 7a;' + llx^+ ... (5) 2 + 2a; + 8x- + 20uj'+... (0) l + 3a;+12a;' + 54a;'+... and employ the last to prove that the integer next gi-eatcr than (^3 + 1)^" is divisible by 2""^', n being any integer. 310. The generating function of the recurring series whose first four terms are a, 6, c, d^ is ah' -ca' + x (a^d - 2ahc + h^) h' ~ ac + x (ad - be) + x^ (c" - bd) ' 311. If the scale of relation of a recurring series be «„-7a„_, + 12a„_, = 0, and if ?/p = 2, ^l^ = 7, find ti^ and the sum of the series w^ + w, + . . . + ?«,,_, . 312. Prove that, if Oo? «i> f'* ••• ^„ ^^ ^^^ ^ ^- ^^^^^ ^v ^i» •■•b^^ao.r., the series «o*o> ('A' ■■■ ",/'.,<•• will bo recurring series. 4—2 52 ALGEBRA. 313, Tlio scries ?*„, n^, v,, ... , and i;,, i\, v^, ... arc both recixrring series, the scales of relation being prove that tlie series tijv^, u^v^, ■zy^, ... is a recurring series whose scale of reUition is **,.+2 -i^.^i^n + i + (pX + I'lh - -P/1'^ ^-. -P^V2M^K-^ +P2VK-2 = 0- [It is ob-vdous that the series ic^ + i\, w, + Vj, it^ + r^, ... is a recurring series whose generating function is the sum of the generating functions of tlie two series.] 314. Prove that the series 1- + 2' +3'' +... + «-, r + 2'' + 3^+ ...^n' are recurring series, the scales of relation being between 4, 5, ... r + 2 teiTQS respectively. 315. Find the generating functions of the recurring series (1) \+2x+5x'+lQx^+\1x* + 1Qx'+ ... (2) l + 3a: + 4sc' + 8a;=*+12a;* + 20aj*+... (3) 3 + Ga;+14ce' + 36«^ + 98£c* + 276 ;«*+.., (4) 3-a; + 13a;^-9a;=' + 41a;'-53a!*+... and prove that the n^^ terms of the series are respectively (1) l+w-l"^, (2) i{4(-l)" + (29-3w)2"-^},(3) 1""' + 2""'+ 3""', and (4) 2»i-l-(-2)". 316. Find the generating function of the recurring series 2 + 9a; + 6a;' + 45a;' + 99a;' + 1 89a;' + . . . and pi'ove that the coefficient of a;"~^ is one third the sum of the n^^ powers of the roots of the equation s' - 3;2; - 9 = 0, and that the co- efficients of oc'"~^ and of a;'" are each divisible by 3". 317. If the terms of the series a^^, «,, rt^, ... be derived each from the preceding by the formula PI "'^' P + ^-'^n prove that {%-p)2f-{%-q)q" [If we assume a = — — , we get at once a scale of relation for W„ M„ w^. pqu^^^- [p + q) M,_^ , + u^^ = 0, so tliat ii^ = -;r^ + -„^\ -J ALGEBRA. 53 XIV. Convergent Fractions. [If — be the w"' convergent to the continued fraction in a, a, «, 6. +6. + 63+... we have the eqiiations and for the fraction a, a, ^3 the equations Pn ^ ^ J^n-i - «ni^„-2' ?„ ^ ^«?,-i - «„!/„-,• The solution of each equation, a^_, 6^ being functions of n, must involve two constants, since it is necessary that two terms be known in order to determine the remaining terms by this formula. These constants may conveniently be taken to be p^, 2^0, l^i I2 respectively. The fraction — thus determined will not generally be in its lowest terms. We will take as an example the question, "To find the n^^ con- vergent to the continued fraction 1 1 4 12 '2n{n-\) 1 -3-6- 9 -... 'in -..." Take u^ to represent eWier p^^ or q^ (since the same law holds for both), then w^^j = 3?im, — 2n (« — l)^*„_i ; or M^^, - 2nu^ = n{u^ — 2 (n— 1)^^^,}. So n„ - 2 (n-l) u„_^ = {n-\) {u^_^ - 2 {n - 2) ?/„_J, W3 - 4i,.+2 - («^ + 2) P, + 1\-, = 0, (?,.+, - (a6 + 2) (7„ + 7„_, - 0. ALGEBKA. 323. Prove that the i)roclucts of the infinite continued fractions , J. 1 1 1 1 ^^1 1 1 ,0) 1 1 i 1 1 ^^^1 111 a + b + c+d+a+...' ' c + b + a + d+... ,-, l+bc .-, b + d+bcd 324. Prove that the differences of the infinite continued fractions (l)illl iilll a + b + c + a+ ...' b + a+c + b + a+ ...' .,.11111 iilll a + b + c + d + a + ...' c ■{■ b + a + d + c + ... arc (I) ^-^ (2) ^("^-") one T£ a b a b . b a b a 62o. If a; = Y t t t > and ?/ = - - - - 1+1+1 + 1 + ...' -^ 1 + 1+1+1 + ..., then will x-y = a — b, and a;?/ + — = « + &+ 1. a;y o,/,Ti! a b a b . b a b a 62b. Li x=— r, — p , and2/=r/ — p -/ . a + 6 + a' + 6 + . . .' ^ 6 + « + 6 + a + . . .' then -will «'« - b'y — a — b. and a;?/ + — = a + 6 + «'i'. xy 327. If a; = v ^ ^^ - and 7/ = - 1+1+1+1 + .. . "^ 1+1 + 1+1...' then will ic - v =-:; — = , and a: (1 + ?/) = -\ — ~ . ^ 1 + 6' ^ "^^ 1 + 6 328. Ifa; = v r t t t and y = t r r r r > 1 +1 + 1 + 1 + 1 + ... -^ 1+1+1+1 + 1 + ...' ii i. /I \ l + c + (Z , ,, . ,l+a + 6 prove that x{\-k-y)=a-. — -. , and y(l -^-xS^d-^ — ; . ^' 1 + 6 + c ■'^ ' 1 + 6 + c 329. The converfjents to the infinite continued fraction 12 12 1 5-1-5-1-5- recur after eight. 5G ALGEBRA. 330. The continued fractions 4 4 4 o ^ 1 1 * + 8+8 + 8+...' ■^'^4 + 4+4+...' each to 7i quotients are in the ratio 3:1. [This can be readily proved without calculating either,] 331. Prove that 71 n -1 n-2 2 1 1 M+1 + • • • + -o T « = n+n-\ +n--2 2+l+2~?i + 2' 332. Prove that, if «> 1, the infinite continued fraction 1 a a+\ 1 a - a+\ — a + 2- ... a-1' 1 a a a+1 a + 7i-\ that a + i +a+a+i -t-a-t-yi-t-.., + vu + it 1 1 1 to M + 2 terms, a + 1 +a+a+l 4-a + 2+.., + a + ?i 1 1 a+1 (a + l)(a + 2) (a+l)(a + 2)(a + 3) and that on reducing this to a single fraction the factor a + w + 1 divides out. 333. The n^^ convergent to 1 - ^ _ j _ is equal to the {2n - 1)*^ convergent to 1111 1+2+1+2+ Pn 334. If — be the w*^ convergent to r z^ , then q^ ° ?--l+?' + l+r-l+... will " (»'-l)A^. + l=»•(^2n+,-»•^2n-l)> ('•-IK',, =»•(*'+ l)2'2,.-l' noK T^ 71 71+1 n + 2 , ,, ^ 335. lix = - z X to 00 , prove that w + M + 1 + ?i + 2 + . . . ' ^ 111 1 H 7 TY-7 ?r; - ... to 00. 71 + X 71 w(?l+l) ?l (»l + 1) (?i+ 2) 336. Prove that the value of the infinite continued fraction 12 3 1 + 2 + 3 + .. . ^-L;andthatc = 2 + i^i^|_^|_^___tooo. ALGEBRA. 57 337. Prove that 1 n n{n+\) n{n + 2) n{n-{-r-\) T+r+ 2 + 3 +...+ r +... is equal to 2 3 n ". »». 1 r + + 1 \n-v\){n + 2) (w + l)(?i + 2)(w + 3) ■' and thence that the value of the fraction continued to infinity is for ?i= 1, 2, 3, 4, 5 respectively €-1 e^' + l 5£'-2 176^3 , 329c*-24 , and 338. Any two consecutive terms of the series a,, a,, ... a,, ... satisfy the equation _w(?i+ 1) find a^ in terms of a^ ; and prove that when n is indefinitely increased the limit of 1 - a. 339. Having expressed ^(n' + a) as a continued fraction in the form n + ^r- -x— , -- is the r"' convergent ; prove that 271 + 2)1 + ... q^ & } f 340. Prove that *y(»t' + a) can be expressed as a continued fraction an the form w + - - ; and that if — be the w + ?4 + w + ?i + . . . q^ r"* convergent, where a, /3 are the roots of the equation in x x' - 2 (n* + a)x + a {)f + a) = 0. 1+33 341. If ■ _ — -^E 1 + «ja; + aga;'+ ... +a„a;"+ ... , and .j — 72^ 1 + b,x + bjx''+ ...+b x''+ .... 1 - 2x-ar ' * " prove that »/ - 2^>,/ = (- l)""'. 58 342. If and prove til lit 343. If aiid prove that ALGElillA. - x ■ — -i -, = 'I + a,x + a,x- + ... +a X + . . . , 1 - ix + af ' ^ " .; -. J E 1 + h,x + h„x- + . . . + hx" + . . . , 1 - 4a; + a;- x 2 r - X = r + a,x + a,x' + ... +a x + ... l-2rx + x" 1 ^ E 1 + h^x + h.,x- + . . . + h^x" + . . . , 1 - 'Irx + ar a^-{r"-\)h;--^\. 344. In the equation X* - 2nx^ -x-¥{n-\)n = 0, prove that ct* = ± Vii ±Jn + x, and find all the roots of the equation. Prove that V? -Vl + v^T-yTTto 00 = 2, and express the other roots of the biquadratic in the same form. 345. Prove that ^ p+Vp+Jp+ ... Vl) - \/p - Ji^- ... Vi)-\/p + ^^ ... - vi^ + V i^ - Vi^ + J wi* + m^n^ + «* , where » = ; — ~-^ , and m > n. ■' 4«i «^ 346. Prove that r(r + l) »-(r+l) () ) — ^^-j — ^ — ^-^j — ^ to ?i quotients 2??i« m^ — mn + if 2mn m^ — mn - - n" 2mn »i^- - mn — m" 2mn _r(r+l)"+' + (r + l)(-r)"+' (2) 1 1 3- 5' {2n-iy 1 + 2+ 2 + 2 + ...+ 2 1 1 1 (-1)" 3 5 2« + 1 ' (3) ALGEBEA. 59 12 6 12 n(n+l) 2+2+ 2+ 2 +...+ 2 ^ I L _1_ _ (-^)" - 1.2 2.3"^ 3.4 •■•^(m+1)(m+2)' ... 1 1 r+1 r+2 (4) V - T ^ to n quotients V / 1 + ?• + r+1 + ?-+2+ ... ^ _ 1 1 1 1 — i ^ + 7 TT~7 — ~7r\ ~ / ' 1 T/ ?rr7 fr^ + • • • to h tonus, r + 1 (y + 1) (r + 2) (*• + 1) (r + 2) (r + 3) ' ,_,, X X 4,'K U'X (•5) 1 + 2 - a; + 3-2x- ...+n + l -nx x' x' (-l)V^' = '-'" 2""^ 3"-'*' n+1 ' /6) ^ -^^ JM' (2n-l.xf 1 + 3 - ar + 5 - 3a;-+ ... + 2n+ 1 - 2ji- 1 x' -^ -g+r, •••+ 271+1 ' ^ 1 r r(r+l) r(r + ?i-l) ^'^ T+1+ 2 +...+■ ^^^ = 1 ^+7 TT-7 =r. -...+ ^ ' r + 1" (r + l)(r+2) ••• " (r + 1) (r + 2) ... {r + n)' ,,, I 1 16 81 n* , tt'' <^^ T+3+y+T'+... + 2^^TT+...*°=* = l2' (9) J i ^ ^ ^ to 00 ^log2, ^ ' 1 + 1 + 1 + 1 + ... + 1 + ... ..468 2?i o ^' ~ 1 (10) T -T o 1 to CO =2 -5 r , ^ ' 1 + 2+3 + ...+5J-1 + ... €-+1 ,„, 3= 3.4 3.5 3n ^ c 2£'' + 1 (1^> T+-2-+-T-+...+;^2 + ...*°^=^5?3-2' ,,.. 4^ 4.5 4.6 40i+3) ^ ,^ Se^-l 347. Prove that (1) T T T T to vi quotients := , , ^ 1-4-1-4-.. . ^ n+l (^) i_I_4_T_... ton quotients =2(^1)' ,.,,114 9 ?r , 1 1 1 ^^) l-3-5_7-...-2,7+l-^^2+3-^--^^l' t)() ALGEBRA. 1 1 9 25 (2k-\)'_ 11 1_ 1 _r' (r + 1)^ (r + Tt-l)' ^'^^ r-2r+l-'2r + 3 "" 2(r + 7i)-l 1 1 1 E- + r+ ... + r r + 1 r + ?i. 1 a' {a + by (a + n-lby ^^ a-~2a + b-2a + 3b-...-2a + {2n-l)b 1 1 1 = - + r+ ... a a + b '" a + nb' { / ) — ^ — - — - J - — = — - to n quotients a(n + 2a-l) na . ,, E — ^^ = or ^ ,^ as n is odd or even, n+l n-2a + 2 ^g^ aja^l) ^ a(a-l) ^ a{a^) ^ ^^ ^^ ^^^^.^^^^ a 1 = a ;; , or a -„+j — ; (9) {a + 1) (1 - a-y -a (1 - cr') ' - 1 XX 4:X n"x 1 ~2 + x — 3 + 2x— ... -n + l +nx x' x' x"^' 2 3 n+\ 14^ (^'-1)' _ n{n+l){2n + \) (11) 1-5-13-. ..-n^ + (?^ + ir~ 6 2 3 8 ?i'-l ^ n {n + 3) (12) 1 _5_7_..._2^^i:i- 2 ' 1 r 2' n* (13) 1 _i» + 2*_2^ + 3^_..._^. + ,,+ l^ 1 1 1 ,, ,,1 cc a + 1 . , .'i (14) T r ^ to w + 1 quotients ^ ' 1 -a + 1 -a+ 2- ... = l+a + a{a + l) + a{a+l){a + 2) + ...+a{a+l) ...(a + n-l), aS) 113 5 2n-l V^*^^ l_2-4-6-...- 2« E2 + 1.3 + 1.3.5 + ... + 1 .3.5...2;i-l, ALGEBRA. 61 2n 2n - 2 4 2 ^^^^ 2.-1 -2^^:^-...- 3 -1=2'^' i^n^ 112 3 n ^ ' 1 - 3 - 4 - 5 - ... -?t4-2- ... ^ l-r+2-r+3-r + 4-... .^-. r-- (r + iy {r + 2)' (19) 7) T —> k o F tooo=r, ^ ' 2r + 1 - 2r + 3 - 2r + 5 - ... /nf\\ r r+l r + 2 r-1 (20) - r to 00 = ^ ^ ' r-r+l-?-+2-... r- 2 XV. Poristic Systevis of Equatimis. [Any system of algebraical equations ^_ + K.. + +/(..+ ^.) + i, (1 + 1 ) + S (I + 1-) = 0, — ^ + K-.aJ„ + c +/('^'„-i + a;„) + Sf ( + - ) + A (^' + -^ ) = 0, + hxx + c +/(« +x.)v g[- +-) + /* (^-" + -' ) = 0, is poristic if a certain relation holds between the coeflScients a, b, c, f,g,h: when w = 3, this relation is M E h^ -ah-ch +fy = ; when 7i = 4, it is - iV= ahc + 2^/i - aj^ - hg^ - ch^ = 0, and when n = 5, it is where L = c — ih. For any number of such equations, if there is one solution for which ai^, x^,...,x^ are all unequal there is an infinite number of such solutions, but this cannot be the case unless a certain relation hold, which relation involves L, M, N only. See Proceedings of London Math. Soc. Vol. iv. page 312.} 348. If x^, x^ be the roots of the equation + 6+ — = 0, (l-7rt)(l-x) mx 1-1 <* T " rt then will -7z r-7^; r + H = 0. (l-a;,)(l-a;J x^x^ 02 ALGEBRA. 349. If x^f x^ be the two roots of tlie equation / \ \ X m ^ a' + a mx H — p= — H — + 1, \ VIXJ 'III X / \ \ X X tlion will rt' + a ( xx^ +- — .)=-i + -? + l, V ' - x^xj x^ x^ X 1 1 1 and x.x^ + m (cc, + xj = + — H = - a. x,x„ vtx, mx. 350. If x^, x^ bo the two roots of tlie equation ft^(l + m") (1 + a;*) + a{m + x) {mx - 1) = mx, tlion will a- (1 + ce,") (1 + x,^) + a {x^ + a;J (a;,.'c^, - 1 ) = •'«,*'3» 111 1 and m + a\ + ic,, + - + - + - = mx.x,, + . m £Cj .f., 351. If it'i, a-'a be tlie roots of the equation (l-??r)(l-a;^) Amx _ {\ + m') {\ + x"") b — G c — a a — h ' 6-c c~a a—o 352. If the (Quantities x, y, z be all unequal and satisfy the equations a{ifz' + l)+y'' + z' a{z\v'+\) + z' + x' ^ a {x' y' + \) + xrif _ yz zx ~ xy ' each member of the equations = «"-!, and xyz {yz + zx + xy) = xyz. 353. Having given the equations I b 1 b 1 b Q/z-i cix — = zx + mi — = .r?/ -i az — : *^ yz X zx y "xy z prove that, if x, y, z be all unequal, a6 = 1, and each member of these equations = 0. 354. Having given the equations y' + z' + ayz = 2;° + cc' + azx = x' + y^ + axy, \)YO\Q that, if X, y, z be all unequal, «=1, and x + y-¥z = 0. 355. The system {a^ - x') {¥ + yz) = {(C - f) (/y-' + zx) = {a' - z') {b' + xy) is |)oristic if // ^ cr ; in which case each member will be equal to xyz {y + z) {z + x) {x + y) {x + 7/ + zf ALGEBRA. 63 356. Prove that the system of equations X {a - y) =2/ (a - z) = z (a -x) = l/ can only be satisfied \i x = ij = z\ unless ly = a", in which case the equations arc not independent. 357. Prove that the system of equations M (•2a -x) = x (2a -y)=y (2a -z)^z (2a -u) = b^ can only be satisfied if 7i = x = y=^z; unless a* = 26^, in whicli cas*^ the equations are not independent. 358. Prove that the system of equations «-•, (1 - ^,) = ^, (1 - --^a) == ■ • • = ^u (1 - ^,.+ ,) = ^n^, (1 - ^\) = « can ouly be satisfied if x^~x,, = x^= ...=x^^^, unless ii be a root of the equation {1 +Jl -4:11)"'*'^ = (l - Jl -■iu)"'^^ different from |, iix ■which case the equations are not independent. [By putting 1 — 4:U = - tan'' 0, it will appear that the roots of tlie auxiliary equation are 1 o TT 1 „ 27r . ?i w - 1 , -T sec" ^ , -r sec" =- , .... to ^ or — ^r— terms, 4 n + l 4: 71 + 1 ' 2 2 359. Prove tliat the system of equations a h a b + - = r • + — =C 1 - a;, x^ 1 - x^ .^■, can only be satisfied if a;, = x^, ixnless c = a + b, in whicli case tlie equa- tions are not independent. 360. Prove that the system of equations a b a b a b T + — =T +- =1 +- = c 1 - a,', X,^ 1 - iC„ 2^3 1 - a^g X^ is poristic if (a + 6 — c)" = ab, and the system a b a b a b n b l-.r, x.^ l~x^ x^ 1-x^ x^ l-x^ a,-, if {a + b- cY = 2ab. 361. In general the system of equations a b a b a b a b :j + -=1 +-=... = - + = :; +--C 1-a;, x^ \-x^ x^ l-a;„ a;„^, l-a^„4, ^^ n+l _ ^" + 1 is ix)ristic if ^ — = 0, where a, ft arc tlio roots of the quadratic X-' + (a + b- c)x + ab = 0. (14. ALGEBRA, XVI. Properties of Numbers. 362. If n be a positive whole number, prove tliat is divisible by 9, 240, 64, 64, 2^" + 15n+l (2?i+l)*-2?i-l 3«"+«_8h-9 3'"+' + 40/1 -27 3'"+' + 160«''-56«- 243 512, 3''"*' + 2''+' 7, 02n+2 . QBn + l 34«+2^43n+3 3 . 5^""^' + 2^'"^' 11, 17, 17. 363. If 2/» + l be a prime number, (l^)^ + (-l)'' will be divisible by 2;;+l. 364. If x> be a prime number, p_iC'„ + (- 1)""' will be divisible by p. 365. If J) be a prime number > 3, ^ _,Cp - 1 will be divisible by p^. 366. If n-\ and w+1 be both, prime numbers >5, n must be of one of the forms 30<, or 30<±12, and ?^^(?^^+16) will be divisible by 720. 367. If n-1 and % + 2 be both prime numbers > 5, oi must be of one of the forms 30i + 15 or 30i ± 9. 368. Prove that there are never more than two proper solutions of the question " to find a number which exceeds p times the integral part of its square root by q" ; that if q be any number between rp + r^ and (r + l)^j + r* the two niunbers p{p> + r) + q and ^^ (^:)+ r- 1) + g' are solutions ; but if q be any number from rp + (?• - 1)* to r/> + r* there is only the single solution p \p + r—\) + q. 369. If n be a whole number, n+\ and n^ -n + \ cannot both be square numbers, ^ itm.-\f\^ - ' ^^ ^ ~> '^'' ' ' s-\ 370. The whole number next greater than (3 + ^5)" is divisible by 2". 371. The integral part of -7o(a/3 + ^5)^'-', and the integer next greater than (^3 + ^5)*", are each divisible by 2"'^'. ALGEBRA. 05 372. If n, r be wliolo numbers, the integer next greater than {Jn-i-\ + Jn-iy is divisible by 2'*'; as is also the integer next greater than ^ {Jn + 1 + Jn- 1)-'"^' and the integer next less than J)i+ 1 sjn- 1 373. The equation x* - 2y" = ± 1 cannot be satisfied by any integral values of x and y different from unity. 374. The sum of the squares of all the numbers less than a giveii number N and prime to it is the sum of the cubes is and the sum of the fourth powers is -^{\-a^){\-h^){\-c')...; where a, b, c, ... are the different prime factors of J^. 375. The product of any r consecutive terms of the series 1-c, l-c^ l-c",... is completely divisible by the product of the first r terms. XYII. Prohahilitie!'. 376. A and B throw for a certain stake, each one throw with one die; ^'s die is marked 2, 3, 4, 5, 6, 7 and ^'s 1, 2, 3, 4, 5, 6 ; and equal throws divide the stake : prove that ^'s expectation is 44 of the stake. What will J.'s expectation be if equal throws go for nothing ? [fi of the stake.] 377. A certain sum of money is to be given to the one of three persons A, B, C who first throws 10 with three dice; supposing them to throw in the order named until the event happen, prove that ^'s chance of winning is (j~\ , Ea ~ , and (7's (^ . 378. Ten persons each write down one of the digits 0, 1, 2, ... 9 at random ; find the probability of all ten digits being ^vritten. w, p. 5 GO ALGEBRA. 379. A tlirows a pair of clico eacli of which is a cube ; £ throws a pair one of which is a regular tetrahedron and the other a regular octa- liedron whose faces are marked from 1 to 4 and from 1 to 8 respectively ; whicli throw is likely to be the higher? (The number on the lowest face is taken in the case of the tetrahedron.) If A throws 6, what is the chance that B will throw higher ] [The chances are even in the first case ; in the second B's chance ^■^■] 380. A, B, C throw three dice for a prize, the highest throw win- ning and equal highest throAvs continuing the trial : at the first throw A throws 13, prove that his chance of the prize is "623804 nearly. 381. The sum of tAvo positive quantities is knowTi, prove that it is an even chance that their product will be not less than three-fourths of their greatest possible product. 382. Two i^oints are taken at random on a given straight line of length a : prove that the probability of their distance exceeding a given length c (< a) is ( j 383. Three points are taken at random on the circumference of a cii'cle : the probability of their lying on the same semicircle is f . 384. If q things be distributed among p persons, the chance that every one of the persons will have at least one is the coefficient of a;' in the expansion of [£ (e^ - I)''. 385. If a rod be marked at random in n points and divided at those points, the chance that none of the parts shall be greater than - tli of the rod is —„ . n 386. If a rod be marked at random in ^^ - 1 points and divided at those points, prove that (1) the chance that none of the parts shall be 1 / ?^V~' -c — th the whole is ( 1 — — ) , (w > 1^) '• (2) the chance that none of m \ vij ^ ^ ' the parts shall be > - th the whole is ^ n to r terms where r is the integer next greater than j'J - «, {n<2^) i ^^ ^^^ equivalent to r terms where r is the integer next greater than n. Also (3) the AXGEBRA. 67 chance that none of the parts shall be < — th and none greater than - th of the whole is n (i-ZT' .JiJl^AT' ^H^zi)(i_?z.2-?y--... Til/ ~~' 7) to r teiTOs where r is the integer next greater than n ', provided that — + -' > 1. If — + < 1, and none of the parts be > - th in n m n -^ n the whole, it follows that none can be < — th the whole, so that the m case is then reduced to (2). 387. At an examination each candidate is distinguished by an index number ; there are n successful candidates, and the highest index number is 'ni + n: prove that the chance that the number of candidates exceeded m + ?i + r - 1 is |m + r |m + n— 1 |m |m + ?^ + r — 1 * [It is assumed that all numbers are a iwiori equally likely. 388. There are 2ni black balls and m white balls, from which six balls are drawn at random ; jirove that when wi is very large the chance of drawing four white and two black is t/jj, and the chance of drawing two white and four black is -o^"^. 389. If n whole numbers taken at random be multiplied together, the chance of the digit in the units' place of the product being 1, 3, 7, or 9 is (I)", and the chances of the several digits are equal j the chance 4" - 2" of its being 2, 4, 6, or 8 is — — — , and the chances are equal; the o 5" _ 4" chance of its being 5 is ; and of its being is 10" -8" -5" + 4- 10" 390. If ten things be distributed among three persons the chance of a particular person having more than five of them is iV^ir> ^^"^'^ ^^ ^^ having five at least is i VeVa - 391. If on a straight line of length rt + 5 be measured at random two lengths a, b, the probability that tlte common pai't of these lengths c shall not exceed c is -y , (c < a or 6) : and the probability of the smaller h lying entii-ely within the larger a is . a 5—2 08 ALGEBRA. 392. If on a Btraight line of length a + b + c he measured at random two lengths a, b, the chance of their ha\ing a common part not greater than d is 7^ — ,-. > Ui < « or h) ; the chance of their not having a (c + a) (c + b)' ^ ' ' ° common iiart greater than d is , — ^^ — r— . ,, ; and the chance of the ^ ® (c + a) (c + h) smaller b Iving altogether within the larger a is , ' ° ° ^ a+c 393. There are m+2) + q coins in a bag each of which is equally likely to be a shilling or a sovereign ; p + q being drawn 2^ are shillings and g sovereigns : prove that the value of the exjiectation of the remain- ing sovereigns in the bag is — — r-^. If ni = 5, w = 2, q=l, find the chance that if two more coins be dra^\^l they will be a shilling and a sovereign, (1) Avhen the coins previously drawn are not replaced, (2) when they are replaced. [In case (1) |, in case (2) ^|.] 394. From an unknown number of balls each equally likely to be white or black three are drawn of w^hich two are white and one black : if five more balls be drawn the chances of ckaMing five white, four white and one black, three white and two black, and so on, are as 7 : 10 : 10 : 8 ;5 : 2. 395. A bag contains ten balls each equally likely to be white or black ■ three balls lieing di-awn turn out two white and one black ; these are replaced and five balls are then drawn, two white and three black : prove that the chance of a draw from the remaining five giving a white ball is -j^. 396. From a very large number of balls each equally likely to be white and black a ball is drawn and replaced ]} times, and each drawing gives a white ball : prove that the chance of drawing a white ball at the next draw is — ^ , 2) + 2 397. A bag contains four white and four black balls ; from these four are drawn at random and placed in another bag ; three di-aws are made from the latter, the ball being replaced after each draw, and each draw gives a white ball : prove that the chance of the next draw giving a black ball is 'SS. 398. From an unkno-vvn number of balls each equally likely to be white or black a ball is drawn and turns out to be white ; this is not replaced and 2n more balls are drawn : prove that the chance that in 3n + 2 the 271 + 1 balls there are more white than black is -. ^ . If the first An + 2 draw be of three balls and they turn out two white and one black and 2)1 more balls be then drawn- from the remainder, the chance that the ALGEBRA. 69 maiority of the 2)i + 3 balls are white is '^ — — • and the chance that in the 2/t balls there are more white than black is Un' + lSn 4 {2n + 1) {-In + 3) " 399, From a large number of balls each equally likely to be white or black p + q being tb-awn turn out to be j) white and q black : prove that if it is an even chance that on three more balls being drawn two will be white and one black ^ = 1.7:2 nearly, p and q being both large. 400. A bag contains m white balls and n black balls and fi'om it balls are dra%\ai one by one until a white ball is dra^\^l ; A bets B at each draw x : y that a black ball is drawn ; prove that the value of ^'s expectation at the beginning of the drawing is ~'L- - x. If balls be m+ 1 drawTi one by one so long as all drawn are of the same colour, and if for a sequence of r white balls A is to pay B rx£, but for a sequence of r black balls B is to pay A ry£, the value of ^'s expectation will be — '—r — , : and if A pay B x for the first white ball ch-a\\Ti, rx for the second, x for the third, and so on, and B pay A y for the r (r + 1 ) first black ball di-awn, ry for the second, — ^^ — ' y for the third, and so on, the value of -4's expectation at the beginning of a drawing will be |m + ?i + r— 1 |m 'n , '~ {w (w + 1) ... in + r) y — m(m-^\\ ... (m + r) x]. 401. From an unknown number of balls each equally likely to be red, white, or blue, ten are dra^\^l and turn out to be five red, tlu'ee white, and two blue ; prove that if three more balls be drawn the chance of their being one red, one white, and one blue is ^"--j ; the chance of three red is -^r; \ of three white is ,,\ ; and of three blue is -y--j- ; and the chance of there being no white ball in the three is 3^. PLANE TRIGONOMETRY. I. Equations. [In the solution of Ti-igonometrical Equations, it must be re- membered that when an equation has been reduced to the fonns (1) sin£c = suia, (2) 008 0; = cos a, (3) tan a; = tan a, the solutions are respectively (1) cc = ?i7r+ (- l)"a, {2) x = 2mr^a, {Z) x = nir + a, where n denotes a positive or negative integer. The formulse most useful in Trigonometrical reductions are 2 sin ^ cos ^ E sin {A+B) + sin {A - B), 2 cos ^ cos ^ = cos {A-£) + cos (A + £), 2 sin ^ sin ^ E cos (.4 -B)- cos {A + B), and (which are really the same with a different notation) . , . p__ . A+B A-B sm A + sm jBe 2 sin — ^ — cos — ^ — , . „ . A-B A+B cos A + cos B=2 cos —^ — cos — - — , . „__ . B-A . A + B cos-d -cosi) E 2 sm — ^r — sm — ^ — j which enable us to transfonn products of Trigonometrical fimctions (sines or cosines) into sums of such functions or conversely sums into products. Thus to transform sin 2 (^ - (7) + sin 2 (C - ^ ) + sin 2 (i - B). "We have sin 2 (C - ^) + sin 2 (^ - ^) E 2 sin {fl-B) cos {B+C-2A) and sin2(i?-(7) E 2sin(5-C)cos(5-C), whence the sum of the three E 2 sin (5 - (7) {cos (5 - C^ - cos (5 + (7 - 2 J ) }, E - 4 sin (5 - (7) sin (C - ^) sin (^ - ^). PLANE TRIGONOMETRY. 71 Again, to transform cos {B - C) cos (C-A) cos (^1 - £), we have 2 cos (O-A) cos (A -B) = cos {C-B) + cos (B + C-2A), whence 4 cos (i? - C) cos (C- ^) cos (vl - i?) E 1 + cos 2 (i? - C) + cos 2 (C - ^) + cos 2 (^ - B).] 402. Solve the equations 2 sin a; sin ox= 1, cos X cos 3.^' = cos 2x cos Gx, sin 5a; cos 3a; = sin 9x cos 7a;, sin 9a; + sin 5a; + 2 sia^ x = l, cos mx cos nx = cos («i + js) a; cos (n - p) x, sin mx sia ?ia; = cos (??i + jo) a; cos (?i +p) x, tan^ 2a; + tan^ a; = 10, cos X + cos (x-a) = cos (a; - j8) + cos (x + (3 - a), 2 sia* 2a; cos 2a; = sin" 3aj, 2 cot 2a; - tan 2a; = 3 cot 3a;, 8 cos X = -—^ — + sm X cos X sin 2a; + cos 2a; + sin x - cos a; = 0, (1 + sin a;) (1 - 2 sin a;)* = (1 - cos a) (1 + 2 cos a)*, sin a cos ()3 + x) tan yS sia ft cos (a + a;) tan a ' cos 2a; + 2 cos x cos a — 2 cos 2a = 1, sin a cos 3a; — 3 sia 3a cos x + sia 4a + 2 sia 2a = 0, cos^ a sin^ a , + -. = 1, cos X sin x (cos 2a; - 4 cos a; - 6)* = 3 (sin 2a; + 4 sin a;)*, (cos 5a; - 10 cos 3a; + 10 cos a;)^= 3 (sin 5a; - 10 sin x)' 403. If cos (x + 3i/) = sin (2.^; + 2i/), and sin (3a; + y) = cos (2a; + 2y) ; a;=(5m-3«)|+fj then will [• ; or a; - ?/ = 2r7r + ^ , V = (5;i - 3m) ^ + ^ m, n, r being integers. 72 PLANE TRIGONOMETRY. 404r. The real roots of the equation tan^ x tan ^ = 1 satisfy the equation cos 2a; = 2 - J^. 3 /3 405. Given cos 3a; = - ; ,-, , prove that the three values of cos x are 4 V:^ fZ . TT IZ . IT /3 . V2^^i0' V2^^^G' -V2^" 10 406. If the equation tauTr = r ^ have real roots, a^ > 1. ^ 2 tan ic + a + 1 407. Find the limits of )-^ 1 for possible values of x. tan {x — a) ^ ... 1— sin 2a , 1 4- sin 2a' it cannot lie between , -. — ^- and 1 + sin 2a 408. The ambiguities in the equations 1 - sin 2a A . A I.. ; -T A . A I- ; J cos ^ + sin - = ± ^1 +sui^, cos -^ - sin — = ± ^ 1 - sm A, may be replaced by (— 1)'", (- 1)", where m, n denote the gi'eatest . ^ + 90" ^ + 270" ,. , integers m -ggQir-j "3^0""" ^'^^P^^^i^ely. 409. The solutions of the equation sec" a; + sec" 2a; = 1 2 are x = W7r±^, x-niT^i^— , x = h, cos ' (- ^). a + e 410. The roots of the equation in 6^ tan— ^^ tan ^ = m, are all roots of the equation sin a sin ^ {1 - cos (a + B)] + m^ cos a cos ^ {1 + cos (a + ())] = m sin^ (a + 6). 411. The equations tan («9 + /3) tan (^ + y) + tan {6 + y) tan (^ + a) + tan {6 + a) tan (^ + ^) = - 3, cot {0 + /3) cot (^ + y) + cot {9 + y) cot (^ + a) + cot {6 + a) cot (^ + ^) = - 3, will be satisfied for all values of 6 if they are satisfied by ^ = 0. ^- cos (a + 6) cos IB + ^) cos (y + 6) ^ , . , 412. If }—, ' = r^T^ = ^^^3 ^ «> A y ^emg unequal sm^ a sm^ /3 sm" y ' "^' ' =» and less than tt, then will a + ^ + y = tt, and 3 - cos 2a - cos 2/3 - cos 2y _ 1 + cos a cos /8 cos y sin 2a + sin 2;8 + sin 2y sin a sin /3 sin y 413. Ifa+/3 + y = 7r and 6 be an angle determined by the equation sin (a - 6) sin (^ - ^) sin (y - ^) = sin' 0, then will sin(a-e) _ sin(^-e)_ sin(y-^)_ sin^ cos g sin' a sin' (i ~ sin' y ~ sin a sin /3 sin y ~ 1 + cos a cos /? cos y ' tan^ = PLANE TRIGONOMETRY. 70 [These equations occur wlien a, ft, y are the angles of a triangle ABC and is a point such that i OBC - l OCA = z OAB = 0.] 414. If a, /?, y, 8 be the four roots of the equation sin 1Q - m cos -n sin ^ + r = 0, then will a + /3 + y + S = (2^j + 1) TT (7; integral) ; sin a + sin /? + sin y + sin S = ??i, cos a 4- cos /3 + cos y + cos S ^ «, sin 2a + sin 2/3 f sin 2y + sin 28 = 2inn - 4r, cos 2a + cos 2/3 + cos 2y + cos 28 = n' - wr. 415. If two roots ^,, ^2 of the equation cos {0 - a)-e cos (2^ - a) = m (1 - e cos ^)' satisfy the equation tan -^ tan — * = * / , then will me sin" a = cos a. 416. Find X and y from the equations X (1 + siir - cos ^) - ?/ sin ^ (1 + cos ^) = c (1 + cos ^), 2/ (1 + COS" B) - a; sin ^ cos Q = c sin ^ ; also eliminate B from the two equations, [a a The results are a; = c cot" ^, ?/ = c cot -, if^cx. 417. The equation 2 sin (a + /? + y - 6) + sin 2^ - sin (/? + y) - sin (y + a) - sin (a + )8) 2 cos (a + ^ + y - 6) + COS 2^ - cos (/? + y) - cos (y + a) — COS (a + /?) 2 sin 2^ + sin (g + y3 + y - ^) - sin (a + g) - sin (/3 + g) - sin (y + ^) ~ 2 cos 26 + COS (a + ^ + y - ^) - cos (a + 6) - cos {ft + 6)- cos (y + B) is satisfied if 6 = a, ^, or y. or if cot — -— + cot + cot - = 0. 2 ^ ^ 418. The equation 2 sin (g + )8 + y + g) + sin g - sin (/3 + y + g) - sin (y + g + g) - sin (g 4- j8 + g) 2 cos (g + i8 + y + 6) + cos ^ - cos ()8 + y + 6) - cos (y + g + 6^) - cos (g + )8 + 6/) _ 2 sin 6 + sin (g + j 3 + y + g) - sin (g + g) - sin {(3 + 6)- sin (y + 6 ) ~ 2 cos 6* + cos (g + /? + y + 6) - cos (a + B)- cos {(3 + B)- cos (y + ^) is independent of ; and equivalent to . g . )3 . y/ ^g ^)3 sin - sin Sr sin '''■-+ ■ ---* 2 2 I (cot " + cot ^ + cot X) = 74- PLANE TRIGONOMETRY. 419. Having given the equations x + y cos c + z cos h ]/ + ~ cos a + x cos c z + x cos 6 + y cos <* _ o cos (s - a) cos (s — b) cos (s — c) ' where 2s=a + b + c; prove that iB _ 2/ _ * _ ''^ sin a sin 6 sine sins' 420. If a, (3, y, 8 be the four roots of the equation a cos 29 + b sin 2$ - c cos ^ - tZ sin ^ + e = 0, and 25= a + ^ + y + 8, then -will ah c cos s sin s cos (s - a) + cos (s - /3) + cos (s — y) + cos (s - 8) d sin (s - a) + sin {s - P) + sin (s — y) + sin (s — 8) e cos (s - a - 8) + cos {s — (3-S) + cos (s — y - 8) ' 421. Reduce to the simplest forms (1) {x cos 2a + 2/ sin 2o - 1) {*' cos 2^ + ?/ sin 2/3 - 1) - {it* cos a + 13 + y sin a + {3- cos a - ^}*, (2) (a;cosa+/?+2/sina+/3-cosa-:/8) (a;cosy+S+2/ sin y+S- cos y-8) - (x cos a + y + 2/ sin a + y - cos a - y) (cc cos ^ + 8 + ?/ sill ^ + 8 - cos /? - 8). [(1) {x^ + f-l)sm^a-l3), (2) (a;^ + 2/^-l)sin(/3-y)sin(a-S).] 422. If y5, y be different values of x given by the equation sin (a + x) = m sin 2a yS-Y cos — ^ =t m sm (/? + y) = 0. 423. The real values of x which satisfy the equation sill f - cos a; j = cos f ^ sin x j are 2m: or 2wir ± - ; w being integral. 424. \ix^ y be real and if ^vclx %vcLy + sin^ {x + y) = (sin a; + sin y)", X or y must be a multiple of tt. [The equation is satisfied if sin a; = 0, sin ?/ = 0, or cos (x + y) + cos X cos y = 2\ and this last can only be satisfied if cos X = cos y = ± 1, and cos {x + y) = l.] PLANE TRIGONOMETRY. 75 425. If a, /3, y be three angles unequal and less than 27r which satisfy the equation a b + —. — + c = 0, cos X SUl X then will sin (/3 + y) + sin (y + a) + sin (a + ^) = 0. 426. If (3, y be angles unequal and less than tt which satisfy the equation cos a cos X sin a sin x 1 + T =-, a be then will (6" + c' — a") cos (3 cos y + (c" + a^ — &^ sin (3 sin y = «- + J- - c*. 427. If a, /3 be angles inieqiial and less than tt which satisfy the equation a cos 2x + & sin 2x = 1, and if (I cos-2a + ?/i sin-2a) (I cos" 2^ + m sin^2y3) = {^ COS" (a + yS) + ??i sin" (a + ^)}^, then will either ^ = m, or a^ — &" = ^ . m+ ^ 428. If a, /?, y, 8 be angles unequal and less than tt, and if (3, y be roots of the equation a cos 2a; + 6 sin 2x = 1, and a, 8 roots of the equation rt' cos 2a; + b' cos 2.'B = 1 ; and if {loos' {a + f3) + m sin" (a + /?)} {I cos" (y + 8) + 7?i sin* (y + 8)} = {I cos^ (a + y) + m sin" (a + y)} {^ cos^ (/? + S) + Wi sin" (^ + 8) }, then will either I = «i, or ««' — bb' = 1 . 0)1+ I II. Identities cmd Equalities. 429. If tan-^ = 1 + 2 tan' B, then will cos 25 = 1 + 2 cos 2A. 430. Having given that sin (B+ C-A), sin {C+A-B), sin (i +5-C) are in A. p. ; prove that tan A, tan B, tan C are in A. P. 431. Having given that 1 + cos (/8 - y) f cos (y - a) + cos (a- f3) = 0; prove that ^ - y, y-a, or a - /3 is an odd multiple of tt. 432. If cot a, cot^, coty be in A. p. so also will cot (J3- a), cot/3, cot (ft-y) ; and sin(/3 + y) sin(y + a) sin (a + i8) , . , . — -•-; > • — -■ — o~~ > • — — ) oe respectively in a. p. sm a sin /3 sin y "^ 7(; PLANE TRIGONOMETRY. 433. Having given 6 6' B cos ^ = cos a cos (3, cos 6' = cos a' cos (3, tan - tau ^ = tan - ; prove that 434. If tan 2a = 2 sin* (3 = (sec a — 1) (sec a - 1) ab + cd and tan •-^;3 2 "'^^'^ then will tan (a - /?) l)c equal to , or to c-b a + d 435. If a, P, y l>o all unequal and less than 27r, and if cos a + cos P + cos y = sin a + sin P + sin y = then will cos'a + cos'/? + cos^y = sin^a + sin^/8 + sin^y = f cos (/3 + y) + cos (y + a) + cos (a + /3) = sin (^ + y) + . . . = ; cos 2a + cos 2/3 + cos 2y = sin 2a + sin 2/3 + sin 2y = ; and generally if n be a whole number not di^^sible by 3, cos 7m + cos w/3 + cos ny = sin na + sin nl3 + sin ny = 0. 436. Prove that 437. If (7 E 2 cos ^ - 5 cos'^ + 4 cos'^, -S'e 2 sin ^ - 5 sin'^ + 4 sin' 6 ; then will C cos 3^ + ^ sin ?>d E cos 29, and C sin 3^ - S'cos 3^ E sin d cos 0. 438. Having given a; cos ^ + 2/ sin ^ = a; cos ) + b tan ^ tan ^ = 0, _^(«6-/r) « + 6 (1 + tan $ tan a; cos — ^ = cos 6 cos cos — -~ , 6-4, - a ■ J. ■ ^ + y cos — ^ = sm 6 sni^ sin — ^y- , c = a cos ^ cos <^ + i sin ^ sin <^, = hc-\- ca + ah ; prove that a V + 1/)/ - c^ Also eliminate 6, (f> from the first three equations when the fourth equation does not hold. [The resultant in general is iafy^ = z' {x^ + y* + s* - 1)*, where ,, ^ (6 + c)(c+ft)(rt + 6) |_1 a^ y*_| _ * " 6c + ca + a6 (a + 6 6 + c c + a j ' \ 6c + crt + aft / \ 6c + ca + rt6 / ^ -^ 78 PLANE TRIGONOMETRY. 446. Ilaving given (a; - a) cos 6 + ysin6= (x- a) cos 6' + 2/ sin 6' = a, tan — - tan -^ — le; prove that if = 2ax - (1 - e*) x^, Of 6' being unequal and less tlian 2ir. 447. If (1 + sin6) (1 +sin^) {1 + sini//) = cos^cosc^cost/', then will each member = (1 - sin 6) (1 - sin 0) (1 - sin i/^), and sec^^ + sec^^ + sec"i// - 2 sec 6 sec <^ sec «/^ = 1. 448. Eliminate a from the equations cos $ sin B = -. — = m. e + cos a sm a [The resultant is 1 = 2e m cos 6 + iii" (1 - e").] 449. If 9^, 9„ be the roots of the equation in 6, (cr - ¥) sin (a + P) {rt'cos a (cos a - cos 6) + 6" sin a (sin a - sin 0)] = 2«-Tsin(/3-^), then will „. a + B, . a+B, ,, a + ^, a + B„ . crsm — ~ sm — ^ + o" cos — ^ cos — ^ = l>. 450. The rational equation equivalent to (m°- 2m?i cos ^ + n^Y + (9i^ - Inl cos i? + ^-)- + (f - 2lm cos C + wr)" = 0, where ^+^ + (7=180", is («m sin A + ?i^ sin ^ + Z??i, sin C)" = 4(^ni?i (/ cos A + vi cos 5 + w cos C). 451. Having given the equation cos A = cos B cos (7 =t sin B sin C cos ^, prove that cos B = cos C cos -4 ± sin G sin ^ cos B, cos C = cos A cos j5 ± sm A sin ^ cos C ; and that sec^^l + sec"^ + sec" (7 - 2 sec^ sec 5 sec C= 1, tan U^' ± ^ tan ^45" ± |) tan ^45° =^ f ) = 1- 452. If tanM tan A' = tan"j5 tan B' = tan-C tan C" = tan J^ tan B tan C, and cosec 2^ +cosec 25 + cosec 2C = ; then will tan {A - A') = tan {B - B') = tan (C - C) = tan u4 + tan B + tan G. ^ftSE LI PLANE TRIGONOMETRY^^ ^ , ' -79 453. Having given •■o o- X sill 3 (/3 - y) + 2/ sin 3 (y - a) + ,:; sin 3 (a - /?) = ; prove that X sin (/? - y) + y sin (y - a) + 2; sin (a - ^) X cos ()3 - y) + 2/ cos (y - a) + « cos (a - yS) sin 2 (yS - y) + sin 2 (y - a) + sin 2 (a - /?) cos 2 ()3 - y) + cos 2 (y - a) + cos 2 (a - /S) 454. Having given the equations x'=^ft' + y'-2Bycos6 1 , , , x + 2/ + s = 0) y- = y^ + a^ _ 2ya cos + a(3 sm i// = 0. 455. Reduce to its simplest form the equation {x cos (a + (3) + y sin (a + ^) - cos (a - )S)} {x" cos (y + 8) + y sin (y + S) - cos (y - 8)} = {x cos (a + y) + 2/ sin (a + y) — cos (a — y)} {x cos (y8 + 8) + y sin (j8 + 8) - cos {(3 - 8)}. [The reduced equation is sin (/3 - y) sin (a - 8) (1 - of - ^) = 0.] 456. Ha\-ing given the equations yz' - y'z + zx - z'x + xy' -xy=0, A + B+C = l 80", xx' sin^A + yy' ^ystB + zzl sin^C = {yz' + y'z) sin 5 sin C cos -^1 + (zx' + :ix) sin (7 sin A cos -5 + (ccy' + xy^ sin ^ sin B cos C ; prove that either x = y = z; or x' = y' ~ z'. 457. Having given the equations {yz - y'z) sin A + {zx' - z'x) sin B + (x-^/' - x'y) sin (7=0, xx' + yy + zz — {yz + y'z) cos A + [zx' + z'x) cos ^ + {vy + tc'^/) cos (7, tcsin (>S'-^) + ysin (^S'- ^) + ssin (6^- C) = = x' sin (.S' -A)+y'sm{S-B)-^ z sin (,S' - (7), where 2;S'=.( +^ + 6'; prove that ^ - ^ ^ and ^' •'^' ^' sin ^ sin ^ sin 6^ ' sin A sin i> sin C ' [It is assumed that cos S is not zero.] 458. Having given the equations y^ + z^ — 2yz cos a s* + x' - 2^.^ cos /? .r* + »/ - 2.r;/ cos y sin-'a sin"^ sin^y 80 PLANE TRIGONOMETRY. ])rovc that, if 2s denote a + /3 + y, one of the following systems of equa- tions will hold : — cos (s - a) cos (s - fi) cos (s - y) ' cos s cos (s - y) cos (s-fi)' y cos {s — y) cos s cos (s — a) ' .r y z cos (s - ^) cos (s - a) cos s ' 459. Having given the equation cos a sin a ^ cos ^ sin ^ ' cos^^ sin^^ , prove that H — : = 1. ^ cos a sm a 4 GO. Eliminate 6, cj> from the equations OC 9/ cc ?/ - cos + ~ sin6= - cos d) + "f sin d) = 1, a b a . 6— d) a — 6 a — di _ 4 cos cos — pr- cos — ^r — = 1. A 2 Z [The resultant equation is ( — cos a] -i-f^-sinaj =3.] 4G1. Having given the equations «^ + b^ - 2ab cos a = c^ + d' - 2cd cos y, b^ + c" - 2bc cos (3=a^ + d^ - 2ad cos S, a6 sin a + cd sin y = bc sin /8 + of/ sin 8 ; proA-e that cos {a + y) = cos (^ + 8). [These are the equations connecting the sides and angles of a quadrangle.] 462. Having given the equations sin $ + sin <\i= a, cos 6 + cos f}> = b ; prove that (1) tan- + tan-^ ^,^^,^^^ , Sab (2) tan e + tan «^ = (3) cos ^ cos ^ = (a' + by-ia' 4 (rt^ + 6=) (4) sin $ sin ^ PLANE TRIGONOMETRY. 81 {a' + hy-W 4:{a' + ¥) ' (5) cos2^+cos2d)= ^ 4^^ — J-, ' . ^ a +0 (6) cos 36 + cos 3 = sin' 6 + sin' 26 + sin' 35-3 sin 6 sin 25 sin 36. 467. Prove that sin 2a sin (ft-y) + sin 2)8 sin (y - a) + sin 2y sin (a - ft) sin (y — ft) + sin (a - y) + sin {ft - a) = sin (/3 + y) + sin (y + a) + sin (a + ft) ; cos 2a sin (ft-y) + cos 2^ sin (y - a) + cos 2y sin (a - ft) ^ sin (y- ft) + sin (a - y) + sin {ft - a) = cos {ft + y) + cos (y + a) + cos (a + ^) ; sin 4a sin {ft-y) + sin ift sin (y - a) + sin 4y sin (a - ft) ^ sin {y- ft) + sin (a - y) + sin {ft - a) = 22 sin (2a + /? + y) + 2 sin 2 (^ + y) + 2 sin {3ft + y) ; w. p. 6 82 PLANE TRIGONOMETRY. COS 4a sin (^ - y) + cos 4/3 sin (y — a) + cos 4y sin (a - P) (4) sin (y - /8) + sin (a - y) + sin (/8 - a) = 22 cos (2a + ^ + y) + 2 cos 2 (/3 + y) + 2 cos {S/3 + y). 468. Prove that ,, , sin 3tt sin (B-y) + two similar terms (1) 5 — r-^^5 — '-{ — r-^j E tan (a + ^ + y), cos oa sm (p - y) + two similar terms ' , sin 5a sin (^ - y) + ...+ ..._ sin (3a + )8 + y) + ... + .. . ' ' cos 5a sin (/J - y) + ... + ... ~ cos (3a + j8 + y) + ...+... * sin 7a sin (/3 - y) +... + ... _ sin (a + 3^ + 3y) + sin (5a + /? + y) + . . . ^ ' cos 7a sin (;3 - y) + ... + ... ~ cos (a + 3/3 + 3y) + cos (5a + fi + y)+ ... ' . , , sin^ a sin (^ - y) + ... + ... _ . . » , (4) 3 r-)% — '-{ = - tan (a + /3 + y). ^ ' cos^asin(/3-y)+ ... + ... \ r- // 469. Prove that, if n be any positive whole number, sin na sin (/8 - y) + sin n^ sin (y — a) + sin ny sin (a - /8) sin (y — jS) + sin (a - y) + sin (/? — a) = 22 sin (pa + q(3 + r(3) ; where p, q, r are three positive integers whose sum is n, no two vanish- ing together, and the coefficient when one vanishes being 1 instead of 2. TT TT TT [If we write a + — , ft + -^ , y + -^ for a, /?, y we get a similar ^7i A7}j JiTi equation with cosines instead of sines as the functions whose argument is the sum of n angles.] 470. Prove that, if n be any odd positive whole number, sin na sin (B — y) + ... + ... _, . , - , -• — o / o\ - 2 sm (pa + qP + ry) ; sm 2(y-/8)+ ... + ... ^^ ^'^ ' ' where p, q^ r are odd positive integers whose sum is n. [The same remark as in the last.] 471. The resultant of the equations X cos 6 -^ y ainO = x cos ^ + y sin =!, a cos Ocos(f) + b sin^sin<^ + c+/(cos^ + cos ^) + ^(sin ^ + sin<^) + ^, .sin(^ + <^) in {c-b)x' + {c-a)y' + a+b + Ify + Igx + Ihxy = 0. 472. Prove that 1 1 1 1 _ sin na +/) sin («- 1) a 2 cos a - 2 cos a - 2 cos a - ... - 2 cos a + p ~ sin (?n- l)o+^sinwa ' there being n quotients in the left-hand member. PLANE TRIGONOMETRY, S'.] 473. Prove that sec^ a sec' a sec* a ^ , • , sin 7ta -, — -^j — -, to n quotients = -; — -. ^- — ; . 4 - 1 - 4 - sin (n + 2) a + sin wa rT,> ,1 ,1 -1 , 1 sin(w4- l)a If we call the dexter u , 1 -u _, =—— r- , whence the equa- ■- " " ' 2 sm na cos a ^ sin (a - y) 1 sec^ cL tion M„ (1 - u_,) = ~. 2- = —J— .1 n\ n-U 4 COS* a 4 ^ 474. Prove tliat, if sin a + sin /5 + sin y = 0, ■ P + y ■ /o \ • //5 + y ^ • sm ^^-^ sm [p — y) + sm ( — ,— ' - o.) sm y . /3 + y . . T . /j8 + y \ . o sin (a - /3) * sm ' sm (y - p) + sm ( — r-^ — a jsinjS ^ ^' 475. From the identity (a; - S) (,«^c) ( ■r-c)(.r- ft) ( a; -«)(«;- 6 ) ^ "^ (f, _ b){a-c) (b- c) {b -a) (c - a) {c - b) ' ' deduce the identities ^ , . , sin (^ — ;8) sin (0 - y) , , cos 2 (6 + a) .-) ^^^— ) \^ + ... + ... = cos 4^, ^ ' sm (a — /S) sm (d - y) . „.. ^ sin (5 - /3) sin (^ - y) . ,. ' sm (a — p) sm (a - y) 476. Prove the identities (1) cos 2 (/3 + y - a - 8) sin (j8 - y) sin (a - S) + cos 2 (y + a - /? - 8) sin (y - a) sin ((3 — 8) + cos 2 (a + /3 - y - 8) sin (a - /3) sin (y - 8) =~8K, (2) cos 3 (y8 + y - a - 8) sin (/? - y) sin {a - 8) + two similar terms = — 16 A'Z, (3) cos (|8 + y - a- 8) sin^ (^ - y) sin' (a - 8) + . . . + ... E A'Z, (4) cos(^-y)cos(a-S)sin'(y8-y)sin'^(a-8)+ ... + ... =KL, (5) cos (j8 + y - a - 8) sin" (/3 - y) sin'" (a - 8) + ... + ... E -16A'/:; ■where /^ denotes sin (/3 - y) sin (y - a) sin (a - (3) sin (a - 8) siu (/? - 8) sin (y - 8) ; and L denotes cos (/3 4- y - a - 8) + cos (y + a - /3 - 8) + cos (a + /3 - y - 8). [The first is deduced from the identity (6'c' + a'cT) (b-c) {a ~ d) + two similar terms = {b ~ c) (c - a) (a - b) (d - a) (d - 6) (d - c) by putting cos 2a 4- r sin 2a for a and the like : the same substitutions iu other identities of (155) give (2), (3), (4), (.i).] e— 2 S4 PLANE TRIGONOMETRY. 477. Prove the identities sin 4a sin 4(3 (1) (2) sin a - /8 sin a - y sin a- 6 sin /3 - y sin (i - 8 sin fi - a + ... + ... = - 8 cos (a + ^ + y + S), COS 4a cos 4^ + -: — , — _ " " ^ _ . sin a - )8 sin a - y sin a - 8 sin j8 - y sin (3 - 8 sin (3 -a + ... + ... = 8 sin (a + )S + y + 8). [In the following questions, up to (484) inclusive, A, B, C are the angles of a finite triangle.] 478. Having given the equations 2/' + «* + 2i/z cos A z' +i)c^ + 2zx cos B x' + i/'+ 2xy cos C _ sinM " sii?J " suFO ' prove that either x sin -4+2/ sin B + z sin (7 — 0, or X sec -4 = 2/ sec ^ = « sec C. 479. Prove that 1+cos^cosC, B-C — ; — fT-. — 7^ — tan — „— + two similar terms sm B sm C 2 ^_(7 (7-^, ^-^ = — tan — ^ — tan — - — tan — ^ — , l-cos.4 , B-C „, B-C^ C-A A-B , tan — K f- ...+... = - 2 tan — - — tan — ^ — tan sin i^ sin C 2 2 2 2 ' 480. Prove that (m sin C + TO sin ^ cos ^4 ) (n sin -4 + ^ sin C cos -6) (I sin .S + 7;i sin A cos C) + (n sin B + m sin (7 cos -4) (Z sin C + ?i sin A cos 5) (ni sin A + I sin 5 cos 6') = {l + m +n) (mn sin' -4 + w? sin' B + /m sin* (7) sin A sin ^ sin C. 481. If a?, y, 2; be real quantities such that 3/ sin C - 2; sin -B s sin ^ - a; sin C x-y cos C — z cos -5 2/ ~ '^ cos -4 - a? cos (7 ' ,, . , , « sin .5 - V sin -4 , , , , prove tliat each memoer = jr-^ -. , and that z-x cos B - y cos ^ sin .4 sin B sin C ' [The given equation is also satisfied if sin C - 0, but this is excluded by the condition that A, B, C are angles of a finite triangle. It may be noticed that each term must be of the form - .] PLANE TRIGONOMETRY. 85 482. Prove that (3 + 2 cos 2A + S cos 25 + 3 cos 2C + cos 2B^^)' + (3 sin 25 - 3 sin 2C+ sin 2B - C)' •= 8 (cos 2A cos 2B cos 2C + 3 cos 5 - C cos C-AcosA-B - 1 5 cos A cos B cos C - 1 ). 483. Prove that sin* A (sin* B + sin^ C - sin^ ^) + . .. + ... = 2 sin* -i sin" B sin' C (1+4 cos A cos 5 cos C) ; (sin ^ + sin 5 + sin C) (- sin -4 + sin B + sin C) (sin A - sin 5 + sin C) (sin ^ + sia B - sin C) = 4 sin* ^ sin* B sin* C. 484. Prove that sin A (sin A — sin 5) (sin A - sin (7) + ... + .. . = sin ^ sin 5 sin C (3 - 2 cos A -2 cos 5-2 cos C) ; sin* ^ (sia A - sin 5) (sia A - sin C) + ...+ .. . = (cos A + cos 5 + cos C - 1)* (cos* -4 + , . . - cos B cos C - . . . ) ; sin* 5 sin 2C- 2 sin 5 sin Csin (5 - (7) - sin* (7 sin 25 = 0; sin* 5 cos 2C - 2 sin 5 sin Ccos (5 - C) + sin* C cos 25 = sin* A. III. Poristic Systems of Equations. [In all the examples under this head, solutions arising from the equality of any two angles are excluded, and all angles are supposed to lie between and iir. A system of n equations of which the type is a cos (a^ - a^^,) + h cos (a^ + a^^,) + c + 2/ (sin a^ + sin a^^,) + 2g (cos a^ + cos a^^j) + 2/i sin (a^ + a^^J = 0, •where r has successively integi-al values from 1 to n, and a^^^ = aj, is a poristic system, when solutions in which angles are equal are excluded : that is, the equations cannot be satisfied unless a certain relation hold connecting the coefficients a, b, c, /, g, h, and if this relation be satisfied the number of solutions is infinite, the n equations being equivalent to w - 1 independent equations only so that there is one solution for each value of ttj. All the examples here given are reducible to this type.] 485. Having given tan /8 cot ^-^^ = tan y cot , w 2 /? + v prove that each = tan a cot — ; and that sin ()8 + y) + sin (y + a) + sin {a + ft') = 0. 86 PLANE TRIGONOMKTUV. 486. Having given a + 6 P + 6^ tan — jr- tan (5 - tan - - tan a, a. 4- R prove that each = - cot —-^- tan $, and tliat sin (a + 6) + sin (j8 + 6) = sin (u + P). 487. Having given the eqiiations (B+y + e^ , y + a + ^, _ ^ a + /3 + 6 ^ tan ^' tan a = tan - — ^r tan /i - tan ^r tan y - ^ A A 'It, prove that sin (,5 + y) + sin (y + a) + sin (a + P)- 0, cos /? cos y sin f3 sin y 1 w. &c. and that ^ = tt. = 0, 488. Having given the system of equations a cos P cos y + b sin /3 sin y = a cos y cos a + b sin y sin a = a cos a cos ft + b sin a sin /? = c, prove that bc + ca + ab = 0, and that the given system is equivalent to any one of the following systems : (1) tan - _ - cot a = tan ^-— - cot B = tan — r — cot y = - ; ^ 2 2 2 ' a (3 + y (^» ^- = =f^ cos a cos — ^— ' (3 + y (3) ^^^-2- b (i-y c sin a cos —— (4) cotf cot|+eot|cot^+cot"cot^ By -,26 = tan "^ tan ^ + . .. + ... = 1 + — 2 2 c (5) sin (/S + y) + sin (y + a) + sin {a + ft) = 0, cos {(S + y) + cos (y + a) + cos {a + (3)= r . 1 + cos (/3 - y) + cos (y - a) + cos (a - ^) = - ^ ^.^ ; (6) (7) PLANE TRIGONOMETRY. 87 cos a + COS P + cos y sin a + sin /? 4- sin y a — b C03(a + /8 + y) ~ sin(a + /8 + y) ~a + b' cos a cos )8 cos y — sin a sin /3 sin y 1 6" cos (a + yS + y) ~ a* sin (a + /8 + y) ~ (a + 6)^" a B y a (^ y tan - + tan ^ + tan -^ cot - + cot - + cot ^ _, (8) = 1 + — ; ^ ' ^ a+ B + y a+ B + y a tan -^ — - cot — -' 2 2 ,Q. 6 (tan a + tan )8 + tan y) a(cota+ cot )84- coty)_ , ^ ' {ia + 6) tan (a + /? + y) " (a + 26) cot (a + ^ + y) ~ ' (10) a* (tan /8 tan y + tan y tan a + tan a tan /3 -I) = 6* (cot /? cot y + cot y cot a + cot a cot /3 - 1) = — (a + 6)*; and that a, /?, y are roots of the equation in 6, bcos(a+ B + y) asm(a+B + y) , ^ ^^ — i^ — ^ V-tF — - + a + 6 = 0. cos 6 sin U 489. Having given the equations e cos (j8 + y) + cos ((3 -y)-e cos (y 4- a) + cos (y — a) = e cos (a + P) + cos (a - /3) ; e' - 1 prove that each member of the equations is equal to — x— , and that sin ()8 + y) + sin (y + a) + sin (a + /?) = 0, cos (/3 + y) + cos (y + a) + cos (a + /S) = e. 490. Having given the equations sin (a - ft) + sin (a — y) _ sin (/3 — y) + sin (ft - a) sin j8 + sin y - 2 sin o sin y + sin a — 2 sin ft * prove that sin a + sin ^ + sin y = 0, cos a + cos ft + cos y = — 3e. 491. Having given the equations sin(.-g-^v) .in(;?->-f°) ^ COS ^a + to'^ cos ^^ + 5^^ prove that each member is equal to . / a + ySN sin(^y--2-j cos(^y+-2-j 88 PLANE TRIGONOMETRY. and that cos (/3 + y) + cos (y + a) + cos (a + (i) = 0, sill (/3 + y) + sill (y + a) + sin {a + (3) = - e. 492. If cos(a + ^-g) cos (g + y - g) sin {a + fi) cos'^y sin (a + y) cos^/3 ' each member will be equal to cos (/3 + y - 6) sin (/3 + y) cos^a ' ^. ,^ /,_ sin(^-t-y)sin(y+a)sin(a + /3) and CO P = jy: r , r j ^rr r-jT 75 ^v " cos (yS + y) cos (y + a) cos [a + fi) + sm (a + jS + y) 493. Having given the equations o* cos a cos (B + a (sin a + sin /3) = a* cos a cos y + a (sin a + sin y) = - 1, prove that a^ cos yS cos y + a (sin /8 + sin y) + 1 = 0, that cos a + cos /? + cos y = cos (a + /? + y), 2 sin a + sin (3 + sin y = sin (a + /8 + y) : and that B + y y + a -,a + /3 1 tan — — ' cos a = tan ^— r — cos B = tan —77— cos y = - . 2 2 2 ' a 494. Having given the equations . ^ a+y . , a+^ sm (3 = 7)1 tan — ^ , sm y = m tan — ^ ; prove that sin a = m tan "—^ ; sin a + sin yS + sin y + sin (a + ^ + y) - 0, cos a + cos /3 + COS y + cos (a + ^ + y) = - 2m, and sin a sin fi sin y = m^ sin (a + /3 + y) - - '^* sin {(;8 + y) + sin (y + a) + sin (a + p)]. 495. The system of equations cos (/3 + y) + 7?i (sin /3 + sin y) + w = 0, cos (y + a) + 7?i (sin y + sin a) + n = 0, cos (a + /?) + wi (sin a + sin ft) +n = 0, is equivalent to m^ = 1, m = sin (a + /8 + y), w^i + sin a 4 sin /S + sin y = 0. 496. Having given the equations sin (/3 + y) + A; sin (a + ^) = sin {y + a) + k sin (/3 + ^) = sin (a+/3)+ k sin (y + ^) ; prove that ^■ = 1, and that each member = 0. PLANE TRIGONOMETRY. 89 497. Having given the equations a cos (y8 - y) + 6 (cos f3 + cos y) + c (sin /3 + sin y) +d = 0, a cos (y — a) +b (cos y + cos a) + c (sin y + sin a) + o? = 0, a cos (a - /?) + 6 (cos a + cos P) + c (sin a + sin/3) + cZ = ; prove that a" + 6" + c^ = 2acl, a (cos a + cos (S + cos y) + 6 = 0, a (sin a + sin /? + sin y) + c = 0. 498. Having given the equations (m + cos {3) {pi + cos y) + n (sin j8 + sin y) = (m + cos y) (m + cos a) + n (sin y + sin a) = (m + cos a) {m + cos /5) + ?i (sin a + sin/8) ; prove that each = — n* ; and that cos (a + ;8 + y) - COS a — COS ;8 - COS y = 2wi, sin (a + )8 + y) — sin a - sin ^ — sin y = 2n. 499. Having given the equations cos (6- ft) + COS (^ - y) + cos (,S - y) = cos (^ - y) + cos (d-a) + cos (y - a) = cos (^ - a) + cos {6-ft) + cos {a- ft) ; prove that each = cos (ft — y) + cos (y — a) + cos (a — ft) = -1 ; and that cos a + cos j8 + cos y + cos ^ = = sina + sin/3 + sin y + sin B. 500. Having given m cos a + w sin a - sin (/? + y) m cos /? + n sin ^ — sin (y + a) cos {ft + y) cos (y + a) ' prove that each m cos y + w sin y - sin (a + /S) , ^ \ • i r, \ = '- , ' ^, — -^^ ^' = m cos (a + /? + y) + ?i sin (a 4- 5 + y). cos (a + /3) \ ' / / 501. The three equations 2 2 . , ft-y — — ■ ={m-n) cos — TT- , : sin a ^ ' 2 y + o . V + a m cos ^-jr— n sin ^-^r- ,. - — ; — TT— = (m - n) cos ^-TT— , cos/3 sin/3 ^ '^ 2 a + yS . a + i3 w cos — --^ ?i sin ±_ 1_ , , a-ft •^— — ^ — ■ = {m - n) cos ^ cos y sin y ^ ' 2 are equivalent to only two independent equations. 90 PLANE TRIGONOMETRY. 502. Having given cos (a + 6) cos (f3 + 6) sin (j8 + y) sin (y + a) ' prove that each = . ' y — -f = ± 1. 503. Having given the equations m (cos yS + cos y) — n (sin ^ + sin y) + sin (f3 + y) = m (cos y + cos a) — n (sin y + sin a) + sin (y + a) = m (cos a + cosyS) — n (sin a + sin /3) + sin (a + ^) ; prove that m = sin (a + /? + y), 7i = cos {a + /3 + y). 504. Having given the equations B+y P—y y + a Y — a a + 5 a — 5 e cos -~~ + cos — ^' e cos --— 4 cos '—^r-~ e cos — ^ + cos ^ .J ^ ^ ^ ^ • y^ + Y • y3 — y . y+a . y — a . a + iS . a — B' e sm — — ^ + sin -^- e sin ^-jr h sin ~— e sin — ^ + sin — ^— Z Z Z Z u u prove that each = ± / — 3 + e'' 505. Having given the equations g'sin )8 sin y + e (cos )8 + cos y) + 1 e^sin y sin a + e (cos y + cos a) + 1 „ ^-y 2 y-a cos — - — • cos -^--- A 2 _ e^sin a sin y8 + e (cos a + cos /3) + 1 cos-y^ prove that each = or 4. If each = 0, sin a + sin y8 + sin y + sin (a + yS + y) = 0, and e {cos a + cos yS + cos y + cos (a + )8 + y)} = - 2 ; and, if each = 4, e' 2e sin a + sin /? + sin y sin {/S + y) + sin (y + a) + sin (a + y3) sin a + sin yS + sin y + sin (a + y8 + y) ' Also the given system is equivalent to the system »-/^ + y (S-y . ./iS + y \ « Bin -?^ cos a - e cos ^-^-^ sin a = sin I ^— -' - « ) • and the two corresponding equations. PLANE TRIGONOMETRY. 91 506. Having given the system of three equations whose type is a cos yS cos y + b sin j3 sin y + c +/(sin /? + sin y) + g (cos y8 + cos y) + h sin (^ + y) = 0, prove that ab — k' — be -/' + ca - g'. 507. Prove that any system of three poristic equations between a, (S, y is equivalent to two independent equations of the form ^(cosa+cos^+cosy) + 7?iCOs(a + ^ + y) + n(cos/:^+y+cosy + a + cosa+/i) = p, /(sina+sin^+siny) + ?/isin(a+/3 + y) + w(siny8 + y + siny + a + sin a+(3) = q. [The equation between /?, y will be 2 (n" - 1') cos (/3 - y) + 2 {Im + np) cos (/3 + y) + m' + n' - jf - q^ - I' + 2lq (sin ^8 + sin y) + 2 {mn + Ip) (cos (3 + cos y) + 2nq sin (/? + y) = 0.] 508. The condition for the coexistence of four equations of the type in (506) between a, /? ; /?, y ; y, 8 ; and 8, a respectively is A=abc + 2/gh - af- - bg^ - ch^ = 0. 509. Having given the system of five equations - cos a cos p + r ^^'^ a sin p = - , a b c and the like equations between P, y ; y, 8 ; 8, c ; c, a resjiectively ; prove that a' + b" + c^ = {b + c){c + a) (a + b). [An equivalent form is 4o6c + (6 + c — a) (c + a - 6) (a + 6 - c) = 0.] 510. Prove that the system of n equations a+a a+a a. + a. tan " - " cot a, = tan ^ ' cot a - tan -^^ * cot a = ... J J J = tan -J — ^ " ' cot a = - , 2 "a is equivalent to only n - 1 independent equations. IV. Inequalities. Q 511. Prove that cot ;r > 1 4- cot 6, for values of 6 between and tt and that, for all values of ^, — ^ — < 2 + cos ^. 92 PLANE TRIGONOMETRY. _ , - , , - cc' — 2x cos a + 1 , . , , 512. Prove that, for real values of x, -^ — ^r z^ , lies between ' a;' - 2a; cos /8 + 1 1 — cos a 1 1 + cos a 1 - COS p 1 + COS p 513. If X, y, z be any real quantities and A, B, C the angles of a triangle, prove that x' + 2/* + s" > 2yz COS A + 2zx cos B + 2xy cos C, unless X cosec A=y cosec B = z cosec C. 514. Under the same conditions as the last, prove that {x sin^ A+y siu^ B + z sin^ Cf > i {yz + zx + xy) sin^ A sin* B sin* C, unless X tan A=y tan B = z tan (7. 515. If A, B, C be the angles of a triangle, prove sin B sin C + siii C sin ^ + sin A sin jB sin* ^ + sin* B + sin* C lies between the values ^ and 1 : and sin A sin ^ sin C r between and 1 + cos A cos B cos C 1 ^/3• 516. Having given the equation sec P sec y + tan /? tan y = tan a, prove that, for real values of P and y, cos 2a must be negative j and that tan P + tan a tan y cos /? tan y + tan a tan jB cos y ' 517. Prove that, A, B, C being the angles of a triangle, ABC i >sin- sin-^ sin -2>(1 -cos^)(l -cos-S)(l -cos C)>cosA cosBcosC; and that ABC cos 2- cos — COS -^ > sin A sin BsmC> sin 2^ sin 2^ sin C ; except when A — B = C. 618, Prove that, A, B, C being the angles of a triangle, ,. A B C/ / 1 + cos -^ COS ^ cos -7 \ 8 sin ^ sin B siaC .y/3 sin ^ sin B sin C. PLANE TRIGONOMETRY. 9^ 519. On a fixed straight line AB is taken a point C such tli.it AC — 2CB and any othei- point F between A and C ; prove that, if CP =. C A sin. A, AP . BP' will vary as 1 + sin 3^ and thence that AP . BP^ has its greatest value when P bisects AC. 520. If a + ^, ^ + 5, y + C be the angles subtended at a point by the sides of the triangle ABC, then will ''sin^ a sin^ fi sin' y\^ 2 sin^ a sin^ yS sin^ y /sin a \sin A sin .5 sin (7/ . A . B . C ' sin -^ sm -^ sm — Ji Ji Z except when the point is the centre of the inscribed circle. 521. Having given the equations cot /3 cot y + cot y cot a + cot a cot /8 = tan j8 tan y 4-tan y tan a + tan atanyS ; prove that cos 2 (/5 - y) + cos 2 (y - a) + cos 2 (a - /?) > - |. 522. If a, /?, y be angles between and - , and if tana tan /? tan y= 1, then will sin a sin /3 sin y < ^ — j^ , unless a = yS = y. 523. Prove that rcos' (g - 6) sin' (a - 6)} |cos' (a + 6) sin' (a + e) \ cannot be less than — ^^ ; and can never be equal to it unless tan* a a'b- lie between -^ and =-5 . or b' 524. If o), X, y, z be any real quantities, and a, h, c, a', b', g cosines of angles satisfying the condition 1—1, c', b', a\—0; I C, - 1, a, b \ b', a, - 1, c a, b, c, —\ prove that 03° + 2/' + 2* + w* > 2a'yz + 2b'zx f 2c'xy + 2aojx + 2buiy + 2c 6 cos B -\- c cos C, b > c cos C + a cos .4, and c> a cos ^4 + 6 cos B. 527. If a ti'iangle A'B'C be drawn whose sides are h + c, c + a, a + h respectively, and if the angle A' -A, then will '2a lie between 6 + c and 2 (6 + c), and B-C , . A . 3A cos -^— = 4 sm - - sin -2" . 528. If $, (j), xf/ be acute angles given by the equations cos if = i , cos d> = , cos \1/ = J ; b + c ^ c + a ^ a + b then will tan'' . + tan= ? + tan'' | = 1 ; and tan - tan - tan -- = tan ^ tan — tan ^ . ^ ^ ^ ^ Z ^ 529. If sin J, sin ^, sin C be in harmonical progression so also will be 1 - cos A, 1 — cos B, 1 - cos C. 530. From the three relations between the sides and angles given in the forms a^ = 6^ + c^ - 2bc cos A, &c. deduce the equations sin^^sjn^^sma ^ ^ ^ + C =180"; a b c assuming that each angle lies between and 180". 531. In the side BC produced if necessary find a point P such that the square on PA may be equal to the sum of the squares on PB, PC ; and prove that this is only possible when A, B, are all acute and tan A < tan B + tan C, or when B or C is obtuse. When possible, prove that there are iii general two such points which lie both between B and C, one between and one beyond, or both beyond, accoi'ding as A is the greatest, the mean, or the least angle of the tiiangle. PLANE TRIGONOMETRY. 532. The sides of a triangle are 2pq+p*, p^-^pq-vq', and p^ - q* \ prove that the angles are in A. p., the common difference being 2tan-YJ-^^ y2p + qj 533. The line joining the middle points of BC and of the perpen- dicular from A on BC makes with JBC the angle cot"' (cot B - cot C). 534. The line joining the centres of the inscribed and circumscribed circles makes with BG the angle cos j5 + cosC- r _i /cos B + cos C - IN \ sin B — sin C J 535. The line joining the centre of the circumscribed circle and the centre of perpendiculars makes with BC the angle _, /tan^tanC-3\ \ tan B — tan C J ' 536. The line joining the centre of the inscribed circle and the centre of perpendictdars makes with BC the angle ^ ,( C-B con A tan" )cot — j: — + 2 ^ . . B . C . B-cy 2 sin - sm — sm — ~ 537. In a triangle, right-angled at A^ prove that r, = r, + ?-3 + r. 538. \i BA = AC ^-IBC and BC be divided in in the ratio 1 : 3, then will the angle ACQ be double of the angle AOC. 539. If the sides of a triangle be in A. p. and the greatest angle exceed the less by (1) 60", (2) 90", (3) 120", the sides of the triangle will be as (1) v/13-1 (2) v/7-1 (3) V5 - 1 ^/13: V13 + 1, v/7: v'7 + 1, s/5 : ^5 -Hi. [In general if the sides be in A. p. and the greatest angle exceed the least by a, the sides will be as Jl — cos a + ^\ — cos a : ^1 - cos a : ^1 - cos a — ^ 1 — cos a.] 540. If be the centre of the circumscribed circle and AO meet BG in A OD : AO = cohA : cos (B-C). 541. Three parallel straight lines are drawn through the angular points of the triangle ABC to meet the opposite sides in A', B', C : prove that A'B.A'C B'C.B'A C'A . C 'B AA' ■*" BB'' ^ CC*'"' on PLANE TRIGONOMETRY. the seginputs of a side being affected with opposite signs when they fall on opposite sides of the point of section. [The convention stated in the last clause ought always to be attended to, but it is not yet so sufficiently recognised in our elementary books as t(i make the mention superfluous. BC + CA + AB=0 ought always to be an allowed identity.] 542. The perimeter of a triangle bears to the perimeter of an inscribed circle the same ratio as the area of the triangle to the area of the circle, which is .-^ ,^ f^ cot -jr cot -X cot jr- : IT. Ii 2i 2i [The first part of this proposition is true for any polygon circum- scribed to a circle, and a similar one for any polyhedron circumscribed to a sphere.] 543. A triangle is formed by joining the feet of the perpendiculars of the triangle ABC, and the circle inscribed in this triangle touches tlie sides in A', B', C : prove that EC CA' A'B' ^ , r, r. 544. A circle is drawn to touch the circumscribed circle and the A . . sides AB, AC ; prove that its radius is rsec^-^ : and if it touch the cir- oumscribed circle and the sides AB, AC produced its radius is /j sec* — . \i B = C and the latter radius = R, cos A = \. 545. Having given the equations cy + h'z = a^z + c^x = h'x + u^y; ,, , X y z prove that — — ^-r = - — —r. = -. — ^7^ . ^ sm 2A sm 'IB sm 2C 546. Determine a triangle having a base c, an altitude h, and a given difference a of the base angles ; and if 6^, 6^ be the two values 4-7/ obtainable for the vertical angle, prove that cot 6. -f cot 6 = — ^—^ . *= ' ^ ' ^ c sin a Prove that only one of these values corresponds to a proper solution ; and if this be 0^, that , ^, JW + c' sin'a - 2h tan -^ = ^ J- r . 2 <^ (1 - cos a) Account for the appearance of the other value. 547. Determine a triangle in which are given a side a, the opposite angle A, and the rectangle m* under the other two sides : and prove that no such triangle exists if 2m sin -^ > a- 548. Find the angles of a triangle in which the greatest side ia t^vice the least, and the greatest angle twice the mean angle. Prove that a triangle whose sides are as 17156 ; 13.S95 : 8578 is a very approximate solution. PLANE TKIGOXOMETRY. 97 549. A triangle A'B'C has its angles respectively complementary to the half angles of the triangle ABC and its side B'C equal to BC ; prove that A.I 'B'C : Ayl^C = sin ^ : 2 sin f sin | . 550. Two triangles ABC, A'B'C are such that cot A + cot A' = cot B + cot B' = cot C + cot C ; a point P is taken within ABC such that its distances from A, B, C are as B'C : CA' : A'B : prove that the angles subtended at P by the sides of the triangle ABC are B' + C , C + A\ A' + B' ; also that cot A' + cot B' + cot C = cot A + cot 5 + cot C. 551. If ^', ^', C be the angles subtended at the centroid of a triangle ABC by the sides, cot A - cot A' = cot B — cot ^' = cot C - cot C" = f (cot ^ + cot 5 + cot C). must lie between the acute angles cos ' f ^^ ) , and the difference 552. If 8 cos A cos B cos C = cos'a, each angle of the triangle ABC /I =t sin a\ between the greatest and least angles cannot exceed a. 553. If a straight line can be drawn not intersecting the sides of a triangle ABC, and sxicli that the perpendiculars on it from the angular points are respectively equal to the opposite sides, then will ^ ,A ^ ^B ^ „C ^ A ^ B ^ C ^ tan^ + tan — + tan' — = tan — + tan — + tan -^ = 2, ABC cos^ -rr + cos^ — + cos^ ^ = sin A + sin B + sin C. [Of course these equations are equivalents.] 554. With A, B, C as centres are described circles whose radii are acos^l, bcosB, ccohC respectively, and the internal common tangents are dra\vn to each pair : prove that three of these will pass through the centre of the circumscribed circle and the other three thi'ough the centre of perpendiculars. 555. The centroid of ABC is G, a triangle A'B'C is drawn whose sides are GA, GB, GC, and circles described with centres A, B, C and radii a' sin A', h' sin B', c sin C respectively : prove that three of the internal common tangents to a pair of circles intersect in G, and the other three in the point of concourse of the lines joining A, B, C to the corresponding intersections of the tangents to the circumscribed circle at A, B, C. \y. P. 7 •)8 PLANE TRIGONOMETRY. nSG. If a triangle AJiC be inscribed in a circle and the tangent at A meet J56' in A', then a straight line drawn through A', perpeudicnlar to the internal bisector of the angle A, will meet the circle in two points (■^> Q), whose distance from ^ is a mean proi)ortional between the distances from B, C respectively ; and z PAB - i CAQ = \{B ~ C). Also if rt' > 46c the straight line drawn through A' parallel to the bisector will meet the circle in two other points having the same property. If B, C be fixed points and A any point such that a' = ibc, the two last points will coincide, and its locus will be a rectangular hyperbola whose foci are B, C. 557. If be the point within the triangle ABC at which the sum of the distances of A, B, C is a minimum, straight lines drawn through A, B, C at right angles to OA, OB, 00 respectively will form the maximum equilateral triangle which can be circumscribed to ABC; and if A'B'C be this maximum triangle, then will OA' : OB' : OC = BC :CA : AB. Prove also that OA siir A sin {B - C) + OB sin= B sin {C-A) + OC sin- C sin {A-B) = 0. 558. If X, y, z be perpendiculars from the angular points on any straight line ; prove that a" {x -y){x-z) + h- {y -z){y- x) + c~ (z- x) (z-y) = {I^ABCf, any perpendicular being reckoned negative which is drawn from its angular point in the opposite sense to the other two. 559. If perpendiculars OD, OE, OF be let fall from any point on the sides of the triangle ABC, and x, y, z be the radii of the circles AEF, BFD, CDE, respectively ; prove that 1 6a^ (af - y') {x'-z')+...- Sabc (ax' con A + ...) + a'b'c' = 0. 560. _ If be the centre of the circle inscribed in ABC, OD, OE, OF pei'pendiculars on the sides, and x, y, z radii of the circles inscribed in the quadrilaterals OEAF, OFBD, OBOE; prove that (r - 2a;) (r - 1y) (r - 1z) = r^ - ixyz. 561, A triangle A'B'C is circumscribed to the triangle ABC, prove that when its perimeter is the least possible BC =^B'C' Jl - sin ^' sin C ; and, if x, y, z be the sides of the triangle A'B'C, that x^ - ft' _y^-b' z^ -c^ (« + 2/ + 2;) {y + z - x){z + x - y){x + y - z) X y z A^xyz 562. If/Jj, p„, 2^3 ^6 ^^^ perpendiculars of the triangle 1111 cos A cos B cos C 1 P. 1\ Pz »* P, 2\ Ih ^^ also ^), is a harmonic mean between r,,, r.y UNJV PLANE TRIGONOMETRY. ~ 09 563. The distances between the centres of the escribed cii-cles being a, )8, y; prove that Va + Vi + 'V, 2jcT (o- - a) (o- - )8) (o- - y) ' where 2o- E a + /3 + y. Prove also that 564. The distances of the centre of the inscribed cii'cle from those of the escribed circles being a, (3', y' ; prove that r^-r r^-r r^-r that 32i2' - 2 i? (a" + jS" + y") - a'/S'y' = 0. 565. Prove that the area of the triangle = T T A / 1 = Tr A / I =r T A / — ' — - oe the radius of 566. Taking p to be the radius of the polar circle of the triangle, prove that the area and that 4p* (r, + r^ + 7-3 - r) + (r^ + r3 - r, + ?•) (7-3 + »-, - r^ + ?•) (r, + r^ - r, + r) = 0. 567. The cosines of the angles of a triangle are the roots of the equation 472V - 472 (7? + r) x' + (2r' + 47?r - p"-) x + p' = 0; and the radii 9\, r^, r^ are roots of the equation x' (472 + r - fc) = (x - r) (^RT~r - p'). 568. Prove that — - = f^iri A sin B sui 6, «py and ^^i'lf'} = ( 1 + cos .1 ) ( 1 + cos 7?) ( 1 + cos C). a/jy 7—2 100 ri.ANE TRIGONOMETRY. 509. Prove that ^^ A ■ b" d' . a ■ . b cos -^ COS ^ cos - sin ^^ sin — y sin ,^ 570. Tlie radii of two of the circles which touch tlie sides of a triangle are p, q, and tlie distance between their centres 8 ; prove that J r^ — 1 , the lower sign being taken when either circle is the inscribed circle. 571. The points 0, 0' are the centres of the circumscribed circle, and of a circle which touches the sides of the triangle in the points A\ B, C; L is the centre of perpendiculars of the triangle A'B'C ; prove that 0, 0', L are in one straight line, and that O'L : C>0' = r(orr,,r„, rj : E. 572. If an isosceles triangle be constructed whose vertical angle is cos"' (i), the inscribed circle will pass thi'ough the centre of perpen- diculars. [In general the inscribed circle will pass through the centre of per- pendiculars if 2 cos A cos B cos C = (1 - cos A) (1 - cos B) (1 - cos 6').] 573. li 0, o be the centres of the circumscribed and inscribed cii'cles, and L the centre of perpendiculars OL'-2oL' = E'-Ar'; and if o^ be the centre of the escribed circle opposite A, 574. If the centre of the inscribed circle be equidistant from the centre of the circumscribed circle and from the centre of perpendiculars, one angle of the triangle must be 60"; and with a similar property for an escribed circle, one angle must be 60" or 120". 575. The cosine of the angle at which the circumscribed circle in- tersects the escribed circle opposite A is 1 + cos A - cos B - cos C 2 ' and if a, /3, y be the three such cosines (^ + y)(y-l-a)(a+/3) = 2(a+^ + y-l)l 576. If P be any point on the circumscribed circle, PA auiA+FB sin B + FCsmC=0; a certain convention being made in respect to sign : also PA^ sin 2A + PB' sin 2B + PC sin 2C = i^ABG. PLANE TRIGONOMETRY. 101 577. If P be any point in the plane of the triangle, and the centre of the circumscribed cii'cle, PA^ sin 2^1 + ... + ... - 40P' sin A sin 5 sin C = 1\ABC. 578. If P be any point on the inscribed circle PA^ sin A + PB^ sin B + PC" sin C will be constant ; and if on the escribed circle opposite A^ - PA' sin A + PB' sin B + PC sin C will be constant. 579. Prove that, if P be any point on the nine points' circle, PA' (sin 2B + sin 2C) + PB' (sin 2C + sin 2A) + PC (sin 2 A + sin 2B) = 8P- sin A sin i? sin (7(1+2 cos -4 cos B cos C). 580. If P be any point on the polar circle, PA' tan A + PB" tan 5 + PB' tan C will be constant. If p be the radius of this circle and S the distance of its centre from the centre of the circumscribed cii'cle, then will and if 8' be the distance of its centre from the centre of the inscribed circle, then will 8'- = p' + 2r^ 581. The straight line joining the centres of the circumscribed and inscribed cii'cles will subtend a right angle at the centre of perpen- diculars if 1 + (1 - 2 cos ^) (1 - 2 cos ^) (1 - 2 cos C) = 8 cos A cos B cos C. 582. If P be a point within a triangle at which the sides subtend angles A + a, B + (3, C + y respectively, PA'^=PBi^ = PC'^^. ■ sm a sin fi sin y 583. Any point P is taken within the ti-iangle ABC and the angles BPC, CPA, APG are A\ B, 6" respectively; prove that A^PC (cot A - cot A') = i^CPA (cot B - cot B') = ^APB (cot C - cot C). 684. Having given the equations cos' ^ (p sin" e + g- cos' 6) = cos' P (^ sin' C + ^ + .7 cos' G + d) = cos' C (p sin' B-e+q cos' C - 0) ; prove that each = (p + q) cos A cos B cos C ; and that (P + 9)' _ 4 cos' J cos' B cos' C ^5* (cos A - cos P cos 6') (cos P - cos G cos ^4 ) (cos C — cos .^4 cos B) ' 1Q£ PLANE TRIGONOMETRY. IV. Heights and Distances. Polygons. 585. At a jjoint A are measiii-ed the angle (a) subtended by two objects (points) P, Q in the same horizontal plane as A and the distances h, c at right angles to AP, AQ respectively to points at which PQ sub- tends the same angle (a) ; tind the length of PQ. 586. An object is observed at three points A, B, C lying in a hori- zontal straight line which passes dii-ectly underneath the object ; the angular elevations at A, B, C are $, 20, 39, and AB = a, BC = b ; prove that the height of the object is '^^J{a + b){3b-a). If cot e = 3, « : &=13 : 5. 587. The sides of a rectangle are 2a, 2b, and the angles subtended by its diagonals at a point whose distance from its centre is c are a, j3 : prove that -1 /» 07 2 2 -r^ — ^ 5-j = a^ (tan a - tan BY + b" (tan a + tan BY : (w" + b - c-y ^ ' la being that side which is cut by the distance c. 588. The diagonals 2a, 2b of a rhombus subtend angles a, y8 at a point whose distance from the centre is c : prove that ¥ (a- - cy tan' a + a' (6" - c")' tan'' (3 = ia'b'c'. 589. Three cii'cles A, B, C touch each other two and two and one common tangent to A and B is parallel to a common tangent of A and C : prove that if a, b, c be the radii, and p, q the distances of the centres of B and C from that diameter of A which is normal to the two parallel tangents pq r= 2a^ = 8&C. 590. Three circles A, B, C touch each other two and two, prove that the distances from the centre of A of the common tangents to B and C are equal to • 2bc (b+ c) -a(b - cf ± 46c J a {a + b+ c) (6T^r ^' and that one of these distances = if a (6 + c ± 2 V26c) = 26c. 591. Circles are described on the sides of the triangle ABC as diameters : prove that the rectangle under the radii of the two circles, which can be described touching the three, is 4i?' (1 + cos A) (1 + cos B) (1 + cos C) cos A cos B cos (7 (l+cos^)(l+cos^)(l+cosC)-(cos5-cosC)^-(cosC-cos^)^-(cos^-cos^)*' 592. Four points A, P, Q, B lie in a straight line and the distances AQ, BP, AB are 2a, 2b, 2c respectively; circles are described with diameters A Q, BP, AB : prove that the radius of the circle which touches the three is c {c — a) {c - b) <^ — ab PLANE TRIGONOMETRY. 103 593. A i)olygon of n sides inscribed in a circle is such that its sides subtend angles 'la, 4a, Ga, ... 2/ta at the centre ; prove that its area is to the area of the inscribed regular w-gon in the ratio sin na : n sin a. 594. A point F is taken within a parallelogram ABCJJ ; prove that the value of A.4PC cot APC - ^BPD cot BPD is independent of the position of P. 595. The distances of any jjoint P on a cii-cle from the angular points of an inscribed regular w-gon are the positive roots of the equation d being the diameter of the circle and 6 the angle subtended at the centre by any one of the distances. Prove that if we take d = 2, the equation may also be written [2 1^ + (-l)"2(l-cos«e) = 0. 596. The sides of a convex quadrilateral are a, h, c, d and 2s is their sum : prove that Js {s — a- d) [s — b- d) (s - c — d) cannot be greater than the area. 597. The equation giving the length x of the diagonal joining the angles (a, d), (b, c) of a quadrilateral whose sides taken in order ai-e a, b, c, d, is {x- (ab + cd) — (ac + bd) [be + ad)Y sin" a + {x^ (ab - cd) - {ac — bd) (be — ad)Y cos* a = ia'b^c-d- sin® 2a ; where 2a is the sum of two opposite angles. [This equation, being a quadi'atic in cos 2a, leads to the equation giving the extreme possible values of x ; which can be reduced to the form 3 (x' ~ a' - by {x' -c'- dj - 4 {a;* (ab + cd) - {ac + bd) {be + ad)]' + 1 GaVc'd' - 0.] 104- PLANE TRIGONOMETRY. 598, 111 any quadrangle ABCD, the vertices are E, F, G, (the intei-sections of BC, AD ^ 6'J, BD ; and AB, CD respectively) ; prove that {EB . EC - EA . EDf _ {FC . FA - FB . FP f EA. EB. EC. ED sin' E ~ FA .FB.FC.FD sin* F ~ GA.GB.GC.GDsm'G' VII. Expansioiis of TrigonometHcal Functions. Inverse Ficnctions. 599. By means of the equivalence of the expansions of 2e' sin X X e* cos x, and e"' sin 2x ; prove that s;:; sta (>'-'■) 7 cos ';;^ a-sm^ t GOO. Prove by comparing the coefficients of $-" ' that the expansions of sin 6 and cos 6 in terms of $ satisfy the identity 2 sin 6 cos 6 = sin 2$. 601. Prove that (n 2)(n-3) ^^0^+1)^ 2" - (« - 1) 2'--' + ^^^"^^'^ ^-^ S"- - . . . = -i , (n integral). - ri) 602. Prove, from the identity 1 1 2isme 1-xz 1 - xz~^ 1 - 2a; cos ^ + ar ' that '. — ^h — ^r-^ = (2 cos 6) - in - 1) (2 cos 6) sm ^ "• ^(«^-2)|^-3)^2cos^)--..., , , - ,. • c af_s\ii{ii+\)6-sinln-\)6\ and deduce the expansion of cos m { = ^ ^^— ^ — ~^ '— J terms of cos 9 when n is a positive integer. 603. From the identity log (1 + xz'^) + log (1 — xz) = log (1 - 2ix sin - x^, or from the identity 1 1 ^ 2 {I -ix sin 6) l-xz 1 +xz-'~ l-2ixsine-x" PLANE TRIGONOMETRY. 105 deduce the equations cos n9 = 1 - o" s"^ ^ + — ^ ■ < s"i ^ ^ rci sin* ^ + . . . , J !* sin ?i^ - n am 6 ^r^ ^ sin 6 + — ^ -A— ' sm' 6- . . . , 71 being an integer, even for the first and odd for the second. Also prove that, if 6 lie between — ^ and ^ , both are true for all values of 7i ; and thence deduce the true expansions of cos n$ and sin oiO in terms of sin 6 for any value of 6. 604. From the expansion of (sin 0)-"^^ in temis of sines of multiples of 6, prove that - - ,_ ,, 2n(2n-3) 2n{2n-l)(2n-5) 0= l-(2?i-l)+ — ^ '- ^^ -p ...to w+1 terms. if L2 605. Prove that , n-l . n-l 2)1-1 -- n-l2n-13n-l „. 1 cos 6 + ;,^ cos 2d H o cos 3& + ... to CO u 11 2)1 ?4 ^M 6)1 n-\d cos n '2i (^2cos^j if 6 lie between - tt and tt. 606. Prove that 1 A, n^(!izMizl) {n-3){n-i){n-b) = (-l)''sin2(vi + l)J-sin^. o o 607. Prove that, if tan (= t) be less than 1, - „- ?i(?i+l)(w+2) - w(w + l)...(?i + 4) , Bmw^cos"^=?«« 5^ r^ ''f+ ^ % ^ '-f- ... and cosngcos"g^l-!i^^v "("-^^^(^:;'')^"^^) ^--... [These results are obviously true when w is a negative whole number.] 608. The sum of the infinite series 1 1 1.3 1 4- K cos 2^ - -—7 cos i9 + ^^— ^ — ^ cos 6^ - . . . 2 2.4 2.4.6 is J cos 6(1 + cos 6), if 9 lie between - -^ and ^ . lOG PLANE TRIGONOMETRY. 609. Prove that the identity |2 \i may be deduced from the identity 2 cos n9 E (2 cos 6)" - n (2 cos 6)"-' + ^li'izA) (o cos 0)"-* - ... when n is an even integer, by Avriting T)~ ^ ^o^' ^ '^^^'^ taking the terms ill reverse order ; and similarly for sin uO when n is odd. GIO. If i' (h) = 1 - — + — ^^- ^ :^ -i^ ' + . . . to yif 1 terms, ^ ^ [2 [4 [0 prove that F(2n) = (- l)''i^(w). [i^(7<^) = cos-^ for all values of n.] o 611. If the constants a^, a,^, a^...a^^ be so determined in the ex- pression ttj sin X + a., sin '2x+ ... + a ^ sin nx + sin (w + 1) a; that the coefficients of x, x^, x^,...x^"~^ shall vanish, the value of the expression will be 2" sin x (cos x — 1)" ; and if, in ctj cos X + a^ cos 2x + ... + « ^ cos nx + cos(?i + l)a; the coefficients of all the powers of x up to x""'' inclusive vanish, the value will be 2" (cos a; - 1 )" ( cos x H =- ) ^ ' V n+\J 612. Prove that =" sin^ ra cos^"~Va _ {2n + \)x 2 =1 x^ + tanVa '" (1 + xf'^' - (1 - xf"*' where (2n + l)a= ir. 613. From the identity X n f\ -J- . . 6-ix . 6 + ix e - 2 cos + e = iaui — - — sm — - — , resolve the former into its quadratic factors. ■ [The result is 4 sin^ |(l + fJ) (l + p^.) (l + p^) -all x^ factors of the form 1 +7:77- — zri being taken where r is a positive or negative integer. PLANE TRIGONOMETRY. 107 Similarly " — = the product of all factors of the form A i " 4 cos -y 1 + , — '- ; where r is an odd positive or negative integer. 1 614. From the result in the last question deduce (1) :^. = C ' 3sin*e - r=-^{r7r + 6)" zero being included among the values of r. [By equating coefficients of x'" in the results, it appears that Oi—l -j-^ = sum of the products a together of all expressions included |2« sin" 6 ^„ for integi-al values of r from - oo to oo including zero.] {rir + &) 615. Prove that 1 1 _1 1 1_ 1 _1 sin^~^'*"7r-^ TT + d 'lir-a^ -lir+e^ Zir-d •••*°^' - - ^ ^ ^ . . to 00 . 2sin^ Tv'-e' {2Try-6'- (37r)'-6^ 616. Prove that tan e 1 1 1 ^80 " TT-' - id' ^ (Stt)^ - i6' ^ (Stt)^ _ ^^, + . . . to CO . 617. Prove that, if ^ be an angle between — -- and - , 4 4 ^ . „ ^ 2 sin* e 2 . 4 sin" e 6- = sm- 9+-^ —V— + .5—^ --~^— + ... to 00 o 2 3.5 o ^ _ /, l\tan*^ /, 1 iXtan"^ = tan-^- (^1 + -j^_ + (^1 + _ + 5) -3— ... to 00 [The former is true for all values of $.] 618. Prove that (1) V T 3' r+1 2''+i 3^+1 • 27r ■n -It c — C /ON 1' 2* 3* ^ 47r' r+1 2V1 3^-fl c''v'-i + e-''^'2-2cos(7ry2) 108 PLANE TRIGONOMETRY. G19. In a triangle the sides a, h, ami the angle ir-d opposite h arc given, and is small : prove that, approximately, c , a6' a(3a'+3ab-b') e* b-a~ b 2 b' \i' G20, Prove that the expansion of tan tan . . . tan x is \o \0 O \ I when the tangent is taken n times. 621. Prove that the expansion of sin sin. . . sin x is a;-w^' + 9i(5«-4)^-| (175^^ + 336/1 + 162)^'+...; the sine being taken n times. 622. Prove that the expansion of tan"' tan"'... tan~'a; is x-2n~ + hi{57i + l)~-^il75n' + Shi + U)^ + ... [That is, the expansion of tan"" a; might be deduced from the ex- pansion of tan" X by putting — n for n, the index applied to the function denoting repetition of the functional operation.] 623. Prove that (1) tan-'^V = ta^~'F3+ta^~'4TT» (2) tan"'yV = tan-'-3-iT + tan-'2i8> (3) = tan"' ^ - tan-' ^V - tan"'.Jg- + tan"'^!^? (4) 2 = tan" ' I + tan-' i + tan"' ^ + 2 tan" ' ^^ + tan"' Jj-, = tan"' ^ + tan"'i + tan~'i + tan"'| - tan"' J^, = 3 tan~'^ + tan"'^ + tan"' yttst- 624. The convei-gents to ^2 are 1, |, |^, ...-^ ...; pr-ove that tan"' tan"' — = tan" ^21 1 2n 1 2n-\-\ _ 1 1 _ / 1 ' and that tan ' tan"' = tan '/ ^ 2 2n+l J? 8n+l \ n 4- PLANE TRIGONOMETRY. 109 625. Find x from the equation cof'o; + cot~'(/r-a;+ 1) =:cot~'(w- 1); aiul fiutl tlie tangent of the angle tan-' 3 + 3 tan"' 7 + tan"' 26 - ^ . 4 626. Prove that, if tan (a + i/8) = z, a, /? being real, that a will be indeterminate and )8 infinite. 627. Prove that if cos (a + i;8) = cos c;^ + i sin ^, where a, yS, ^ arc real, sin (/> = ± sin'^ a, and that the relation between a and /8 is €^-e"^ = ±2sina. 628. Prove that, if tan (a + i/3) = cos <^ + t sin ^, and a, /3, be real, TT TT , o , /I + sin <^\ " = '* o + 7 ' 4/3 = log ^-^ . 2 4 '^ ° \1 - sin ^/ 629. Prove that, if tan (a + ?/5) = tan ^ + i sec ^, and a, /?, ^ be real, o '^ _i ^ o 1 /I + cos ^\ VIII. Series. [In the summation of many Trigonometric series in which the 7'*'' term is of the form «_. cos r8, or a^ sin rO, a^ being a function of r, it is convenient to s\im the series in the manner exemplified by the following solution of the question : — " To find the sum of the series 1 + 2 cos ^ + 3 cos 26 + ... +n cos n — 16." Let C denote the proposed seiies and /S the corresponding series with sines in place of cosines, namely, ;S'e 2sin^ + 3sin25+ ... + oi sin n-l6, the first term being sin . ^ or 0, then, if cos 6 + i sin 6=.z, n -cf 1 c oo „-i 1 —s" (n+1 —oiz) C + i;S E 1 + 2.~ + 3;;- + . .. + nz" ' E f^ — ^ _ 1 — (cos nd + i sin n6) {n +1) +n (cos )i + I 6 + i sin n+ 16) ~ (1 — cos^- tsin^)^ _ same numerator (2sin-j f sin --icos-^ j E-(2sin^j [cos^ + isin^j 6 — 1 sin 6 — {n + 1) (cos n-\B -\-r sin n - 1 ^) + n (cos n6 ■\- i sin n 6) -(2si„|)' 110 PLA'IW: yRIGONOMETRY. whence, equating possibles and imjigs^bles on tlie two sides, _ (n+l) cos (n-\)6 — cos 6-n cos 7iO ^ 2(1 -cos^) ' and also o _ ('* + !) ■'^"'^ (n-l) + sin -n sin nO 271-008^) ■ It is obvious that in general if /(x) = «o + ct^x + a^ + . . . , and z have the same meaning as above, f{xz) +f{xz~^) = 2 (a^ + a^x cos B + ... + a^x" cos nO + ...), f{xz) —f[xz~^) = 2i (a^x sin 9 + ... + ax" sin nO ■¥ ...). Some doubt may often arise as to the limiting values of the angle 6 beyond which results found by this method may not be tnie, but this can always be cleared up by the use of the powers of x as coefficients as in the forms just given. Thus, to take a very well-known case, to sum the infinite series sin ^ — | sin 2^ + ^ sin ?>6 ... . Take x^ x^ C = x cos 0-'-^ cos 26 + ..., and S = x sin 6 — -- sin 2^ + ... and we have C + iS=xz- I x'^z^ + ^ofz^ - ... = log(l +xz) = log p (cos (f) + i sin (since the series manifestly vanishes when x=0), and when a; =1, the B B result will be ^ if ^ lie between those limits, or B between - tt and tt. So also the corresponding series in cosines cos B-\ cos 2B + \ cos 3^ - . . . = log y 2 ( I + cos B)^\ log U cos' -) , (B\ 2 cos - ) without the proper limitation that cos ^ must be positive. The series will be convergent only if 01? be less than 1 ; and this will be generally the case. PLANE TRIGONOMETRY. Ill Many series also may be summed by the same method as was exphxined under the corresponding head in Algebi-a : that is by obtaining the r"' term (n^) of the pro])Osed series in the form f7^^.^ — U^. Thus, to sum the series cosec X + cosec 2x+ ... + cosec 2"~^x, we have sin2''~'a; sin (2'' -2''"') a; cosec 2x = —. — - -, — -. — ^~ = —. — ^~— j — -. — '^—- = cot 2 'x - cot 2''x, sin 2 X sin 2 x sm 2 x sm 2x so that U^_^_^ = — cot 2'~^x, and S^ — f7_^, - TJ^ -cot-- cot 2"~^x. Such being the method, it is clear that giving the answer would, in these cases, amount to giving the whole solution.] G30. Sum the following infinite series, and the corresponding series in sines, (1) cose+ Jcos2^+ icos3^ + ..., ,^. . cos 3^ cos 5^ (2) cos^ ^^^5 ■■■' (3) cos^-^cos3^ + icos55-..., (4) 1 - 92, cos ^ + -^^ — - cos 20 i ^^ ' cos 3^ + . . ., ... , - cos 26 cos 36 (5) l-cos6+-^ ^-+-.-, ,-, . 1 cos 36 1 . 3 cos 56 (6) C0S6 + I ^-+27-4-5-+-. „ , COS 36 1 . 3 cos 56 (/) cos 6 -^-^— + 2;-^—^ ..., ,-, . „ cos 26 2. 4 cos 36 (8) cos6 + |-^ + 3-^— g— + ..., (9) a; cos 6 + 1 03^ cos 26 + ^ :>? cos 36 + . . ., when x = cos 6, /iA\ /I 1 o cos 36 1.3 3 cos 56 , „. (10) aj cos 6 + i 03" — ;r — + - — 7 X — 3 — + . . ., when x = cos 26. ^ ' ^ 3 2.4 5 [(1) C = -llog(4sin^|), ,?=tan-'(^^^^)="--^f 61iebe- tween and tt, (2) C=|sin(cos6){«*'"^ + £-s'n«), ,S' = l cos (cos6) {eS'"^- c-s'"^), (3) C* = ± ^ being of the sign of cos 6, -S' = \ log f y^Jin $) ' 112 PLANE TRIGONOMETRY. (4) C=f2cos-j con 71-, S=(2cos-^\ sinw-, 6 being be- tween - 7r and n, (5) (7= 6-<"osecos(sin^), ,S'= - e-cose^in (sin^), (a a \ COS - + sin - + ^siu 6^ ) , if sin be positive, (7) (7 =log rycos^ + ^/2cos-j , and cos^aS' = cos ^, if cos ^ be positive, (8) C = p'' coa 2(f), /S = p'sin2<]i, where p cos ^ = cos"' [. /sin-j, cos - + sin - + . / sin - j , 6 being between and 27r, (9) C=-|log(cosec'^), /S' = tan-'(cot^), (10) cos^ ■ = sin 6 (sin + cos 6), if cos 26 be positive.] ,ycos 26 631. Sum the series . ,. sin^3^ sin" 3"-^^ 3- cos'Se , ,,„_i cos'S"-'^ cos^*^ __+...+(_l)" '- ^„_, , cos 9 cos SO cos S"~^$ sinTS^'^sm^'*""" '^ sin 3"^ ' sin 20 3 sin 60 3""' sin 2 (3"-'^) 1 + 2cos2^ l + 2cos6^ "■ 1 + 2 cos 2(.3"-'^) ' 2 cos g - cos SO 2"cos3"-'^-2''-'cos3"^ sm3^ + • . . + siiTS"^ ' 1 + 2 cos 20 1 + 2 cos 2"'-'0 sin 4^ "^""'"^ sin 2^'"^ 1 - 2 cos 20 1-2 cos 2"e sin 20 ^ ■■■"'" sin 2"6 ' 5sin3g-3sin 5g 5 sin 3 (4""'^) - 3 sin 5 (4"-'^) cos 3^ - cos 5^ + . . . + 4 ^^^ f(^ln-f0^_ ^Qg 5 ^^n-.^^ , 1 + 4 sin g s in 3^ _, r+ 4 sin ^'-'0 sin3 (4""'^) sin iO '" sin 4"^ ' 3 sin - sin 30 3 sin 3""'^ -sin 3"^ cos~3^ ' "^ ■■■ "^ 3"-' cos 3"^ ' PLANE TRIGONOMETRY. IIH 632. Prove that sec + sec(— + 6) + 5ec(—+e) + ...+ sec (2 («i - 1) - + o] is equal to 0, or to (— 1) ^ msec 7n9, according as m is even or odd : also that sec-^ + sec^l — + 6]+ ... +sec^- 2 (m-1)- +$> \m J ( ^ ' m ) is equal to ;;; , or to m" sec^ mO, according aa m is even or odd. [The equation which expresses cos mO in terms of cos $ wnll be satisfied by cos ^, cos ( — + ^] , ..., cos-|(m- 1) — ^ + ^1; the results of the question follow on finding the sura of the reciprocals of the roots, and the sum of their squares.] 633. Prove that cos 9 + cos (a + ^) + cos (2a + ^) + . . . + cos (??i — la + ^) = 0, if ma = 4:7r, m being any positive integer except 2 ; that sec- - + sec — + . . . + sec n n (^0^^('*■-^>■ if w be any even positive integer except 2 ; and that Stt „ 9i — 1 tt w' — 1 sec - + sec — + sec 1- . . . + sec n n n ' 2 n 2 if n be any odd positive integer except 1 . 634. Prove that, if 71 be a whole number > 4, sin - cos sm — cos — + sin — cos — ... = 0, n n n n n n the number of terms being — - — , or h — 1> as n is odd or even. [The roots of the equation (1 + x)" = (1 - a:)" are the values of i tan — , where r may have all integral values from to n — 1 , omitting 7: if w be even. Hence -; ri = 2 f , where r has 2 (l+x)--(l-x) ^^,^^,r. n all integral values from Ito— „— or--l and A^ is (-1)'"' sin' — cos""*—.] MM 71 71 w. p. 8 lU PLANE TRIGONOMETRY. G35. Prove tliat n sin 7»0 sin sin cos n — cos n9 cos <^ — cos cos (^ — cos (a + 6) sin d) sin — cos n9 cos ^ — cos 9 cos <^ — cos (a + ^) sin (2a + 9) sin (?i — 1 a + ^) cos ^ - cos (2a + ^) ■■■ cas^-cos(M-la + ^) where w is a positive integer, and na = 27r. CONIC SECTIONS. GEOMETRICAL. I. Parabola. [The focus and vertex are denoted always by S and A respectively.] 6.36. Two parabolas having the same focus intersect : prove that the angles between their tangents at the two points of intersection are either equal or supplementary. 637. A chord PQ of, a parabola is a normal at P and subtends a right angle at the focus : prove that SQ is twice SP, and that PQ subtends a right aaigle at one end of the latus rectum. 638. A chord PQ of a parabola is a normal at P and subtends a right angle at the vertex : prove that SQ is three times SP. 639. Two circles each touch a parabola and touch each other at the focus of the parabola : prove that the angle between the focal distances of the points of contact is 120". 640. Two parabolas have a common focus and axes at right angles, a circle is drawn touching both and passing through the focus : prove that the points of contact are ends of a diameter, or subtend an angle of 30" at the focus. 641. Two parabolas have a common focus, a circle is described touching both and passing through the focus : prove that the angle between the focal distances of the points of contact will be one thu-d f)f the angle between the axes, or one thii'd of the defect from four right angles of this angle. 642. Two parabolas A, B have a common focus and axes at right angles : prove that any two tangents drawn to A at right angles to each other will be equally inclined to the tangents drawn to B from the same point. 643. In a pai'abola AQ \^ drawn through the vertex A at right angles to a chord AP to meet the diameter through P xrx Q : prove that Q lies on a fixed straight line. 8—2 lir> roxrc RFCTIONS, GKOMETRTOAL. G 44. Tlirou_<;li any ])oiiit P of a parabola is drawn a straight line QPQ' perpcnilicular to the axis, and terminated by the tangents at the onortional between QP, P(/. G45. A circle touches a paral)o]a at a point whose distance from the focus exceeds the latus rectum, and passes through the focus : prove that it will cut the j)arabola in two points, and that the common chord will cut the axis of the paral)ola in a fixed point at a distance from the focus equal to the latus rectum. 64G. A paraljola is described touching a given circle, and having its focus at a given point on the circle : prove that if the distance of the point of contact from the focus be less than the radius of the circle, the circle and parabola will have two other common tangents whose common j)oiut will lie on a fixed straight line which bisects the radius drawn from the focus. 647. With a given point as focus is described a parabola touching a given circle : prove that the ]ioint of intersection of the two other common tangents lies on a fixed circle, such that the polar of the given point with respect to it passes through the centre of the given circle. [If the given point lie on the given circle, the locus degenerates into the straight line bisecting at right angles the radius through the given jioint.] G48. On the tangents drawn from a point are taken two points P, Q such that SP, SO, SQ aie all equal : prove that PQ is ])er2)en- dicular to the axis and its distance from is twice its distance from A. G4t). Two equal parabolas have a common focus /S'and axes opposite, and fSPQ is any straight line meeting them in P, Q ; with centres P, Q are drawn circles touching the respective tangents at the vertices: prove that these circles will have internal contact, and that the rectangle under their radii will l)e fixed. 650. On a focal chord PQ as diameter is described a circle which meets the parabola again in P' , Q' : proAe that the circle P'SQ' will touch the parabola. 651. A circle touches a parabola in /*, passes through S and meets the parabola again in Q, Q' ; a focal chord is drawn parallel to the tangent at P : prove that the circle on this chord as diameter will pass through Q, Q', and that the focal chord and QQ' will intersect on the directrix. 652. Two parabolas whose foci are S, S' have three common tangents, and the circle circumscribing the triangle formed by these tangents is drawn : jjrove that SS' will subtend at any point on this circle an angle equal to that between the axes of the parabolas. 653. Fiom any jioint on the tangent at any point of a parabola perpendiculars are let fixll on the focal distance and on the axis : prove that the sum, or the difference, of the focal distances of the feet of these perpendiculais is equal to half the latus rectum. CONIC SECTIONS, GEOMETRICAL. 117 654. The noi'mal at a poiut P is produced to so that PO is bisected by the axis : pi-ove that any cliord througli sul)teuds a riglit angle at P ; and that tlie circle on PO as diameter will have double contact with tlie parabola. 655. From a fixed point is let fall OQ perpendicular on the diameter through a point P of a parabola : ])rove that the ])erpendicular from Q on the tangent at P will pass through a fixed point, which remains the same for all equal jiarabolas on a common axis. 656. A circle is drawn through two fixed points R, ,% and meets a fixed straight line through P again in P : prove that the tangent at P will touch a fixed parabola whose focus is S'. 657. Two fixed straight lines intersect in : prove that any cii'ch; tlu'oiigh and through another fixed point >S meets the two fixed lines again in points such that the chord joining them touches a fixed parabola whose focus is S. 658. The perpendicular AZ on the tangent at P meets the parabola again in Q : prove that the i-ectangle ZA, AQ is equal to the square on the semi latus rectum and that PQ passes through the centre of curva- ture at A. 659. Two parabolas have a common focus and axes at right angles : l)rove that the directrix of either passes through the point of contact of their common tangent with the other. 660. Through any point P on a parabola is drawn PK at right angles to AP to meet the axis in K : })rove that AK is equal to the focal chord parallel to AP. Explain the result when P coincides with A . 661. A circle on a double ordinate to the axis PP' meets the parabola again in Q, Q' : prove that the latus rectum of the parabola which touches PQ, P(/, P'Q, P'Q' is double that of the former, and its focus is the centre of the circle. 662. Three points A, B, C are taken on a parabola, and tangents drawn at them foi-ming a triangle A'B'C ; a, b, c are the centimes of the circles PC A', CAP', ABC: prove that the circle through a, b, c will pass through the focus. 663. Two points arc taken on a parabola, such that the sum of the parts of the normals intercepted between the points and the axis is equal to the part of the axis intercepted between the normals : pi'ove that the dirterence of the normals is equal to the latus rectum. 664. The perpendicular ^S'}' being drawn to any tangent, a straight line is di-awn through Y parallel to the axis to meet in Q the straight line through >S' parallel to the tangent : prove that the locus of ^ is a parabola. 6G5. If X be the foot of the directrix, SY peri)cndicular from the focus on a chord PP', and a circle with centre *S' and radius equal to X Y nieet the chord in QQ': prove that PP, QQ' subtend equal angles at S. 118 CONIC SECTIONS, OEOMETllICAL. 6G6. A given straight line meets one of a series of coaxial circles in A, B: prove that the parabola which touches the given straight line, the tangents to the circle at A, B, and the coiniuou radical axis will have another fixed tangent. [If K be a ])oint circle of the system, L the intei'section of the given straight line with the radical axis and KO drawn at right angles to KL to meet the radical axis in 0, the fixed tangent is the straight line through perpendicular to the given straight line,] G67. Two tangents TP, TQ are di-awn to a parabola, OP, OQ aie tangents to the cii'cle TPQ : prove that TO wUl pass through the focus. 6C8. A triangle ABC is inscribed in a circle, Aii' is a diameter, a parabola is described touching the sides of the triangle with its directrix passing through A' and S is its focus : prove that the tangents to the cii-cle at B, C will intersect on SA'. 6G9. The normals at two points P, Q meet the axis in p, q and the chord PQ meets it in 0: pi'ove that straight lines drawn through 0, p, q at right angles respectively to the three lines will meet in a point, 670. Normals at P, P' meet the axis in G, G', and straight lines at right angles to the normals from G, G' meet in Q : prove that PG :GQ = FG' : G'Q. 071. The tangent to a parabola at P meets the tangent at Q in T and meets SQ in R ; also the tangent at Q meets the directrix in A": prove that Pl\ TR subtend equal or supplementary angles at K. 672. Two equal parabolas have a common focus and axes inclined at an angle of 120": prove that a tangent to either ciu've at a common jjoint will meet the other in a point of contact of a common tangent. 673. The chord PPi. is normal at P, is the centre of cui"vature at P and U the pole of PR : prove that U will be peiiDendicular to SP. 674. From a fixed point is dra^vn a straight line OP to any point P on a fixed straight line : prove that the straight lines drawn through P equally inclined to PO and to the fixed sti'aight line touch a fixed parabola. 675. A parabola whose focus lies on a fixed circle and whose directrix is given, always touches two fixed parabolas whose common focus is the given centre, and whose directrices are each at a distance from the given directrix equal to the given radius; and the tangents at the points of contact are at right angles. 676. The centre of curvature at P is 0, PO meets the axis in G and OL is drawn perpendicular to the axis to meet the diameter through P; prove that LG is parallel to the tangent at P. 677. The straight lines Aa, Bh, Cc are dravsm perpendicular to the sides BC, CA, AB of a triangle ABC : prove that two parabolas can be drawn touching the sides of the triangles ABC, abc respectively, such that the tangent at the vertex of the former is the axis of the latter. CONIC SECTIONS, GEOMETRICAL. 119 678. A right-angled triangle is described self-conjugate to a given parabola and with its hypotenuse in a given direction : prove that its vertex lies on a tixed straight line parallel to the axis of the parabola and its sides touch a fixed parabola. 679. Two equal parabolas have their axes in the same straight line and their vertices at a distance equal to the latus rectum ; a chord of the outer touches the inner and on it as diameter is described a circle : prove that this will touch the outer parabola. 680. Tangents are drawn from a fixed point to a series of con- focal parabolas : prove that the corresponding normals envelope a fixed parabola whose directrix passes through and is pai'allel to the axis of the system, and whose focus S' is such that OS' is bisected by S. 681. A point on the directrix is joined to the focus S and SO bisected in F ; with focus i^is described another parabola whose axis is the tangent at the vertex of the former and from two tangents are drawn to the latter parabola : prove that the chord of contact and the corresijonding normals all touch the given parabola. 682. Prove the following construction for inscribing in a parabola a ti-iangle with its sides in given directions : — Draw tangents in the given directions touching at A, B, C, and chords A A', BB\ CC parallel to BC, CA, AB ; A'B'C will be the i-equired triangle. [The construction is not limited to the parabola, and a similar construction may be made for an inscribed polygon.] 683. Two fixed tangents are dra^v^l to a parabola : prove that the centre of the nine points' cii-cle of the triangle formed by these and any other tangent is a straight line. 684. At one extremity of a given finite straight line is drawn any circle touching the line, and from the other extremity is drawn a tangent to this cii'cle : prove that the point of intersection of this tangent with the tangent parallel to the given line lies on a fixed parabola, and those with the tangents perpendicular to the given line on two fixed hyperbolas. 685. Two parabolas have a common focus and from any point on their common tangent are drawn other tangents to the two ; jirove that the distances of these from the focus are in a constant ratio. 686. Two tangents are drawn to a parabola equally inclined to a given straight line : prove that their point of intersection lies on a fixed straight line passing through the focus. 687. Two parabolas have a common focus S, parallel tangents di'awn to them at F, Q meet their connnon tangent in F, Q' : prove that the angles FSQ, FSQ' are each e(pial to the angle between the axes. 688. Two parabolas have parallel axes and two parallel tangents are drawn to them : prove that the straight line joining the points of contact passes through a fixed point. [A general property of similar and similarly situate figures.] 120 CONIC SECTIONS, GEOMETRICAL. 689. On a tangent are taken two points equidistant from the focus : prove that the other tangents drawn from these points will intersect ou the axis. 690. A circle is described on the latus rectum as diameter and a straight line through the focus meets the two curves in F, Q : prove that the tangents at J\ Q will intersect either on the latus rectum or on a straight line parallel to the latus rectum and at a distance from it equal to the latus rectum. 691. A chord is dra^vn in a given direction and on it as diameter a cii'cle is described : prove that the distance between the middle points of this chord and of the other common chord of the circle and parabola is of constant length. 692. On any chord as diameter is described a circle cutting the parabola again in two points : prove that the part of the axis of the parabola intei'cepted between the two common choi-ds is equal to the latus rectum. 693. Two equal parabolas are placed with their axes in the same straight line and their vertices at a distance equal to the latus rectum ; a tangent drawn to one meets the other in two points : prove that the circle of which this chord is a diameter touches the parabola of which this is a chord. 694. A parabola is described having its focus on the arc, its axis parallel to the axis, and touching the directrix, of a given parabola : prove that the two curves will touch each other. 695. Circles are described having for diameters a series of parallel chords of a parabola : prove that they will all touch another parabola related to the given one in the manner described in the last question. 696. A circle is described having double contact with a parabola and a chord QQ' of the parabola touches the circle in P : prove that QP, Q'P are respectively equal to the distances of Q, Q' from the com- mon chord. 697. The locus of the centre of the circle circumsciibing the tri- angle formed by two fixed tangents to a parabola and any other tangent is a straight line. 698. The locus of the focus of a parabola touching two fixed straight lines one of them at a given point is a circle. 699. Two equal parabolas A, B have a common vertex and axes opposite : prove that the locus of the poles with respect to A of tangents to B is A. 700. Three common tangents PP\ QQ', PR' are drawn to two pai-abolas and PQ, P'Q' intersect in L : prove that LP, LP' are parallel to the axes. Also prove that if PP' bisect QQ' it will also bisect PP', and PP' will be divided harmonically by QQ', PP'. CONIC SECTIONS, GEOMETRICAL. 121 701. Two equal parabolas have a common focus and axes opposite ; two circles are described touching eacli other, eacli with its centre on one parabola and touching the tangent at the vertex of that i)arabola : prove that the rectangle under their radii is constant whether the contact be internal or external, but in the former case is four times as great as in the latter. 702. Two equal parabolas have their axes parallel and opposite, and one passes through the centre of curvature at the vertex of the other : prove that this relation is reciprocal and that the parabolas cut at right angles. 703. From the ends of a chord PP' are let fall perpendiculars PM, P'M' on the tangent at the vertex : prove that the circle on PP' as diameter and the circle of curvature at the vertex have PP' for radical axis. [The analytical proof of this is instantaneous.] 704. A parabola touches the sides of a triangle ABC in A', £', C", B'C meets £G in P, another parabola is di'awn touching the sides and P is its i^oint of contact with PC : prove that its axis is parallel to £'C". 705. The directrix and one point being given, jirove that the para- bola will touch a fixed parabola to which the given straight line is tangent at the vertex. 706. The locus of the focus of a parabola which touches a given parabola and has a given directrix parallel to that of the given parabola is a circle. 707. A triangle is self-conjugate to a parabola, proA-e that the straight lines joining the mid points of its sides touch the parabola ; and that the straight line joining any angular point of the triangle to the point of contact of the corresponding tangent will be parallel to the axis. 708. Four tangents are drawn to a pai-abola : prove that the three circles whose diameters are the diagonals of the quadrilateral will have the directrix as common radical axis. 709. A circle is drawn meeting a parabola in four points and tangents drawn to the parabola at these points : prove that the axis of the parabola will bisect the diagonals of the quadrilateral so foi-med. 710. The tangents at P, Q meet in T, and is the centre of the circle I'PQ : prove that 02' subtends a right angle at aS' and that the circle OPQ passes through 6'. 711. Three parallels are drawn through A, B, C to meet the opposite sides of the triangle ABC in A', B', C : prove that a parabola can be drawn through A'B'C and the middle points of the sides, and that its axis will be in the same direction as the tlirec parallels. 122 CONIC SECTIONS, GEOMETRICAL. 712. A chord LL of a circle is bisected in 0, and // is its pole; two pai'abolas are described with their focus at 0, theii' directrices passiiii^ through //, and one of their connnon points on the circle : prove that the angle between their axes is equal to LllL'. II. Central Conies. [In these questions, unless other meanings are expressly assigned, S, IS' are the foci of a central conic, C the centre, AA', BB' the major and minor axes, T, t and G, g the points where the tangent and normal at a point P meet the axes, and CD the semi-diameter conjugate to Crl\ 713. If S". 714. If 'S'F, SZ be drawn pei-pendicular respectively to the tangent and normal at any point, YZ will pass through the centre. 715. A common tangent is drawn to a conic and to the cii'cle whose diameter is the latus rectum : prove that the latus rectum bisects the angle between the focal distances of the points of contact, 716. If a triangle ABC circumscribe a conic the sum of the angles subtended by BC at the foci will exceed the angle A by two right angles. 717. Two conies U, F have a common focus ^S*, the tangents to U zX two common points meet in P and to F in Q : prove that PQ passes throiigh 8 (the common points being rightly selected when there are four). 718. Perpendiculai's SY, S'Y' are drawn on any tangent and YP, Y'P' are the other tangents from Y, Y' : prove that SP, S'P' will inter- sect on the conic. 719. A circle touches the conic at P and passes through S, PQK drawn perpendicular to the directrix meets the circle in Q : prove that QSK is a right angle. 720. On a tangent are taken two points 0, 0' such that SO-SO' = mnjor axis: prove that the radius of the circle OS'O' is equal to the major axis. 721. Tangents OP, OQ, OP', O'Q' are drawn to a certain circle : prove that the foci of the conic which touches the sides of the two tri- angles OPQ, O'P'Q' lie on the circle. 722. In an ellipse in which BB' = SS' a diameter PP' is taken and circles drawn touching the ellipse in P, P' and passing through S : their second common point will lie on the latus rectum. CONIC SECTIONS, GEOMETRICAL. 123 723. Prove that when SG = PG, SP is equal to the hxtus rectum ; and if PK drawn always at right angles to SP meet the axis major in /t, SK has then its least possible length, 724. A common radius CPQ is dra^vn to the two auxiliary circles of an ellipse and tangents to the circles at P, Q meet the corresponding axes ia. U, T : prove that TU will touch the ellipse. 725. A circle has double contact with a conic : prove that the tangent from any point of the conic to the circle bears a constant ratio to the distance from the chord of contact. 726. The foot of the perpendicular from the focus on the tangent at the extremity of the farther latus rectum lies on the minor axis. 727. The common tangents to an ellipse and to a cii-cle through the foci will touch the circle in points lying on the tangents at the ends of the minor axis ; and the common tangents to an ellipse and to a circle with its centre on the major axis and dividing /S'*b" harmonically will touch the circle in points on the tangent at one end of the major axis. [These two cases are uiidistinguishable analytically.] 728. The tangent at P and normal at Q meet on the minor axis : prove that the tangent at Q and normal at P will also meet on the minor axis and PQ Avill always touch a confocal hyperbola. [The data will not be possible for the ellipse unless SS' > BB'.'\ 729. Prove that at the point P where SP = PG, SP :IIP = BG^ :AC\ 730. A tangent meets the auxiliary cii'cle in two points through which are di-awn chords of the circle parallel to the minor axis : prove that the straight line drawn from the foot of the ordinate parallel to the tangent will divide either chord into segments which are as the focal dis- tances of the points of contact. 731. Two diagonals of a quadrilateral intersect at right angles : prove that a conic can be inscribed with a focus at the intersection of the diagonals. 732. Given a focus ^S' and two tangents, the locus of the second focus is the straight line through the intersection of the tangents per- pendicular to the line joining the feet of the perpendiculars from S on the tangents. 733. Given one focus, a tangent, and a sti-aight line on which the centre lies, prove that the conic has a second fixed tangent. 734. From the foci S, S' are drawn perpendiculars SPY, S'P'Y' on any tangent to the auxiliary circle meeting the conic in P, P' : prove that the rectangle SY, S'P' - the rectangle S'Y, SP = PC\ 735. The length of the focal perpendicular on any tangent to the auxiliary circle is equal to the focal distance of the corresponding point on the ellipse. 124 CONIC SECTIONS, GEOMETRICAL. 73G. Through C is drawn a straiglit line parallel to either focal dis- tance of P, and CD is the radius jiarallel to the tangent at P : prove that the distance of D from the former straight line is equal to BC. 737. Prove that, if an ellipse be inscribed in a given rectangle, the ]>oints of contact will be the angular points of a parallelogram of con- stant perimeter : and investigate the corresjiondiiig theorem when the conic is an hyi)erbola. 738. A straight line is drawn touching the minor auxiliary ciicle meeting the ellipse in P and the director circle in Q, Q : prove that (^P, PQ' are equal to the focal distances of P. 739. Given a focus and two tangents of a conic, prove that the iMnelope of the minor axis is a parabola with its focus at the given focus : also a common tangent to this parabola and any one of the conies subtends a right angle at the given focus. 740. A pei'pendicular from the centre on the tangent meets the focal distances of the point of contact in two points : prove that these points are at a constant distance SC from the feet of the focal perpen- diculars on the tangent. 741. The tangent at a point P meets the major axis in T ] prove that SP :ST ■-.A^' -.AT. 742. The circle passing through the feet of the perpendiculars from the foci on the tangent and through the foot of the ordinate will pass through the centre; and the angle subtended at either end of the major axis by the distance between the feet of the perjiendiculars will be equal or supplementary to the angle which either focal distance makes with the corresponding perpendicular. 743. Given a focus and the length and direction of the major axis, })rove that a conic will touch two fixed 2)arabolas whose common focus is the given focus and semi latus rectum along the given line and of the given length. 744. A conic is described having one focus at the focus of a given parabola and its major axis coincident in direction with and equal to half of the latus rectum of the parabola : prove that this conic wiU touch the parabola. 745. A conic touches two adjacent sides of a given parallelogram and its foci lie on the two other sides one on each : prove that each dii-ectrix touches a fixed parabola. If A BCD be the parallelogram, S, S' the foci on BC, CD respectively, and on AS, AS' be taken AL=AB, AL' = AD, the excentricity of the conic will be the ratio LL' : SS'. 746. Three points A, B, C are taken on a conic such that CA, CB are equally inclined to the tangent at C : pi'ove that the normal at G will pass through the pole of A B. CONIC SECTIONS, GEOMETRICAL. 12.> 747. Given one focus, a tangent, and the lengtli of the major axis, pi'ove that the locus of the second focus is a circle : and determine the portions of the locus which correspond to an ellipse, and those which correspond to an hyperbola in which the given focus belongs to the branch which touches the given straight line. 748. Given a focus, the excentricity, and a tangent : prove that the directrix will touch a fixed conic having the same focus and excentricity, and the minor axis of this envelope will lie along the given tangent. 749. A conic described with its foci at the centres of two given intersecting circles and touching a tangent drawn to either circle at a common point will touch the other tangents at the common points ; its auxiliary circle Avill pass through the common points, and any tangent to the conic will be harmonically divided by the circles. 750. Through any poiiit are drawn two tangents to a conic, and on them are taken two points P, Q so that 0, P, Q are equidistant from *S' : prove that S'O is perpendicular to PQ, and, if jS'O, PQ meet in P, that twice the rectangle S'O, S'P, together with the square on SO, is equal to the square on aS^-S", sign being attended to. 751. In any conic \i PO be taken along the normal at P equal to the harmonic mean between PG, Pg, will be the point such that any chord through it subtends a right angle at ; and if from P perpendi- culars be let fall on any two conjugate diameters the straight line join- ing the feet of these perpendiculars will bisect PO. 752. The triangle ABC is isosceles, A being the vertex, and conies are drawn touching the sides AB, AG and the perpendiculars from £, C on the opposite sides : pi-ove that the foci of these conies lie on either a fixed cii-cle or a fixed straight line, and trace the motion of the foci as the centre moves along the straight line which is its locus. 753. Two diameters PP', QQ' of a conic are drawn, and PR, PR' let fall perpendicular on P'Q, P'Q' ; prove that the chord intercepted by the conic on RR' subtends a right angle at P, 754. A triangle ABC circumscribes a conic, and Sa, Sh, Sc are drawn perpendiculars on the side3 : prove that A She : aS'BC= a Sea : aS'CA= a Sab : aS'AB = A abe : a ABC. 755. From a point on an ellipse perpendiculars are let fall on the axes and produced to meet the corresponding auxiliary circles : prove that the straight line joining the two jwints of intersection passes through the centre. 756. Two conies have common foci S, S' and any straight line being taken another straight line is drawn joining the poles of the former with respect to the two conies : prove that the conic whose focus is S and which touches both these straight lines and the minor axis will be a parabola and that its directrix will pass through .S". 12G CONIC SECTIONS, GEOMETRICAL. 7r)7. All ellipse and hyperbola are confocal, a straight line is drawn parallel to one of their coninion diameters and its poles with respect to the two conies joined to the centre : prove that the joining lines are at right angles, and that the polars of any point on a common diameter are also at right angles. 758. Tangents (or normals) are drawn in a given direction to a .series of confocal conies : prove that the points of contact lie on a rect- angular hyperbola having an asymptote in the given direction and passing through the foci. 759. An hyperbola and an ellipse are confocal, and from any point T on an asymptote are drawn TO^ TP, TQ touching the hyperbola in and the ellipse in P, Q I'espectively, prove that OT is a mean proportional between OP and OQ. 760. The tangents drawn to a series of confocal conies at the points where they meet a fixed straight line through S all touch a fixed para- bola whose focus is S' and directrix is the given straight line, and which touches the minor axis. 761. From a point on any ellipse are drawn tangents OP, OQ to any confocal : prove that the chord of cui'V'ature at in direction of either tangent is double the harmonic mean between OP, OQ ; tangents drawn outside the ellipse being considered negative. 762. An ellipse is described touching two given confocal conies and having the same centre : prove that the tangents at the points of con- tact will form a I'ectangle. For real contact one of the given conies must be an ellipse. 763. An ellipse is described having double contact with each of two confocals : prove that the sum of the squares on its axis is constant, and that the locus of its foci is the lemniscate in which SP . IIP = dif- ference of the squares on the given semi-major axes. 764. From a point on a given ellipse are cbawn two tangents OP, OQ to a given confocal ellipse and a diameter parallel to the tangent at meets OP, OQ in the points P', Q' : prove that the har- monic mean between OS, OS' bears to the harmonic mean between OP, . OQ the constant ratio 0S+ OS : OP' + 0Q\ 765. On any tangent to a conic are taken two points equidistant from one focus and subtending a light angle at the other ; prove that their distance from the former focus is constant. 766. The perpendicular CY on a tangent meets an ellipse in P, and Q is another point on the ellipse such that CQ = CY: prove that the i:)erpendicular from C on the tangent at Q is equal to CP. 767. A tangent to a conic at P meets the minor axis in T, and TQ is drawn perpendicular to SP: prove that SQ is of constant length ; and, PM being drawn perj^endicular to the minor axis, that QM will pass through a fixed point. CONIC SECTIONS, GEOMETRICAL. 127 7G8, Three tangents to a conic are svicli that their points of inter- section are at equal distances from a focus : prove that each distance is equal to the major axis ; and that the second focus is the centre of perpendiculars of the ti'iangle formed by the tangents. 769. A conic is inscribed in a circle and is concentric with the nine points' circle of the triangle : prove that it will have double contact with the nine points' circle. 770. An ellipse is inscribed in an acute-angled triangle ABC witli its foci ^S*, *S" at the centre of the circumscribed circle and the centre of perpendiculars respectively; SA,SB, aS'(7 meet the ellipse again in «, h, c : prove that the tangents at a, b, c are parallel to the sides of ABC, and form a triangle in which *S", S are the centre of the cii-cumscribed ciixle and the centre of perpendiculars respectively. 771. With the centre of perpendiculars of a triangle as centre are described two ellipses, one inscribed in the triangle the other cii'cum- scribing it : prove that these ellipses are similar and their major axes at right angles ; and that the diameters of the inscribed conic parallel to the sides are as the cosines of the angles. 772. "With a focus of a given conic as focus and any tangent as directrix is described a conic similar to the given conic : prove that it will touch the minor axis. If with the same focus and dii'ectrix a para- bola be described it will intercept on the minor axis a segment subtending a constant angle at the focus. [In general if the described conic have a given excentricity e not less than that of the given conic it will intercept on the minor axis a segment subtending at the given focus a constant angle = 2 cos"' ( ~/ ) •] 773. "With a focus ^S' of a given conic as focus and any tangent as dii-ectiTLX is described a conic touching a fixed straight line perpendicular to the major axis, another fixed straight line is drawn parallel and conjugate to the former: prove that the segment of this latter sti-aight line intercepted by the variable conic subtends at S the same angle as the segment intercepted by the given conic. 774. With the vertex of a given conic as focus and any tangent as directrix is described a conic passing through one of the foci of the given conic : prove that the major axis is equal to the distance of either focus of the given conic from its directrix. 775. An ellipse is inscribed to a given triangle with its centre at the cii'cumscribed cii'cle of the triangle : prove tliat both auxiliary circles of the ellipse touch the nine points' circle of the triangle ; and that the three pei-pendiculars of the triangle ai'e normals to the ellipse. 776. A conic touches the sides and passes through the centre of the circumscribed circle of a triangle : prove that the dii-ector circle of the ellipse will touch the circumscribed circle of the triangle. 128 CONIC SECTIONS, GEOMETRICAL. 777. A (Hametor PF being fixed, QVQ' is any chord parallel to it bisected in J', and 7^ T intersects CQ or CQ' in R : prove that the locus of A* is rt parabola. 778. A chord is drawn parallel to the major axis and circles drawn through S to touch the conic at tlie ends of the chord : ])rove that the second common point of the circles is the intersection of the chord with the focal radius to its pole ; and that the locus of this point is a parabola with its vertex at S. 779. With any point on a given circle as focus and a given diameter as directrix is described a conic similar to a given conic : prove that it will touch two fixed similar conies to which the given diameter is latus rectum, its points of contact lying on the radius through the focus. 780. Given the side BC of a triangle ABC and that cos A ■— m cos B cos C, prove that the locus of A is an ellipse of which BC is the minor axis when m is positive, an ellipse of which BG is major axis when m is negative but 1 + m positive, and an liypex'bola of which BC is transverse axis when 1 + m is negative. 781. Given a focus S and two tangents to a conic, prove that the envelope of the minor axis is a parabola of which S is focus. 782. A circle is drawn touching the latus rectum of a given ellipse in aS^ the focus on the side towards the centre and also touches the tangents at the ends of the latus rectum : prove that the two other com- mon tangents will touch the ellipse in points lying on a tangent to the circle. 783. From the foci S, S' are let fall perpendiculars *S'F, S'T on any tangent to an ellipse ; prove that the perimeter of the quadrilateral SY Y'S' will be the greatest possible when YY' subtends a right angle at the centre. [This is only possible when SS' is greater than BB' ; when aS'*S" is equal to or less than BB' the perimeter is greatest when the point of contact is the end of the minor axis.] 784. A conic is described having one side of a triangle for directrix, the opposite vei'tex for centre and the centre of perpendiculars for focus : prove that the sides of the ti-iangle which meet in the centre are con- jugate. 785. The angle which a diameter of an ellipse subtends at an extremity of the minor axis is supplementary to that which its conjugate subtends at the ends of the major axis. 786. Two pairs of conjugate diameters of an ellipse are PP\ DD' ; pp, dd' resi)ectively ; prove that Pp, Pp' ai-e respectively parallel to D'd, D'd\ CONIC SECTIONS, GEOMETRICAL. ^~-:, - 129 787. Tangents TP, T(j are diawii to a conic and clioiils Q(j, Pp parallel to TP, TQ respectively : i)rove that pq i.s jiarallel to P(^. Also prove that the diameters parallel to the tangents form a liarmonic pencil with CT and the diameter conjugate to CT. 788. A chord QQ' of an ellipse is parallel to one of the equal conjugate diameters and QX, Q'N' are perpendiculars on an axis : prove that the triangles QCN, Q'CN' are equal and that the normals at Q, Q' intersect on the diameter which is perpendicular to the other equal con- jugate diameter. 789. Any ordinate XP of an ellipse is produced to meet the auxiliary circle in Q and normals to the ellipse and circle at P, Q meet in R ; KK, RL are drawn perpendicular to the axes : prove that K, P, L lie on one straight line aiid tliat KP, PL are equal respectively to the semi-axes. (The point Q may be either point in which XP meets the auxiliary circle.) 790. On the normal to an ellipse at P are taken two points Q, Q' such that QP = PQ' = CD : proAe that the cosine of the angle QCQ' is CP' — C'D° -jT^ B772 > ^^^^ ^ from Q or Q' be di'awn a straight line normal to the ellipse at R, the parts of this straight line intercepted between R and the axes will be equal respectively to BC and AC. 791. Through any point Q of one of the auxiliary circles is drawn QPP' perpendicular to tlie axis of contact meeting the ellipse in P, P' : prove that the normals to the ellipse at P, P' intercept on the nonnal to the circle at ^ a length equal to the diameter of the other auxiliary circle. 792. Tangents to an ellipse at P, D ends of conjugate diametei-s meet in 0, any other tangent meets these in P\ D' respectively : prove that the rectangle under OP', OD' is double that under PP' , DD'. [The ratio of the two rectangles is constant for any two fixed points P, D, having a value depending on the area cut olf by the seg- ment PD.] 793. A chord PQ is drawn through one focus, L is its pole and the centre of the cii-cle LPQ: prove that the circle OPQ will pass tlu-ough the second focus. 794. Through two fixed points A, B of a conic ai-e drawn chords AP, BQ parallel to each other: prove that PQ always touches a concentric similar and similarly placed conic. 795. A parallelogi'am ABCD cii'cumscribes a given conic and a tangent meets AB, AD in P, Q, and CB, CD in /", Q' : prove that the rectangles BP, DQ, and BP', DQ' are equal and constant. 796. An equilateral triangle PQR is inscribed in an auxiliary cii-ck- of an ellipse and P', Q', R' are the corresponding points on the ellipse : prove that the circles of curvature at J^\ Q', R' meet in one point lying on the ellipse and on the circle P'Q'R'. w. r. y 130 CONIC SECTIONS, GEOMETRICAL. 797. A conic, ccutre 0, is iii8cril)ecl in a triangle ABC and through B, C are drawn straiglit linos parallel to the diameter conjugate to OA : prove that these straight lines will be conjugates. 798. A cliord EF of a given circle is divided in a given ratio in aS'; construct a conic of which E is one point, S a focus, and the given cii-cle the cii'cle of curvature at E. 799. A point P Ls taken on an ellipse equidistant from the minor axis and a directrix; prove that the circle of curvature at P will pass through a focus. 800. An ellipse is drawn concentric with a given ellipse, similar to it, and touching it at a point P ; prove that the areas of the two are as CP^ : SP . S'P ; and their cvirvatures at P in the duplicate ratio of SP.S'P : CP\ 801. Any chord PQ of an ellipse meets the cii-cle of curvature at P in Q' : prove that PQ' has to PQ the duplicate ratio of the diameters of the ellipse which are I'espectively parallel to the tangent at P and to the chord PQ. 802. Two circles are described with S, S' as centres and intersecting in P, P' ; i^rove that with any point on the conic, whose foci are ^S*, S' and which passes throvigh P, as centre, can be described a circle touching both the former, and that all these tangent circles cut at right angles a fixed cu'cle touching the conic in P, P'. 803. Given a focus, a point, and the length of the major axis ; prove that the envelope of either dii-ectrix is a conic having its focus at the common point and excentricity equal to the ratio of the focal distances of the common point. 804. Given a point and the dii'ectrices ; prove that the locus of each focus is a circle, and the envelope of the conic is a conic having the given point for focus and the distances between the directrices for major axis. 805. A circle is described having internal contact witli each of two given circles one of which lies within the other, and the centre P of the moving circle describes an ellipse of which -4 J.' is the major axis; through A is drawn a diameter of the moving cii-cle ; prove that the ends of this diameter will lie on an ellipse similar to the locus of P, and having a focus at A and centre at A'. 806. A conic has one focus in common with a given conic, touches the given conic and passes through its second focus : prove that the major axis is constant. 807. Two similar conies U, V are placed with their major axes in the same straight line, and the focus of U is the centre of V : prove that the focal distance of the point of contact with Z7 of a common tangent is equal to the semi-major axis of V. CONIC SECTIONS, GEOMETRICAL. 131 808. In a conic one focus, the excentricity, and the dii'ection of the major axis are given, and tangents are drawn to it at })oints wliere it meets a given circle having its centre at the given focus : prove that these tangents all touch a fixed conic having the given excentricity and whose auxiliary circle is the given circle. 809. The tangent to a conic at P meets the axes in T, t and the central radius at right angles to CF in Q : prove that the ratio of QT to Qt is constant. 810. Through a given point on a given conic are drawn chords OP, OQ equally inclined to a given direction : prove that PQ passes through a fixed point. 811. A chord PQ is normal at P to a given conic and a diameter LL' is drawn bisecting PQ ; pi'ove that PQ makes equal angles with LP, L'P and that LP + LP is constant. 812. A conic is described through the foci of a given conic and touching it at the ends of a diameter : prove that the rectangle under the distances of a focus of this conic from the foci of the given conic is equal to the square on the semiminor axis of the given conic ; and that the diameter of this conic which is conjugate to the major axis of the given conic is equal to the minor axis of that conic. 813. A conic is inscribed in a triangle ABC and has its focus at ; the angles BOC, COA, AOB are denoted by A', B', C; prove that O^sin^ OB sin B OC sin C ~ — 7~irf 7\ = ~ — TT? — D\ - "■ — TTTf — 7T^ - major axis. sm{A -A) siii{B-B) sm(6 -6) *" Under what convention is this true if be a point without the triangle? 814. Two conies are described having a common minor axis and such that the outer touches the directrices of the inner; MPI^ is a common ordinate ; prove that MP" is equal to the normal at P. 815. Two tangents OA, OB are drawn to a conic and a straight line meets the tangents in Q, Q', the chord AB in R, and the conic in P, P'; prove that QP . PQ' : RP' = QP' . P'Q' : RP'\ and that for a given direction of the straight line each of these ratios is constant. 816. With B the extremity of the minor axis of an ellipse as centre is described a circle whose diameter is equal to the major axis, and the tangents at the end of the major axis meet the other common tangents to the ellipse and circle in P, F, Q, Q' \ prove that B, F, F\ Q, Q' lie on a circle whose diameter is equal to the radius of curvature of the ellipse at B. 817. A chord of an ellipse subtends at «S' an angle equal to the angle between the equal conjugate diameters : prove that the foot of the perpendicular from S", and that the tangent at the vertex of the parabola touches the auxiliary circle. 819. A parabola is drawn through the foci of a given ellipse with its own focus J* on the ellii)se ; prove that the ])arts of the axis of the parabola intercejtted between 7^ and the axes of the eliii)se are of con- stant length, and if through the points where the axis of the parabola meets the axes of the ellipse straight Hues be drawn at right angles to the axes of the ellipse theii* point of intersection will lie upon the normal to the ellipse at /*. 820. A given finite straight line is one of the equal conjugate diameters of an ellipse; prove that the locus of the foci is a lemniscate of Bernoulli. 821. A parallelogram is inscribed in a conic and from any point on the conic are drawn two straight lines each parallel to two sides : prove that the rectangles under the segments of these lines cut off by the sides of the parallelogram are in a constant ratio. 822. Two central conies in the same plane have two conjugate diameters of the one parallel respectively to two conjugate diameters of the other ; and in general no more. 823. In two similar and similarly placed ellipses are drawn two parallel chords FF', QQ' ; PQ, P'Q' meet the two conies in R, S, R', /S" respectiA^ely : prove that RR\ SS' are parallels : also that QQ', RR' and PF', SS' intersect in points lying on a fixed straight Hue. 824. A circle described on the intercept of the tangent at F made by the tangents at A, A' meets the conic again in Q ; prove that the ordinate of Q is to the oixlinate of F as the minor axis is to the sum of the minor axis and the diameter conjugate to P. (As BC : BC + CD.) 825. A point P is taken on a conic and is the centre of the circle SFS', FO is divided in 0' so that FO' : PO = BC' : AC: prove that the circle with 0' as centre and O'P as radius will touch the major axis at the foot of the normal at P. 826. With a fixed point P on a given conic as focus is described a parabola touching a pair of conjugate cHameters ; prove that this parabola will have a fixed tangent parallel to the tangent at F and that this tangent divides CF in the ratio CF^ : (7Z)^ 827. Through a point are drawn two straight lines conjugates with respect to a given conic ; any tangent meets them in P, ^ : prove that the other tangents drawn from P, Q intersect on the polar of 0. 828. A parabola is described having S for its focus and touching the minor axis ; prove that a common tangent will subtend a right angle at *S' and that its point of contact with either conic lies on the dii'Pctrix of the othci-. CONIC SECTIONS, GEOMETRICAL. 133 820. Prove that the two points common to the (.liroctor circles of all conies inscribed in a given quadrilateral may be constructed as follows : take aa, bb', cc the three diagonals of the quadrilateral forming a triangle ABC and let be the centre of the circle AUC, then if F, Q be the required points, 0, P, Q lie in one straight line peri^endicular to the bisector of the diagonals, PQ is bisected by this bisector and the rectangle OF, OQ is equal to the square of the radius of the circle ABC. 830. At each point P of an ellipse is drawn QPQ' parallel to the major axis so that QP = PQ' = SP: prove that Q, (/ will trace out ellipses whose centres are A, A' and -whose ai-eas are together double the area of the given ellipse. If QPQ' be drawn parallel to the minor axis instead of the major, the loci are ellipses whose major axes are at right angles to each other and they touch each other in S and touch the tangents at A, A'. 831. A given ellipse has its minor axis increased and major axis diminished in the ratio ^1 -e : 1, its centre then displaced along the minor axis through a length equal to a and the ellipse then turned about its centre thi-oiigh half a right angle : prove that the whole effect is equivalent to a simple shear parallel to the minor axis by which the major axis is transferred into the position of a tangent at one end of the latus rectum. 832. A point P is taken on- an hyperbola such that CP = CS: prove that the circle PTO will touch CF at P, and, if Q, Q' be two other points such that the ordinate of P is a mean proportional between those of Q, Q', that the tangents at Q, Q' will inteisect on the circle whose radius is CS. 833. An hyperbola is described through the focus of a parabola with its own foci on the parabola ; prove that one of its asymptotes is pai'allel to the axis of the parabola. 834. A parabola passes through two given points and its axis is in a given direction : prove that its focus lies on a fixed hyperbola. 835. Two tangents of an hyperbola U ai-e asymptotes of another V ; prove that if V touch one of the asjTiiptotes of U it will touch both. 836. In an hyperbola whose excentricity is 2, the circle on a focal chord as diameter passes through the farther vertex. Any chord of a single branch subtends at the focus S interior to that branch an angle double that which it subtends at the farther vertex A'. If PSJi' be a choi'd, SPp, SQq chords inclined at GO" to the former, the circles qPPi.', pQli will intei'sect in *S' and A', and if the former intersect the circle on the latus rectum in U, V, the angle A'SU is three times A'SP, and UV is a diameter of the last-mentioned circle. 837. The straight line joining two points which are conjugates with respect to a conic is bisected by the conic : i)rove that the line is parallel to an asymptote. 838. A conic is drawn through two given points with asymptotes in given directions : prove that the locus of its foci is an hyperbola. 134 CONIC SECTIONS, GEOMETRICAL. 839. A straight line is draAvu equidistant from focns and directrix of an hyperbola, and througli any jwint of it is drawn a straight line at right angles to the focal distance of the {)oint : prove that the intercept made by the conic will subtend at the focus an angle equal to the angle between the asymptotes. 840. Two hyi^erbolas U, V are similar and have a common focus, and the directrix of V is an asymptote of U ; prove that the conjugate axis of U is an asymptote of V. 841. In an hyperbola LL' is the intercept of a tangent by the asymptotes : prove that SL . S'L=CL . LL', and SL' . S'L' = CL' . LL'. 842. To an hyj^erbola the concentric circle through the foci is draAVTi : prove that tangents drawn from any point on this circle to the hyperbola divide harmonically the diameter of the circle which lies on the conjugate axis ; and if OP, OP' the tangents meet the conjugate axis in U, U' and P2f, P'M' be perpendiculars on the conjugate axis, TIM', U'M will be divided in a constant ratio by C. 843. A circle is di'awn touching both branches, prove that it inter- cepts on either asymptote a length equal to the major axis ; the tan- gents to it where it meets the asymptotes pass through one or other of the foci, and those meeting in a fociis are inclined at a constant angle equal to that between the asymjitotes ; and the straight lines joining the points -where it meets the asymptotes (not being parallel to the trans- verse axis) will touch two fixed parabolas whose foci are the foci of the hyi^erbola. III. Rectangular Hyperbola. [In the questions under this head, r. h. is an abbreviation for rectangular hyjierbola.] 844. Four points A, B, C, D are taken on a R. h. such that BG is perpendicular to AD : prove that CA is perpendicular to BB and AB to CD. 845. The angle between two diameters of a R. H. is equal to the angle between the conjugate diameters, ■ 846. A point P on a r. h. is taken, and PK, PK' drawn at right angles to PA, PA' to meet the transverse axis; prove that PK=PA'j and PK' ^ PA, and that the normal at P bisects KK'. 847. The foci of an ellipse are ends of a diameter of a R. h,; prove that the tangent and normal to the ellipse at any one of the common points are parallel to the asymptotes of the hyperbola : and that tan- gents drawn from any point of the hyperbola to the ellipse are parallel to a pair of conjugate diameters of the hyperbola. CONIC SECTIONS, GEOMETRICAL. 135 848. Prove that any chord of a r. h. subtends at the en, or if the straight line fx-vqy- 1 pass through the point ( — -, , j , a somewhat different mode of proving the theorem already dealt with. In general the equa- tion of the sti-aight lines joining the origin to the two points determined by the equations ax^ + bi/^ + C + 2/i/ + 2ffx + 2hxi/ = 0, px -i-qy + r = 0, 2 c is ax' + 2hxi/ + hif — -^ {px + qy) {gx +/y) + -r, (jyx + qyY = 0. The results of linear ti'ansformation may generally be obtained from the consideration that, if the origin be unaltered, the expression x' + '2xy cos (Ji + y^ must be transformed into .r^+2xrcoso + r^ if (x, y), (X, Y) I'epresent the same point and w, Q be the angles between the co-ordinate axes in the two systems respectively. Thus if tc= aaf + by^ + C + 2fy + 2gx + 2hxy be transformed into U=AX' + BY' + c + 2FY+2GX + 2HXY, then X (x" + y" + 2xy cos w) + u must be transfonned into X {X' +Y' + 2XY cos a) + U, and if X have such a value that the former be the product of two linear factors, so also must the latter ; hence the two quadratic equations in X c (X + «) (X + 6) + 2fg (X cos w + h) = (X + a)f^ + (X + b) g'' + c{X cos w + hy, and c{\ + A){\ + B) + 2FG{\cosn + II) = {\ + A) F"- + {X + B) G' + c (Xcosil + H)' must coincide ; and thus the invariants may be deduced. Also, by the same transformation, \(x'+ y' + 2xy cos co) + a:f? + by"^ + 21ixy must be transformed into X {X' + r^ + 2X7 cos n) + .Lr^ + BY^ ^ 2IIXY, and if X have such a value that the former is a square, so must the latter; hence the equations (X -f- a) (X 4- 6) = (X cos 0) -f- hy, (X + y() (X + 7?) - (X cos n + //)', 140 CONIC SECTIONS, ANALYTICAL. must coincide, whence a + J - 27i cos o) _ A + B - 211 cos n bin* is obtained by adding the terms {(x - x;} {y - 2/J + {x- X,) {y - y,)} cos w ; each equation being found at once from the property that the angle in a semicircle is a right angle. In questions relating to two circles, it is generally best to take their equations as X' -^ y^ - lax + ^- = 0, x^ + y^-2hx-¥k = 0, the axis of x being the radical axis, and h negative when the circles intersect in real points.] 883. The equation of the straight lines which pass throiigh the origin and make an angle a with the straight line x + y = is x' + 2xy sec 2a + v/* = 0. 884. The equation hx' — 2Iixy + ay^ = represents two straight Hnes at right angles I'espectively to the two whose equation is ax^ + 2hxy + by' = 0. If the axes of co-ordinates be inclined at an angle w, the equation will be (a + b — 2h C0& w) {x' + y^ + 2xy cos w) = (ax- + 2Jhxy -i- b}f) %\vl w. 885. The two straight lines ^ (tan* 6 + cos' Q) - 2xy is^^ Q ^- y' sin* 5 = make with the axis of x angles a, /? such that tan a — tan /3 - 2. .886. The two straight lines (as* + if) (cos' sin" a -t- sin* 5) = {x tan a. — y sin B)^ include an angle a. 887. The two straight lines 03* sin* a cos'' + ixy sin a sin -f- y" {4 cos a - (1 + cos of cos' 5} = include an angle a. CONIC SECTIONS, ANALYTICAL. 141 888. Form the equation of the straight Hues joining the origin to the points given by the equations (x - hy + {i/- kf = c\ kx + hy = 2hk, and prove tliat they will be at right angles if h" + k' = c*. Interpret geometrically. 889. The straight lines joining the points given by the equations ax^ + hi/ + C+ Ify + Igx + 21ixy - 0, fx + qi/=l, to the origin will be at right angles if a + b + 2(/q + g2)) + c (jr + q') = 0; and the locus of the foot of the perpendicular from the origin on tlie line px + qy =\ is (a + b) (x" + y^) + '2fy + 2gx + c = : also the same is the locus of the foot of the perpendicular from the point / 2y _J/\ \ a + 6 ' a + hj' 890. The locus of the eqiiation ^ X- -\ x^ — \ y 2 + 2 + to CO is the parts of two straight lines at right angles to each other which include one quadrant. [The equation gives y = 1 +x when x is positive and y = 1 — x when x is negative.] 891. The formulse for effecting a transformation of co-ordinates, not necessarily rectangular, are x = pX + qY + r, y = p'X + q'Y+r'; prove that (pq - p'q') {pq - p'q) = qq' - pp. 892. The expression ax* + by' + c + 2fy + 2gx + 21ixy is transformed into Ax^ + By'-¥c^ 2Fy + 2Gx + 2IIxij, the origin being unchanged : prove that f + g'-2/gcos h ) ^^^ o^^ the side BC o a segment containing an angle -^ — 6 : prove that the centi-e of the last circle lies on the radical axis of the other two ; each segment being towards the same parts as the opposite angle. 905. There are two systems of circles such that any cii-cle of one system cuts any circle of the other system at right angles ; i)rove that the circles of either system have a common radical axis which is the line of centres of the other system. 906. On a fixed chord AB of a given circle is taken a point such that, P being any point on the cu-cle, OA . OB ~ ± PA . PB : prove that the straiglit line which bisects PO at right angles will pass through one end of the diameter conjugate to AB ; and, if Q be the other point in which the straight line meets the cii'cle, that QO^ = QA . QB. 907. A cuxle U lies altogether within another circle V ; prove that the ratio of the segments intercepted by U^, V on any straight line can- not be greater than Ja'-{b-cy--Ja'-{b+cy : J(a + cY-¥-J{a-cf-h\ where a, b are the radii and c the distance between the centx'es. 90S. An equilateral triangle is drawn with its sides passing through three given points A, B, C : prove that the locus of its centre is a circle having its centre at the centroid of ABC, and that the centres of two equilateral triangles whose sides are at right angles will be at the ends of a diameter of the locus. [The radius of the locus is the difference of the axes of the minimuni ellipse about ABC, the altitude of the maximum equilatei-al triangle is equal to three-fourths the sum of the axes of the minimiini ellipse, and is also equal to the minimum sum of the disttuices of any point from A, B, C] 909. Prove that the equation {x cos {a + ft) + y sin (a -f- /3) — a cos (a — /?)} {x cos (y + 8) -f- y sin (y -f- S) - a cos (y — 8)} = {x cos (a 4- y) -f- 2/ sin (a -I- y) — a cos (a — y)} {x cos {(S + S) + y sin (ft + S)- a cos (/? - S)} 14-4- CONIC SECTIONS, ANALYTICAL. is eciuivalent to the equation a;' + ?/* - a* : and state the property of the circle expressed by the equation in this form. 910. Four fixed tangents to a circle form a quadrilateral whose diagonals are aa, hh', cc, and perpendiculars 7;,;/ ; q,q' ; r,r' are let fall from these points on any other tangent : prove that j8-v a-8 , 7-a fl — S , a—B y-8 pp cos - ' COS — ;^ — = qq cos i-^^ — cos -- n — = TV COS —x COS ^ , , . a-6 . 13-e . y-e . B-e = 4« sin -^y- sm ,^— sm — — sm -- ; the co-ordinates of the points of contact being (a cos a, a sin a), and the like in /3, y, S, 0. 911. The radii of two cii'cles are A', p, the distance between their centres is Jji^ + 2p^ and p < 2E : prove that an infinite number of tri- angles can be inscribed in the first which are self-conjugate with respect to the second ; and that an infinite number can be circumscribed to the second which are self-conjugate to the first. [In general, if S denote the distance between the centres, and the polar of a point A on the first circle with respect to the second meet the first in B, C, the chords AB, AC wUl touch the conic f {2R' + 2p' -8') + {R' + p'- h') {2x' - 2Bx + h'-Ii'- p') = 0, and BC will touch the conic {{x-hy + r\R'^{hx+p'-hy; and these two "will coincide if 8^ = A' 4- 2p".] 912. A triangle is inscribed in the circle x^ + if = R-, and two of its sides touch the circle {x — ^Y + y' = pi'\ prove that the third side will touch the circle .W8 y ^.f 2r-(^-^8-) ,V AR' (R' which coincides with the second circle if 8^= A^± 2i??-. Also prove tliat the three circles have always a conmiou radical axis. 913. Two given polygons of n sides are similar and similarly situated : prove that in general only two polygons can be drawn of th(^ same number of sides circumscribing one of the two given polygons antl inscribed in the other ; but that if the ratio of homologous sides in the two be cos* ^r— : cos'' sin* - , where r is anv whole number less than 2ti 2ii It tt - , there will be an infinite numbei". CONIC SECTIONS, ANALYTICAL. 145 II. Parabola referred to its axis. [The equation of the parabola being taken y' = 4:OX, the co-ordinates of any point on it may be represented by ( — 2 , — ) , and with this notation the equation of the tangent is « = mx H — ; of the normal m mi/ + x = 2a-^ — 5; and of the chord through two points (??i,, m„), 2m^mjx-^(m^+m^) + 2a=0. The equation of the polar of a point (AT) is i/Y=2a{x^X), and that of the two tangents drawn from {X, Y) is {¥' - iaX) (1/ - iax) = [jjY - 2a {x + X)}^ As an example, we may take the following, " To find the locus of the point of intersection of normals to a parabola at right angles to each other." If (X, 3'') be a point on the locus, the points on the parabola to which normals can be drawn from (X", Y) are given by the equation m'Y + m' {X -2a)- a = 0; so that, if m,, m^, m^ be the three roots of the equation 2a -X ^ a m^ + m^ + m^ = — j^ , mjn^ + m,^m^ + m^m^ = 0, m^m^m^ = j. ; and since two normals meet at right angles in (X', Y) the product of two of the roots is — 1 ; let then ')n,^m^ = — 1. Then a Za-X Y or the locus is the parabola y" = a(x — 3a). Again, " Tlie sides of a triangle touch a parabola and two of its angular points lie on another parabola with its axis in the same direc- tion, to find the locus of the third angular point." Let the equations of the parabolas be y' = iax, (y — ky = 46 (x — h), and let the three tangents to the former be at the points m,, m^, m^. The point of intersection of (1), (2) is , a f — H ), and this will lie on the second parabola if and similarly for m^, w^. Hence 77?^^, 7n^ are the roots of the quadratic in^, Hi,-^-}'=K^.-"). hence, if (X, I') be the point of intersection of the tangents at m^, i»^, 46 Y = a(l^l-).2(k-^) W. P. 10 + m. 14(; CONIC SECTIONS, ANALYTICAL. SO that {a Y - UIc)' = 4 (2i - af (aX - ibh), or the third point lies on another parabola with its axis in the same direction as the two given parabolas, and Avhich coincides with the second if a= 46.] 914. Two parabolas have a common vertex A and a common axis, an ordinate XFQ meets them, the tangent at F meets the outer parabola in B, E' and AE, AE' meet the ordinate in L, M ; prove that JVP, i\'^ are respectively harmonic and geometric means between JVL, NM. 915. A triangle is inscribed in a parabola and a similar and similarly placed triangle circumscribes it : prove that the sides of the latter triangle are respectively four times the corresponding sides of the latter. 916. Two tangents p, q being drawn to a given parabola U, through their point of intersection are drawn the two parabolas confocal with U, and A', A" are their vertices : prove that \p+qj 'S^i . A A ^ ^ A, A' being taken on opposite sides of S. 917. An equilateral triangle is insci'ibed in a parabola : prove that the ordinates y,, y^, y^ of the angular points satisfy the equations 3 {y, + 2/3) (2/3 + 2/1) (y, + 2/J ^ 32«^ {y^ + y^ + y^) = 0, (2/, + 2/0 + 2/3)' + 2/^2/3 + 2/32/1 + 2/,2/3 + 48a' = 0; and that its centre lies on the parabola 9?/" — 4a (a? — 8a). 918. An equilateral triangle circumscribes a parabola : prove that the ordinates y^, y^, y^ of its angular points satisfy the equations (2/. + 2/2 + yy = 4 (2/22/3 + 2/32/, + 2/,2/2 + 3«'), 4a' {2/1 + 2/2 + 2/3) + 3 (2/^ + 2/3 - 2/1) {2/3 + 2/1 - 2/2) (2/1 + 2/2 - 2/3) = 0. [The simplest way of expressing the conditions for an equilateral triangle is to equate the co-ordinates of the centroid and of the centre of perpendiculars.] 919. The pole is taken of a chord PQ of a parabola : prove that the perpendiculars from 0, P, Q on any tangent to the parabola are in geometric progression. 920. Four fixed tangents are drawn to a parabola, and from the angular points taken in order of a quadrangle formed by them are let fall perpendiculars p^, p^, p^, p^ on any other tangent : prove that CONIC SECTIONS, ANALYTICAL. 147 921. The pei-pendiculars from the angular points of a tiianglo ABC, whose sides touch a parabola, on the directrix are />», q, r, and on any other tangent are x, y, z : prove that p tan A q tan Br tan x{y-z) y{z- x) z[x-y)' [Of course the algebraical sign must be regarded.] 922. The distance of the middle point of any one of the three diagonals of a quadrilateral from the axis of the inscribed parabola is one-fourth of the sum of the distances of the four jiouits of contact from the axis. 923. Through the point where the tangent to a given parabola at P meets the axis is drawn a straight line meeting the parabola in Q, Q' which tlivides the ordinate at P in a given ratio : px-ove that PQ, PQ' will both touch a fixed parabola having the same vertex and axis as the given one. [If the ratio of the part cut off to the whole ordinate be ^ : 1, the ratio of the latus rectum of the envelope to that of the given parabola will be ^k-.l+h] 924. Two equal parabolas have axes in one straight line, and from any point on the outer tangents are dra^\^l to the inner: prove that they will intercept a constant length on any fixed tangent to the inner equal to half the chox'd of the outer intercepted on the fixed tangent. 925. A tangent is drawn to the circle of curvature at the vertex and the ordinates of the points where it meets the parabola are y^, y^ : prove that i ^1 -1 926. On the diameter thi'ough a point of a parabola are taken points P, P' so that the rectangle OP, OP' is constant : prove that the four points of intersection of the tangents drawn from P, P lie on two fixed straight lines parallel to the tangent at and equidistant from it. 927. The points P, P' are taken on the diameter through a fixed point of a parabola so that the mid-point of PP' is fixed : prove that the tangents drawn from P, P' to the parabola will intersect on another parabola of half the linear dimensions. [In general if tangents to the parabola y* = 4«a; divide a given segment LL' on the axis of x harmonically, their point of ijitersection lies on the conic a where OL + OL' = 2c and LL' = 2m.] 928. A chord of a parabola passes through a point on the axis (outside the parabola) at a distance from the vertex equal to half the latus rectum : prove that the normals at its extremities intersect on the parabola. 10—2 14S CONIC SECTIONS, ANALYTICAL. 929. Tlio sum of the angles which three normals drawn from one point make with the axis exceeds the angle which the focal distance of the point makes with the axis by a multiple of tt. 930. Normals are drawTi at the extremities of any chord passing through a fixed ])oint on the axis of a parabola : prove that their point of intersection lies on a fixed })arabola. [More generally, if a chord pass through (A", Y), the locus of the point of intersection of the normals at its ends is the parabola 2 {2ay +Y{x-X- 2a)Y + {Y' - iaX) { Yy + 2X {x - X- 2rt)} = 0.] 931. Two normals to a parabola meet at right angles, and from the foot of the perpendicular let fall from their point of intersection on the axis is measured towards the vertex a distance equal to one-fourth of the latus rectum : prove that the straight line joining the end of this distance with the point of intersection of the normals is also a noniial. 932. Two equal parabolas have their axes coincident but their vertices separated by a distance equal to the latus rectum ; through the centres of curvatvire at the vertices are drawn chords PQ, P'Q' equally inclined in opposite senses to the axis, P, P' being on the same side of the axis: prove that (1) PQ' , P'Q are normals to the outer parabola; (2) their common point R' lies on the inner; (3) the normals at P\ (?', B! meet in a point which lies on a thuxl equal parabola. 933. From a point are drawn three normals OP, OQ, OR and two tangents OL, OAt to a parabola : px'ove that the latus rectum OP.OQ. OR = 4 OL.OM 934. The normals to the parabola y^ = iax at jioints P, Q, R meet in the point (A'", Y) : prove that the co-ordinates of the centre of perpen- diculars of the triangle PQR are X - 6«, - | Y. 935. Three tangents are drawn to a parabola so that the snm of the angles which they make with the axis is tt : prove that the circle round the triangle formed by the tangents touches the axis (in the focus of course). 936. The locns of a point from which two normals can be drawn making complementary angles with the axis is the parabola y' = a {x — a). 937. Two (equal) parabolas have the same latus rectum and from any point of either two tangents are drawn to the other : prove that the centres of two of the four circles which touch the sides of the triangle formed by the tangents and their chord of contact lie on the parabola to which the tangents are drawn. Also, if two points be taken conjugate to each other with respect to one of the parabolas and from them tangents drawn to the other at points Z, M ; N, 0, respectively, the rectangle under the perpendiculars from any point of the second jiarabola on the chords LN, MO will be equal to that under the perpendiculars from the same point on MN, LO. CONIC SECTIONS, ANALYTICAL, 140 938. Prove that the two parabohis y* = ««, y'=\a{x + a) are so related that if a normal to the latter meet the former iii P, Q and A be the vei-tex of the fomier, either AP or AQ is perpendicular to the uormal. 939. The noi-mals at three points of the parabola y^ = iax meet in the point (A", Y) : prove that the equation of the circle through the three points is 2 (/ + /) - 2x (X + 2a) - yY= ; and that of the cii-cle round the triangle formed by the three tangents is {x-a){x-2a + X ) + y (// + 7) = 0. [Hence if be the point from which the normals are drawn and 00' be bi.sected by S, SO' is a diameter of the circle round the triangle formed V)y the tangents.] 940. In the two parabolas y^=2c(a;±c) a tangent di-awn to one meets the other in two points and on the chord intercepted as diameter is described a circle: prove that this circle will touch the second parabola. 941. On a focal chord as diameter is described a cii-cle cutting the parabola again 'in.P,Q: prove that the cii-cle PSQ will touch the parabola. 942. On a chord of a given parabola as diameter a circle is described and the other common chord of the circle and parabola is conjugate to the former with respect to the parabola : prove that each chord touches a fixed pai-abola. 943. Two tangents OL, OM to a pai-abola meet the tangent at the vertex in. P, Q : prove that ^PQ^OL cos QPL = OM cos PQM. 944. Two parabolas have a common focus and direction of axis, a chord Q VQ' of the outer is bisected by the inner in V, VP parallel to the axis meets the outer in P : prove that QV i^ a. mean proportional between the tangents drawn from P to the inner. 945. Prove that the parabolas y' = 4rtA", y^ + icy + iax — 8ft* cut each other at right angles in two points and that each passes through the centre of curvature at the vertex of the other. If the origin be taken at the mid-point of their common chord their equations will be 7/^ - c- - 4a* = ± {2cy + Uix). [The general orthogonal trajectory of the system of parabolas y- + 2Ay + 4ax = 8a" for different values of X Is ff — iax= Ce^.] 946. On a focal chord PSQ of a parabola are taken points p, q on opposite sides of S so that qS .Sp= QS . SP, and another parabola is drawn with parallel axis and passing through q, p : prove that the common chord of the two pai'abolas will pass through S. 150 CONIC SECTIONS, ANALYTICAL. 947. A clioicl PQ of a parabola meets tlie axis in T, U is the mid- point and the pole of the chord, a normal to PQ tlirough U meets the axis in G and OK is perpendicular from on the directrix : prove that SO is parallel to TK and ^7t to GU. 948. Through eacli point of the straight line x = my + h is drawn a chord of the jiarabola if = 4rt.r, which is bisected in the point : prove that this chord touches the parabola {y-2amY=^a{x-h). 949. Prove that the triangle formed by three nonnals to a parabola is to the triangle fox'med by the three corresponding tangents in the ratio {t, + t^ + t.y : 1, where t^, t^, t^ are the tangents of the angles which the normals make with the axis. 950. Three tangents to the parabola y'' = ia (x + a) make angles a, P, y Math the axis : prove that the co-ordinates of the centre of the circle cii'cumscribing the triangle formed by them are , 1 „ sin (g + /3 + y) cos(a + ^ + y) "T" ■«■ Cv ~~. : .T . . "^ T> C* sin a sin (3 sin y ' ^ siii a sin ft sin y * 951. Three confocal parabolas have tlieii* axes in a. P., a normal is drawn to the outer and a tangent perpendicular to this normal to the inner : prove that the chord which the middle parabola intercepts on this tangent is bisected in the point where it meets the normal. 952. Two normals OP, OQ are drawn to a parabola, and a, ft are the angles which the tangents at /*, Q make with the axis : prove that qP_ OQ a sin a 4- sin ft cos (a — ft) sin ft + sin a cos [ft — a) sin^ a sin^ ft ' 953. From any point on the outer of two equal parabolas with a common axis tangents are drawn to the inner : prove that the part of the axis intercepted bears to the ordinate of the point from which the tangents are drawn a constant ratio equal to that which the chord intercepted on tlie tangent at the vertex of the inner parabola bears to the semilatus rectum. 954. Prove that the common tangent to the two parabolas X' COS" a — ia [x cos a + y sin a), y^ sin' a = — 4a (x cos a + y sin a), subtends a right angle at the origin. 955. Two parabolas have a common focus S and axes in the same straight line, and fi-om a point P on the outer are di'awn two tangents PQ, PQ' to the inner : prove that the ratio cos I QPQ' : cos | ASP is constant, A being the vertex of either parabola. CONIC SECTIONS, ANALYTICAL. 151 956. A pai-abola circumscribes a triangle ABC and its axis makes with CB an angle 6 (measured from CB towards CA): prove that its latus rectum is 2Esm6sm{C-6)dn{B + 6); and that for an inscribed parabola the latus rectum is four times as large. 957. A triangle ABC is inscribed in a given parabola and the focus is the centre of perpendiculars of the triangle : prove that (1 - cos A) (1 - cos B) (1 - cos C) = 2 cos A cos BcohC; and that each side of the triangle touches a fixed circle which passes through the focus and whose diameter is equal to the latus rectum. 958. A parabola is drawn touching the sides AB, AC of a triangle ABC at B, C and passing through the centre of perpendicidars : prove that the centre of perpendiculars is the vertex of the parabola and that the centre of curvature at the vertex is a point on BC. 959. The latus rectum of a parabola which touches the sides of a triangle ABC and whose focus is aS' is equal to >SA . SB . SC-^ li^. 960. A chord LL' of a given circle has its mid-point at and its pole at P ; a parabola is drawn with its focus at and its directrix passing through F : prove that the tangent to this parabola at any point where it meets the circle passes throiigh either L or L'. 961. A triangle, self- conjugate to a given parabola, has one angular point given : prove that the circle circumscribing the triangle passes through another fixed point Q such that OQ iii parallel to the axis and bisected by the directrix. 962. A triangle is inscribed in a parabola, its sides are at distances X, y, z from the focus and subtend at the focus angles 6, , \j/ (always measiu-ed in the same sense so that the sum is 27r) : prove that . -, . , . , sin 6 + sin d> + sin i/^ + 2 tan ^ tan ^ tan ^ sm 6 sm dy sui ip -r r 2 2 J , X 'if Z' I where 21 is the latus rectum. 963. Two points L, L' are taken on the directrix of a parabola conjugate to each other with respect to the parabola : prove that any other conic through LSL' having its focus on LL' will have for the cor- responding directrix a tangent to the pai-abola. 964. An ellipse of given excentricity ^-^, is described passing through the focus of a given parabola y* = 4a.r and with its o\vn foci on 152 CONIC SECTIONS, ANALYTICAL. the panibola : prove that its major axis touches one of the parabolas, foufocal with the given parabohi, and that its minor axis is normal to one of the two i/=ia{l+\'){.c + aX'). 9G5. An ellipse is described with its focus at the vertex of a given parabola ; its minor axis and the distance between its foci are each double of the latus rectum of the parabola : prove that the pole with respect to the ellipse of that ordinate of the parabola with which the minor axis in one position coincides always lies on the parabola and also on an equal parabola whose axis coincides with that of the ellipse. 966. A parabola touches the sides of a triangle ABC in the points A', B', C and is the point of concourse of AA', BB', CC : prove that, under a certain convention as to sign, OA cosec BOC + OB cosec CO A + OC cosec AOB=^0: also, if P be a point such that PA' bisects the angle BPC and PB', PC I'espectively bisect the external angles between PC, PA, and PA, PB, PA=PB + PC. 967. A triangle cii'cumscribes the cii'cle x' + t/^ = a", and two angular points lie on the circle (x — laf + if = 2a^ : prove that the thii-d angular point lies on the parabola y^ = a{x — fa). Prove also that the thi'ee curves have two real and two impossible common tangents. 968. Two parabolas have a common focus, axes inclined at an angle a, and are such that triangles can be insciibed in one whose sides touch the other : prove that ^^ = 2^^ (1 + cos a), l^, l^ being their latera recta. 969. A circle is described with its centre at a point P of a parabola and its radius equal to twice the normal at P\ prove that triangles can be inscribed in the parabola whose sides touch the circle. 970. Two parabolas A, B liave their axes parallel and the latus rectum of A is four times that of B : pi-ove that triangles can be inscribed in B whose sides touch A. If the axes be in the same straight line the normals to B at the angular points of such a triangle will all meet in one point, as will the normals to A at the points of contact, and the loci of these points of concourse are straight lines perpendicular to the axis. [Taking the equations of the parabolas to be if - \%ax, if = ia(x + h), the straight lines will be x=1a, x = ^a-\- /i.] 971. The circle of curvature of a parabola at P meets the parabola again in Q and QL, QM are drawn tangents to the cuxle and parabola at Q, each terminated by the other cui-ve : prove that when L3I subtends a right angle at P, PL is parallel to the axis, and that this is the case when the focal distance of 1 is one-third of the latus rectum. CONIC SECTIONS, ANALYTICAL. 153 972. If the tangent at P make an angle d with the axis, the tangent to the cii-cle at Q will make an angle tt -'iO with the axis ; also the angle between the tangent at F and the other common tangfuit to the parabola and circle will be 2 tan"' (^ tan d), and if ^ be the angle which this common taiigent makes with the axis tan -: tan — = 1. 973. From a point on the normal at P are drawn two tangents to a parabola making angles a, /3 with OP: prove that the radius of curvature at P is 20 P tan a tan /?. 974. The normal at a point of a parabola makes an angle 6 with the axis : prove that the length of the choi'd intercepted on the normal bears to the latus rectum the ratio 1 : sin ^ cos' 6, and the length of the common chord of the parabola and the circle of curvature at the point bears to the latus rectum the ratio 2 siu d : cos" 6. 975. At a point P of a parabola is dra^vn a circle equal to the circle of curvatm-e and touching the parabola externally ; the other common tangents to this circle and the parabola intersect in Q : prove that, if QK be let fall perpendicular on the directrix, S Q - QK _ AS SQ + QK~ AS + SP' III. EllijJse referred to its axes. [The equation of the ellipse in the following questions is always supposed to be -j + p-= 1, and the axes to be rectangular, unless other- wise stated. The point whose excentric angle is 6 is called the point 6. The excentricity is denoted by e. The tangent and normal at the i)oint 6 are respectively X ^ y . ^ . ax hy o it a h cos d sm d the chord through the two points a, /? is X a+B y . a+B a-^ - cos —^ + r sm —^ = cos—-—; a 2 6 J ^ and the intersection of tangents at a, /?, (the pole of this chord) a + B , . a + )3 a COS osin- — r a-^ ' a~(i COS ^^ — COS — - — 1.34 CONIC SECTIONS, ANALYTICAL. xX vY The polar of a point {X, 7) is ^ + ^ = 1 ; and the equation of the two tangents from {X, Y) is It follows from the equation of the tangent that if the equation of any straight line be lx+my=\, and I, m satisfy the equation d'l'+¥m' = \, the straight line touches the ellipse -^ + ^ = 1, a result often useful. The equation of the tangent in the form X cos ^ + y sin ^ = Jar cos' d + h^ sin^ 6 may be occasionally employed with advantage. The points a, yS w411 be extremities of conjugate diameters if a~^ — -. Any two points are called conjugate if either lies on the polar of the other, and any two straight lines if either passes through the pole of the other. If (X, Y) be the pole of the chord through (a, ^) it will be found that sin a sin /8 cos a cos ^ sin a + sin /? cos a + cos ^ 1 "" X^ " —; F" " 2Y " -IX " X^ F ' 1 - -^ 1-72 r — -^ ■*■ r.' a a a o which enable us to find the locus of (X, Y) when a, ft are connected by some fixed equation. Thus, "If a triangle be circumscribed about an ellipse — + |j = 1 and two angular points lie on the ellipse — 75 + ttz = 1> to find the locus of the third angular point." If a, /3, y be the three points of contact and (a, y8), (a, y) be the pairs of points whose tangents intersect on the second ellipse, we have a^ ,a + B If . „a + 3 „a-B -^2 cos' —Y~ + ^z sui —3- == cos- -2- » and the like equation with y in place of ^. Hence /?, y are the two roots of the equation A cos a cos 6 ■\- B sin a sin 6 = C; where ^=--^+- + 1, ^e^,-^-.,+ 1, C = ^,^^,~l; and we have therefore y8 + y . P + y /3 - y 2 2 A cos a j5 sin a 6* ' BO that the co-ordinates of the third angular point are Aa Bh . -y;- COS a, -y^ Sin a, C CONIC SECTIONS, ANALYTICAL. 155 SO that its locus is the ellipse of tf 1 This locus -will be found to coincide with the second ellipse if — , ± 77 ± 1 = 0, and if we so choose the siffiis of «', h' that the relation is ah -j+v, = 1, A = 2r;', B = 2—,, C = — 2 —pn , so that the co-ordinates of the ah a ah thii'd point ai'e —a' cos a, —6' sin a, or its excentric angle is tt + o, and similarly the excentric angles of the other points are tt + {3, ir + y. Hence the ellipses "^+1 = 1 ^ + ^ = 1 a' b' ' a" h" ' will be such that an infinite number of triangles can be inscribed in the second whose sides touch the fii'st, if with any signs to a', h' the relation — + v, = 1 is satisfied and the excentric angle of any comer of such a ah o J triangle exceeds that of the corresponding point of contact by tt. If this condition be not satisfied the two given ellipses and the locus will be found to have four common tangents real or impossible. Again, for the reciprocal problem, "If a triangle be inscribed in the ellipse — , + ^ = 1 and two of its sides touch the ellipse — +'^,= \, to ah' a' h' find the envelope of the thii'd side." Taking a, yS, y for the angular points and (a, )8), (a, y) for the sides which touch the second ellipse, we have «" ,a+ B b'^ . „a+ B „a — iS —, cos"* r— + 777, Sm- — -r— = COS- — ^r— , a" 2 6' 2 2 and a like eqxiation with y in place of (3. Hence, as before, ^,, cos-^ ^„ sm-^ cos- -—^ cos- ■ which, since the third side is (3 + y . P + y COS ~~ sin ^ ■ ^ L. + 2^ L_i a' /3-y h' )8-y~ ' COS -— -^ COS ^ ' proves that the envelope is the ellipse «■ 156 CONIC SECTIONS, ANALYTICAL. •which coiucides with the second ellipse if — J- j^, ± 1 = 0, or if with any signs of ft', h', — + ., = 1. The excentric angles of the points of contact Avill ])e a — IT, /3 — TT, y — TT (or a + tt, )8 + tt, y 4- tt, which are pi-actically the same). If tliis condition be not satisfied the three conies intersect in the same four points real or impossible. The relations between the excentric angles corresponding to normals di-awn fi-om {X, Y) may be found from the equation aX hY „ ,, COS B sm a biquadratic whose roots give the excentric angles of the points to which n normals can be drawn from (A", Y). If tan- = ^, this equation becomes Z*h r+ 2Z^ {aX + a' - b') + 2Z(aX-a' + b') -bY^O. This equation having four roots, there must be two relations inde- pendent of X, Y between the roots, as is also obvious geometrically. These relations are manifest on insjiection of the equation ; they are and the relation between Z^, ^„, Z^ is therefore ZZ + Z^Z, +ZZ= -J^ + --^ 4- ^ z^z- z.^z^ z^z: which is equivalent to sin (/8 + y) + sin (y + a) + sin {a + (3) = 0, if a, /S, y be the coi'responding values of 0. Since 1 - {Z^Z^ +...) + Z^Z^Z.^Z^ = 0, it follows that a + B +y + 8 tan T— ^-^ — - — c/D , or a + /3 4- y + S is an odd multiple of tt. The following is another method of investigating the same question. If the normal at {x, y) to the ellipse j^ass through (A", Y), ccx Y - h'yX = (a' - b') xy. (A) Now if — + V^= 1, and h -V^ = 1, be the equations of two lines a b a b joining the four points to which normals can be drawn from (A", Y), the x' y^ , . fix my S\ fl'x m'y A ,. equation -+^.-l+x(-+-^-l)(-+-^-'-l)=0 can be made to coincide with (A). The identification of the two gives A. = 1, ZZ' + 1 = 0, mm +1^0, whence it follows that noi-mals at the points where the two straight lines a b ' al bm meet the ellipse all meet in a point. The point is given by ax —bya^ — b^ l{l~m') ^ VI (1 - f ) " r+< • CONIC SECTIONS, ANALYTICAL. I.jV If a, P be the two jtoints on the foruicr, and y one of the points on the latter. a + /3 . a4- S cos — -r— sm — - /«' I cosy siny , Aviience sm — ,^ cos — -^ + f cos y sm ^-^ + sm y cos — ^ j cos — — - 0, or sin (/? + y) + sin (y + a) + sin (a + yS) = 0. The equation formed from this by replacing y by 8 must also hokl, whence y+S . y+S y-S cos ^-—— sm '—- cos ' -— ^ -^ -J sin a + sin y8 cos a + cos y3 ~ sin (a + ;8) ' and tan ~— = cot ^ , or a + /3 + y + 8 is an odd multiple of - .] 976. A chord AP is drawn from the vertex of an ellipse of excen- tricity e, along PA is taken a length PP equal to PA -^e", and PQ is drawn at riglit angles to the chord to meet the straight line through P parallel to the axis : the locus of Q is a straight line pei-jiendicular to the axis. Similarly if £P be a chord through a vertex on the minor axis and along £P be taken a length PP equal to PP -f e*, and PQ be drawn at right angles to PP to meet the straight line through P parallel to the minor axis, the locus of (J is a straight line parallel to the major axis. [The equations of the loci, with the centre as origin, are 977. Tangents drawn from a point P to a given ellipse meet a given tangent whose point of contact is in Q, Q' : prove that if the distance of P from the given tangent be constant, the rectangle OQ, OQ' will be constant. Also if the length QQ' be given the locus of Q will be a conic having contact of the third order with the given ellipse at the other end of the diameter through 0; and the conic will be an elli})se, ])arabola, or hyperbola according as the given length QQ' is less than, equal to, or greater than the diameter parallel to the given tangent. 978. Two ellijises have the same major axis and an ordinate NPQ is drawn, the tangent at P meets the other ellipse in points the lines joining which to either extremity of the major axis meet the ordinate in Z, M : prove that NP is a harmonic and NQ a geometric mean between NP, NM. loS CONIC SECTIONS, ANALYTICAL. 979. Tlio equation giving t tlie lengUi of the tangent from (A", Y) to tlie ellipse '-, + n = 1 is AVI UY" YfU ly LVL_l\^_n A'* Y^ where t^= -^ + tj - !• 980. Tlie major and ininor axes of an ellipse being A A', BE', another similar ellipse is described with BB' for its major axis, P is any point on the former ellipse and L the centre of peri)endicula)-s of the triangle PBB' : prove that L will lie on the second ellipse and that the normals at L, P will intersect on another ellipse whose minor axis is 46, 1 . . ^cir + h' and major axis J . (h 981. A given ellipse subtends a right angle at 0, and 00' is drawn perpendicular to and bisected by the polar of : prove that 00' is divided by the axes in a constant ratio, CO' is a constant length, the middle point of 00' is the point of contact of the polar of with its envelope, and the rectangle under the perpendiculars from 0, C on the polar of is constant. 982. The rectangle under the perpendiculars let fall on a straight line, from its pole with respect to a given ellipse and from the centre of the ellipse, is constant (= A) : prove that the straight line touches the confocal -s — r- + ji, — : = 1. d' + X b- + \ 983. The rectangle under the perpendiculars drawn to the normal at a point P from the centre and from the pole of the normal is equal to the rectangle under the focal distances of P. 984. The sum or the difference of the rectangles under the perpen- dicidars upon any straight line (1) from its pole with respect to a given ellipse and from the centre, (2) from the foci of the given ellipse, is constant (= b^) ; the sum when the straight line intersects the ellipse in real points, otherwise the difference; or with proper regard to sign in both cases, the rectangle (2) always exceeds the rectangle (I) by 6^ 985. Through a point are drawn two straight lines at right angles to each other and conjugate with respect to a given ellipse : prove that the arithmetical difference between the rectangles under the perpen- diculars on these lines each from the centre and from its own pole is equal to the sum of the rectangles under the focal perpendiculars, and to the rectangle under the focal distances of the point. 986. On the focal distances of any point of an ellipse as diameters are described two circles : prove that the excentric angle of the point is equal to the angle which a common tangent to the circles makes with the minor axis. CONIC SECTIONS, ANALYTICAL. 159 987. Tlie ordinate A^P at a point F of an ellipse is producod to Q so tliat JVQ : NF :: CA : t'N, and from Q two tangents are drawn to tlie ellipse : prove that they intercept on the minor axis pro- duced a length equal to the minor axis. 988. A circle of radius r is described with its centre on the minor axis of a given ellipse at a distance er from the centre : prove that the tangent to this circle at a })oint where it meets the ellipse will touch the minor auxiliary circle. 989. A point P on the auxiliary circle is joined to the ends of the major axis and the joining lines meet the ellipse again in Q, Q' : prove that the equation of QQ' is (a- + ¥) tjsme + 2b' X cos 6 = 1ah\ where 6 is the angle ACP, and if the ordinate to P meet QQ' in E, R is the point of contact of QQ' with its envelope. 990. From a point P of an ellipse two tangents ai-e drawn to the circle on the minor axis : prove that these tangents will meet the diameter at right angles to CP in points lying on two fixed straight lines parallel to the major axis. 991. Two tangents are drawn to an ellipse from a point P: prove that the angle between them is _^fCP^-AC"--BC^ \ ~SP7S'P / 992. If 2^, q be the lengths of two tangents at right angles to each other p- + q^ ( V q' n 993. li p, q be the lengths of two tangents and 2ma, 2mb the axes of the concentric similar and similarly situated ellipse drawn through their point of intersection 994. The lengths of two tangents drawn to an ellipse from a jwint on one of the equal conjugate diameters are ]}, q : prove that {a' + b') (if - qj (p' +q' + cr + by- = 4 (/ + ^7=')^ (a' - by. 995. If p, q be the lengths of two tangents drawn from a point on the hyperbola '^ = a-b to the ellipse -^ + ^s = 1, and r the central a a b distance of the point, then will b (pq + a') (pq + 6-) pq = r' - a" + ab -b', p -q = 2 (a - b) , , ^ ^ ' ab +pq and (p + q)' = ipq (pq + aby IGO CONIC SECTIONS, ANALYTICAL. 99G. If two tangents be drtiNvn from any point of the hyperbola 'L = a — b to the ellipse -^ + Vs = 1 , the difference of their lengths a b '^ a will be 2 (a—b) (l — 3 — ; rx, ) , where r is the central distance of the \ r^ -{fc- by J point : and if a parallelogram be inscribed in the hyperbola whose sides touch the ellipse and r, , 7'^ be the central distances of two adjacent angular points, then will {r^ -a' + ah - ¥) (r/ - a" + ab - b') = a'¥; tlie lengths of the sides of the parallelogi-am will be Jr,^ + r^^-{a-by^{a-b), and the point of contact on any side will divide that side in the ratio r^ + ab — a" — b' : ab. 997. A circle is described on a chord of the ellipse lying on the straight line p - + q^=l as diameter : prove that the equation of the a b straight line joining the other tAvo common points of the ellipse and circle is X 7/ a~ + b^ 998. In an ellipse whose axes are in the ratio ^2 + 1 : 1, a circle whose diameter joins the ends of two conjugate diameters of the ellipse will touch the ellipse. 999. Normals to an ellipse at P, Q meet in and CO, PQ are equally inclined to the axes : prove that the paii; of PQ intercepted between the axes is of constant length and that the other normals drawn from will be at right angles to each other. 1000. If be the point in the normal at P such that chords drawn through subtend a right angle at P, and 0' be the corresponding point for another point P, 00', PP' will be equally inclined to the axes and their lengths in a constant ratio. 1001. A circle is described having for diameter the part of the nonnal at P intercepted between the axes, and from any point on the tangent at P two tangents are drawn to this circle : prove that the choi-d of the ellipse which passes through the points of contact siibtends a right angle at P. 1002. The normals at three points of an ellipse whose excentric angles are a, /3, y will meet in a point, if sin (/3 + y) + sin (y + a) + sin (a + /3) = 0, which is equivalent to 4. /5 + y * X y + a ,0 , a + yS . tan - ' cot a = tan - — - cot p - tan ^ - coty. CONIC SECTIONS, ANALYTICAL. IGl 1003. If four normals to an ellipse meet in a point the sum of tlio corresponding excentric angles will be an odd multiple of tt. Also two tangents drawn to the ellipse parallel to two chords through the four points will intersect on one of the equal conjugate diameters. 1004. The normals to the ellipse at the points where it is met by the straight lines r^ ^nv _^ « ^ .'/ _ 1 a ap bq will all intersect in one point, / ax hy _ <*' - ^'\ 100.5. From a point P of an ellipse PM, PX are let fall pci-pendi- cular upon the axes and J/iV produced meets the ellipse in ^, g' : prove that the normals at Q, q intersect in the centre of curvature at p, Pp being a diameter. 1006. From a point are drawn normals OP, OQ, OP, OS, and p, q, r, s are taken such that their co-ordinates are equal to the intercepts on the axes made by the tangents at P, Q, R, S : prove that p, q, r, s lie in one straight line. Also, if through the centre G be drawn straight lines at right angles to CP, CQ, CR, CS to meet the corresponding tangents, tiie four points so determined will lie in one straight line. [If X, Y be the co-ordinates of 0, the two straight lines will be xX -ijY^a' - h\ a'Xx + ¥ Yy + a'W = 0.] 1007. The normals to an ellipse at P, Q, R, S meet in a point and the circles QRS, RSP, SPQ, PQR meet the ellipse again in the point.s F, Q', R\ S' respectively : prove that the normals at P\ Q', R' , S' meet in a point. 1008. Normals are drawn at the extremities of a chord parallel to the tangent at the point a : prove that the locus of their intersection is the curve 2 {ax sin aJthy cos u) {ax cos a ■¥ hy sin a) = (a' - by sin 2a cos^ 2a. 1009. Normals are drawn at the extremities of a chord drawn through a fixed point on the major axis : prove that the locu.s of their intersection is an ellipse whose axes are the distance of the given point from the centre being ca. w. p. 11 162 CONIC SECTIONS, ANALYTICAL. 1010. The nonn.il at a point P of an ellipse meets the curve in Q and any other chonl /V is drawn ; QP' and the straight line through P at right angles to PP' meet in R : prove that the locus of R is the straight line X , V • , f'^ + ^^ — cos qt-'r «iii

Avhere <^ is the excentric angle of P. The part of any tangent intercepted between this straight line and the tangent at P is divided by the point of contact into two parts which subtend equal or supplementary angles at P. 1011. A chord PQ is normal at P, PP' is a chord perpendicular to the axis, the tangent at P' meets the axes in T, T', the rectangle TCT'R is completed and CR meets PQ in U : prove that CR. CU^a:-b\ 1012. Along the normal at P is measured PO inwards equal to CZ), and the other normals OL, OM, ON ai-e drawn : prove that the parts of LP, MP, NP intercepted between the axes are eqtial to a + 6 ; the tangents at L, M, N form a triangle whose circumscribed circle is fixed ; and if r,, r„, r^ be the lengths LP, MP, NP, r, + r„ + ?-3 = 2 (a - b), r^r^ + r^9\ + o\ i\ = P0~ - iah, r,r^r^{a-b) = 2ab{ab-P0'); any of the three i\, r^, r^ being reckoned negative when draAvn from a point whose distance from the major axis is greater than . / . CoiTesponding i^esults may be found when PO is measured outwards, but in that case two of the normals will always be impossible unless a > 2b. 1013. The chord PQ is normal at P, and is the pole of PQ: prove that where p is the perpendicular from the centre on the tangent at P. 1014. Perpendiculars ^'^ />2 are let fall from the ends of a given chord on any tangent, and a peipeudicular 2^3 from the pole of the chord : prove that where a, (3 are the excentric angles of the gi\^en points. CONIC SECTIONS, ANALYTICAL. 103 1015. Two circles liave eacli double contact with an ellipse and touch each other : prove that 7*1 , r, being the radii ; also the point of contact of the two circles is equi- distant from the clioi-ds of contact Avith the ellipse. [Only the iipper sign applies when the circles are real; the coiTC- sponding equation for the hypei'bola is formed by putting — b^ for 6^, as usual, wlien the circles touch only one branchy but for circles touching both branches the equation is e- ^' '' e^-1 J 1016. Two ellipses have common foci S, S', and from a point P on the outer are drawn two tangents PQ, PQ' to the inner : prove that QPQ' SPS' . ^ ^ ^. cos — - — : cos — ^r— IS a constant I'atio. 1017. The sides of a parallelogram circumscribing an ellipse are parallel to conjugate diameters : prove that the rectangle under the per- pendiculars let fall from two opposite angles on any tangent is equal to the I'ectangle under those from the other two angles. 1018. The diagonals of a quadrilateral circumscribing an ellipse are aa, bb', cc', and from 6, b', c, c' are let fall perpendiculars ^?i, ^?„, j)^, p^ on any tangent to the ellipse : prove that the ratio 2'>iP2 '■ P3P4 is constant and equal to A, where -V + ^T- + 1 X + 1 «' b' ^-1 /%,^C-i) K.'iC-i a' b' J \a' b and (cc , y,), (.r,, 1/,) are the points a, a. If the points of contact of the tangents from b be L, L', from // be 3f, M', from c he L, M', and from c' be L', Jf, the value of X. is equal to the ratio of [LL'M'M] at any point of the ellipse to its value at the centre. 1019. Prove that the equation |'^cos(a-/?) + |sin(a-;8)-l}.[^'cos(a+/3) + |sin(a + ^)-l} (x y . Y = \ cos a +i-iima - cos B [ [a b '^j is true at any point of the ellipse -2 + rs = 1 ; and hence that the locus 11—2 lG4f CONIC SECTIONS, ANALYTICAL. of a point from wliicli if two tangents be drawn to the ellipse the centre of the circle inscribed in the trianptle formed by the two tangents and the chord of contact shall lie on the ellipse is the confocal ^'~ i?~ a'+b'' 1020. Two tangents are drawn to an ellipse from a point {X, Y) : prove that the rectangle under the perpendiculars from any point of the ellipse on the tangents bears to the square on the perpendicular from the same point on the chord of contact the ratio 1 : X ; where 1021. Four points A, B, C, D are taken on an ellipse, and perpen- diciilars p^, p^, p^, p^ let fall from any point of the ellipse iTpon the chords AB, CD; AC, BD respectively: express the constant ratio pjy^ : 2)^p^ in terms of the co-ordinates (X^, Y^), (X,, Y^) of the poles of BC, AD, and prove that the value of the ratio will be unity if X^X^ _ LL - ^'' ~ ^' 'oT b''~ a- + b'' 1022. A tangent is drawn to an ellipse and with the point of con- tact as centre is described another ellipse similar and similarly situated but of three times the area : prove that if from any point of this latter ellipse two other tangents be dra'^Ti to the former, the triangle formed by the three tangents will be double of the triangle formed by joining their points of contact. 1023. Two tangents TF, TQ meet any other tangent in F, Q' : prove that PF. QQ' = TF. TQ' co^' "-^ ', ■where a, ^ are the excentric angles of P, Q. 1024. Two sides of a triangle ai-e given in position and the third in magnitude : prove that the locus of the centre of the nine points' circle of the triangle is an ellipse; ■which reduces to a limited straight line if the acute angle between the given directions be 60". If c be the given length and 2a the given angle, the axes of the ellipse will be equal to c sin 3a c cos 3a 4 siix" a cos a ' 4 sin a cos^ a ' 1025. The tangent at a point P meets the equal conjugate diameters in Q, Q' : prove that tangents from Q, Q' will be parallel to the straight line joining the feet of the perpendiculars from P on the axes. 1026. The excentric angles of the corners of an inscribed triangle are a, /3, y. prove that the co-ordinates of the centre of perpendicidars are CONIC SECTIONS, ANALYTICAL. 165 — - — (cos a + cos p + cosy) — cos {a + (3 + y), a' + b\ . . - . . a'-b" . , „ > — ^-.^ (sm a + sin (3 + sin y) — sm (a + p + y) ; and those of the centre of the circumscribed circle are a'-b {cos a + cos ^ + cos y + cos {a + (3 + y)] Aa J — {sin a + sin /S + sin y - sin (a + /? + y)}. The loci of these points when the triangle is of maximum area are respectively 4 (aV + by") = {a' - bj, IQ (a'af + by-) = {a' - by-. 1027. The centre of perpendiculars of the triangle formed by tan- gents at the points a, (3, y is the point given by the equations /3-y y-a a-fi iax cos — — - cos ■—- cos — ^ = a' {cos a + cos /? -i- cos y - cos (a + ^ + y)} + 2 (a* + b^) cos a cos ft cos y, A7 fi~y y~°- °-~P AlUU cos ,, ■ cos '—;;;— COS --T— ^2 2 2 = h'' {sin a + sin j8 + sin y + sin (a + /3 + y)} + 2 (a' + ¥) sin a sin ft sin y. 1028. Two points E, IT are conjugate with respect to an ellipse, P is any point on the ellipse, and FH, PH' meet the ellipse again in Q, Q' : prove that QQ' passes through the pole of II H'. 1029. The lines form a triangle self-conjugate to the ellipse : prove that and that the co-ordinates of the centre of perpendiculars of the triangla are a'-b',, b'-a' 1030. A triangle is self-conjugate to a given ellipse and one corner of the triangle is fixed : prove that the circle circumscribing the triangle passes through another fixed point 0', that C, 0, 0' are in one straight line, and that CO . CO' - a' + b\ lOG CONIC SECTIONS, ANALYTICAL. 1031. Ill the ellipses a b a u a tangent to the former meets the latter in P, Q : prove that the tan- gents at P, Q are at right angles to each other. 1032. Two tangents OP, OQ are dra^v^l at the points a, /?: prove that the co-ordinates of the centre of the cii'cle circumscribing the tri- angle OPQ are 2 «■ + («* - h^) cos a cos /3 7^ 2a ' cos — ^— . a + (i 2 6" -f (6" — a~) sin a sin )8 a -Id 26 cos-^ If this point lie on the axis of x, the locus of is a circle (or the axis of x). 1033. Two points P, Q are taken on an ellipse such that the per- jjendiculars from Q, P on the tangents at P, Q intersect on the ellipse : prove that tlie locus of the pole of PQ is the ellipse a'x^ + hY = {a?+hy, and that if R be another point similarly related to P, the same relation will hold between Q, R ; the centre of perpendiculars of the triangle formed by the tangents at P, Q, R will be the centre of the ellipse, and the centre of perpendiculars of the triangle P, Q, R lies on the ellipse a^x' -f- Jfif = («" — 6')". 1034. Three points {x^, y^, (.t;,,, y^, (x.^, y.^) on an ellipse are such that iCj + cCg -t- a?3 = 0, y, + 2/, + 2/3 = : prove that the circles of curvature at these points will pass through a point on the ellipse whose co- ordinates are a' ' 6* • 1035. At a point P of an ellipse is drawn a circle touching the ellipse and of radius equal to n times the radius of cm-vature, and the two other common tangents to the circle and ellipse intersect in {X, Y) and include an angle : prove that X^ Y- and 4n«^6^ tan^ ^ - ^"''^' ^ ''^^' ^'"^^ - "^^- n being reckoned negative when the circle has external contact and X being the semidiameter parallel to the tangent at P. CONIC SECTIONS, ANALYTICAL, 167 1036. A triangle of niiniiniiiu area circumscribes an ellipse, is its centre of perpendiculars and OM, OX perpendiculars on the axes : i)rove that J/xN'is a normal to the ellipse at the point of concourse of the three cii'cles of curvature drawn at the points of contact of the sides. 1037. The tangent at the point whose excentric angle is ^ touches the circle of curvature at the point whose excentric angle is 6 : prove that . 6+e 2 _ 1- e^ cos* sm -^ — ^ ' If P be the point 0, and T the pole of the normal at P, PT will be the least possible when the point lies on the normal at P. 1038. The hyperbola which osculates a given elli})se at a point and has its asymptotes })arallel to the equal conjugate diameters meets the ellipse again in the same point as the common circle of curvature ; and if P be the point of osculation and the centre of the hyperbola, PO is the tangent at to the locus of and is normal to the ellipse 1039. A rectangular hyperbola osculates a given ellipse at a point P and meets the ellipse again in the same point as the common circle of curvature : prove that, if be its centre, PO will be the tangent at to the locus of and will be normal to the ellipse a' b° \d'-hr) 1040. An hyperbola is described with two conjugate diameters of a given ellipse for asymptotes : prove that, if the cur\es intersect, the tan- gent to the ellipse at any common point is pai*allel to the tangent to the hyperbola at an adjacent common point, and the parallelogram formed by the tangents to the hyperbola will be to that formed by the tangents to the ellipse as wrsin"^ : 1, the equation of the hyperbola being -i + -^ cot ^ - ^ = m. a" ah b^ If the common points be impossible the points of contact of the common tangents will lij on two diameters, and the parallelograms formed by joining the points of contact will be for the ellipse and hyperbola respec- tively in the ratio ni^ : sin* 6. 1041. A triangle circxirascribes the ellipse and its centroid lies in the axis of a; at a distance c from the centre : prove that its angular points will lie on the conic (x-3cy 9/ {a' -9c') a' a'b' ~ 1G8 CONIC SECTIONS, ANALYTICAL. 1042. A tiiiingle is inscribed in the ellipse and its centrold lies in the axis of x at a distance c from the centre : prove that its sides will touch the conic 4aV , {2x-3cY [In this and the preceding question the axes need not be rectangular.] 104^. A triangle is inscribed in the ellipse and the centre of perpen- diculars of the triangle is one of the foci : prove that the sides of the triangle will touch one of the circles n' Ja'-by , a'b 2M ^^'Z^TJ.2- +v = a' + h' J ^ {a' + by 1044. A triangle circumscribes the circle x^ + y' = a' and two of its angular points lie on the circle (x - c)' + ?/ = b' : prove that the locus of the third angular point is a conic touching the common tangents of the two circles; that this conic becomes a parabola if (c=t a)^= 6*-«^ ; and that the chords intercepted on any tangent to this conic by the two circles are in the constant ratio # 2a' : J {-lab + ¥ - c'){2ab - b' + c"). 1045. A triangle circumscribes an ellipse and two of its angular points lie on a confocal ellipse : prove that the third angular point lies on another confocal and that the perimeter of the ti'iangle is constant. 1046. Two conjugate radii CP, CD being taken, PO is measured along the normal at P equal to k times CD : prove that the locus of is the ellipse (a-kbf {b-JcaY and this ellipse touches the evolute of the ellipse in four points which h n are real onlv when I: lies between - and r : ^ being negative when PO is a b ° ° measured outwards. 1047. The ellipses 2v2 1 -,+p^i, «^- + &-2/- = (^:^.^ are so related that (1) an infinite number of triangles can be inscribed in the former whose sides touch tlie latter ; (2) the central distance of any angular point of such a triangle will be perpendicular to the opposite side ; (3) the normals to the first ellipse at the angles of any such triangle, and to the second at the points of contact, will severally meet in a })i)int. CONIC SECTIONS, ANALYTICAL. 169 1048. The ellipses are such that the normals to the latter at the corners of any inscribed triangle whose sides touch the former meet on the latter. 1049. The semi-axes of an ellipse U are CA, CB; LCL' is the major axis and C the focus of another ellipse V, LC^BC, CL'^^CA: \)Vo\e that the auxiliary circle of V touches both the auxiliary circles of U ; one of the common tangents, PP\ of U and V is such tliat P lies on the auxiliary circle of V; and PL, PL' ax?, parallel to CA, CB; CP', GL are equally inclined to CA, CB; if the auxiliary circle of V meet U also, in Q, R, /S, the triangle QliS has tlie centre of its inscribed circle at C, and the straight lines bisecting its external angles touch V and form a triangle whose nine points' circle is the auxiliary circle of V, and whose circumscribed circle has its centre at the second focus of V; aloo if the three other common tangents to U, V form a triangle Q'li'S', the centre of its circumscribed circle is C and its nine poiiits' circle is the auxiliary circle of U ; the sum of the excentric angles of Q, B, S is equal to that of the points of contact of the triangle Q'B'S', and if this snm be S the excentric angles of Q, B, S are the roots of the equation tan — ^— = - tan 6, and those of the points of contact of Q'B'S' are the ?) — B a roots of the equation tan — — =7- tan B; the three perpendiculars of the triangle Q'B'S' are normals to U and meet in the second focus of V, OP is normal at P and a circle goes through P and the other three points of contact. The straight lines through Q, B, S at right angles to CQ, CB, CjS will touch V in points q, r, s such that Cq, Cr, Cs make with CA angles respectively equal to the excentric angles of Q, B, S. The normals at Q, B, S meet in a point 0' from which, if the fourth noi-mal O'p be drawn, Pp is a diameter of U ; and the normals at the i)oints of contact of Q'B'S' meet in a point on the same normal O'p such that op : O'p = ah : a' - ah + h'. 1050. A triangle LMN is inscribed in the ellipse '-5 + 4:; = 1 so that ® a b~ the normals at L, M, iV meet in a point 0, and from tlie fourth normal OP is drawn : prove the following theorems. (1) OP will bear to the semi-diameter conjugate to CP the ratio k : 1 where k is given by either of the equations A' = (^-6 - a) cos (a + ft + y), Y^{h- ka) sin (a + /3 + y), where X, Y are the co-ordinates of and a, f3, y the excentric angles of Z, iV, N. (2) The sides of the triangle LMN will touch the ellipse 9 s — -f- -=^ — 1 a* b'" , 170 CONIC SECTIONS, ANALYTICAL, in points whose cxccntric angles arc tt + a, tt + (3, tt + y, if a h' 1 d?{a-kh) b\ka-b) a'-b'' (3) Tlie tangents at L, M, N will form a triangle whose corners lie on the ellipse -., + -^ = 1, at points whose excentric angles are tt + a, TT + (3, TT + y ; whert: A a' = a", lib' = b'. (4) An infinite number of such triangles LJf^ can be inscribed in the ellipse — ^ + y-^, = 1 and circumscribed to the ellipse —^ + jTa = 1, the excentric angles a, j3, y satisfying the two independent equations cos a + cos 13 + cos y = ( T ) cos (a + /3 + y), sin a + sin /? + sin y = ( y ) sin (a + ^ + y), and the relation between the axes being — h ^ = !• Ths ratio k : 1 ° a b remains the same for all such triangles, and if L', M', N' be the jioints of contact of the sides, the ratio of the areas of the triangles L' M' N' , LMN is always the same, being a'b' : ab, the ratio of the areas of the corresponding conies. (5) Four points related to each triangle LMN : (a) the centroid, (^) the centre of perpendiculars, (y) the centre of the cii'cumscribed circle, (S) the point of concourse of the normals, lie each on a fixed ellipse co-axial with the original, and the excentric angle is always the excess of the sum of the excentric angles of X, J/, N above tt, while the several semiaxes are , -. (KO — bb' cm — bb' [The results here given include all cases of triangles inscribed in the ellipse — 2 -!- p^= 1 with sides touching a co-aa»arellipse.] 1051. Triangles are circumscribed to an ellipse such that the normal at each point of contact jjasses through the opposite angular point : prove that the angular points lie on the ellipse (tV by _ {\-ay'^{\-by~' {-) b'\a 'bj3 ■(^ b'\ b bj'y (y) 2a ¥b' 6' a'-b' 26 a' a' CONIC SECTIONS, ANALYTICAL. 171 \ being the greater root of the equation .=1; the locus of the centre of perpendiculars of the triangles is the ellipse and the perimeter of the triangle formed by joining the points of contact is constant. 1052. The two similar and similarly situated conies a- b- a- b^ will be capable of having triangles circumscribing the first and inscribed in the second, if a- 6 1053. A circle has its centre in the major axis of an ellipse and triangles can be inscribed in the circle whose sides touch the ellipse : prove that the .circle must touch the two circles x" + (y ± &)" = ci^. 1054. A triangle iJ/iV" is inscribed in a given ellipse and its sides touch a fixed concentric ellipse : prove that the excentric angles a, {3, y must satisfy two equations of the form sin (/3 + y) + sin (y + a) + sin (a + /8) - 7?i, cos (y3 + y) + cos (y + a) + cos (a + f3) = n, where vi, n are constant ; and that the equation of the ellipse touching the sides is of, , r2- 'f . , -\ . xy /w'4-?i*-l\2- ^(?/r + 9i+l) + ^,K + /i-l) + 4m-^ = (^ Y~)- Also prove that the area of the triangle LMN bears to the area of the triangle foi*med by joining the points of contact a constant ratio equal to that of the area of the ellipses. If (.Vj,, y^ be the centroid, (a;,, yj the centre of perpendiculars, and (.r„ y^ the centre of the cii'cumscribed circle, and a + )3 + y = ^, — ^ = m sin ^ + n cos Q. -/-" = wsin Q - m cos Q; a 2aa3j = m {or + b^) sin B-\-{n {a^ + b') - «^ + 6") cos 0, 2by^ = {n (a* + b^) - a' -I- b"} sin 6 - m (a" + ¥) cos 0; 4aa;g= («* - b'){m sin 6 + (ii + 1) cos 0}, iby^ = («" - 6") {m cos ^ - (n - 1 ) sin ^} ; from which the loci can easily be found. 172 CONIC SKCTIONS, ANALYTICAL. 1055. TrianglfS are inscribed iu a given ellipse such that theii- sides touch a fixed concentric ellii)se of given area j-jr, — rr : prove that this ellijise Avill have double contact with each of the ellipses a' ^9 \ 2 J 1056. A triangle LMX is cii'cumscribed about a given ellipse of focus aS' such that the angles SMN, SNL, SLM are all equal (= 6) : prove that sin ^ = ^r^ , and that L, J/ X lie on one of two fixed cii-cles whose 2a ,. . 2rt" , . sin Z sin J/ sin ^Y , , common radius is - : also tan6 = , _ ..^ ~ ,_, and the point 1 + cos L cos M cos N of contact of MN lies on the straight line joining L to the point of intersection of the tangents at M, N to the circle L2IN. 1057. A triangle is formed by tangents to the ellipse at points whose exceutric angles a, fi, y satisfy the poristic system cos (jS + y) ± (sin [i + sin y) + ?i = 0, kc. : prove that the locus of the angular points is ^(h + 1) + |;(«-1)=.|^ = 0; the envelope of the sides of the triangle formed by joining the points of contact is the parabola ^:=(„-i)(„.i.|) and the centroid of this latter triangle, its centre of perpendiculars, and centre of circumscribed circle lie on three fixed straight lines parallel to the axis of x. 1058. A triangle is formed by tangents to the ellipse at points whose exceutric angles a, ;8, y satisfy the poristic system cos ^ cos y + m (sin /3 + sin y) + vi^ = 0, &c. : prove that the locus of the angular points is m^(<+|;V-^+2«.f+l=0; \a" b' J b- b the envelope of the sides of the triangle formed by joining the points of contact is the hjqierbola 2 3-(f + "0 +1 = 0; the locus of the cen- troid is the ellipse ^ + ( -^ + n^ ) = 1 ; that of the centre of perpendi- culars the ellipse r/V + {by + m ar + h'f = h* ; and that of the centre of the circumscribed ciicle the straight line 2by = in (a' — b^). CONIC SECTIONS, ANALYTICAL. 173 1059. A triangle is formed by tangents drawn to a given ellipse at points whose excentric angles satisfy the equations ^cos(a + ^ + y)+ /»(cos/i +y + cosy + a + cosa + )8) + «(cosa + cos/3 + cosy)^2?, I sin (a + yS + y) + m (sin /S + y+. .. + ...) + » (sin a + ... ) == ^ : prove that the angular points will lie on the conic whose equation is "2 {I + ti- — q' — 7n -1)') + 7-3 (^ -'*■" — 5' - "i +i^0 + "^ ("^' ~ ^^') + \qn y + 4 (Im + »;;) - + Xnm -f = : 6 a ■'00 (which, since its equation involves four independent constants, will be the general equation of the conic in which can be inscribed triangles iC "2/ whose sides touch the given ellipse -^ + j^ = 1.) The loci of the cen- troid, ikc. of the triangle whose angular points are a, ^, y can easily be formed, and it will be fiiuud that the centroid lies on an ellipse similar and similarly situated to the given ellipse ; that the locus of the centre of pei'pendiculars is similar to the given ellipse but turned through a right angle; and that each locus reduces to a straight line when m'=n^y in which case a + /? + y is constant. 1060. The maximum perimeter of any triangle inscribed in a given ellipse is 2/3 a" + 1/ + J a* - a^b^ + b* J a' + b' + 2 J a* -a'b'+b* ' and if 2X, 2 F, '2Z be diameters parallel to its sides A'^ + Y' + Z' =a' + h' + Ja^^^b'Tb*. 1061. A parallelogram of maximum perimeter is inscribed in a given ellipse and 2A', 2F are its diagonals : prove that 1 111 T, + a'+b'-X' a' + b'-Y' a' b" and that the perimeter is 4 J a' + 6^. 1062. A hexagon AB'CA'BC' of maximum perimeter is inscribed , . . . , a" + ab + b'^ , ^ ^ ^ in an ellipse : prove that its pei-imeter is 4 7 — ; tlie tangents at A, B, C and A', B', C form triangles inscribed in the same fixed circle of radius a f 6 ; also, if a triangle be inscribed in the ellipse with sides each parallel to two sides of the hexagon, the sides of this triangle will touch a fixed circle of radius , and its area will be half that of the a f 6 ]74 CONIC SECTIONS, ANALYTICAL. hexagon. Also, if A", }', Z be radii of the ellipse each parallel to two sides of the hexagon, X^ + Y°- + Z' = a^-¥ah + h\ IX YZ = ah{a + h)- and if A", I", Z' be radii each parallel to the tangents at two corners of the hexagon 111111 2 1/11 X" + y* + z" a' "*■ ab ^ b 2 ^1/1 1\ ^" X'Y'Z' ab\a'^ b) 10G3, A hexagon AB'CA'BC is inscribed in the ellipse ^ + Vo = 1, and its sides touch the ellipse -^^ + ^7^= 1; a triangle abc is inscribed in the former ellipse so that he is parallel to B'C^ and BC, /XY + {xX + i/Y - a' - bj = 0. 1074. A fixed point is taken within a given circle, a pair of parallel tangents drawn to the circle, and. 10.4' is a straight line meeting the tangents at right angles. An ellipse is describetl with focns and axis A A', and the other two common tangents to this ellipse and the circle meet in F : prove that P lies on a fixed straight line bisecting at richt ansles the distance between and the centre of the circle. 1075. With the focus of an ellipse as centre is described a circle touching the directrix ; two tangents drawn to the circle from a point F on the ellipse meet the ellipse again in Q, Q' : prove that QQ' is parallel to the minor axis, and that tangents drawn from Q, Q' to the circle will intersect in a point F' on the ellipse so that FF' is also pai'allel to the minor axis. The tangents to the circle at the real common points pass through the further extremity of the major axis, and the points of contact with the ellipse of the (real) common tangents are at a distance from the focus equal to the latus. rectum. 1076. At the ends of the equal conjugate diameters of an ellipse whose foci are given are drawn circles equal to the circle of curvature and touching the ellipse externally : prove that the common tangents to the ellipse and one of these circles intersect on the rectangular hyperbola which is confocal with the ellipse. x^ if 1077. From any point F on the ellipse -j+^2 = l, tangents are a? V^ 1 drawn to the ellipse -^ + — r = 1 '• pi'ove that they meet the former ellipse in points Q, Q' at the ends of a diameter, and that the tangents at Qi Q' "^vi^l touch the circle which touches the ellipse externally at P and has a diameter equal to the diameter conjugate to CF. 1078. A circle is drawn through the foci of a given ellipse and common tangents drawn to the ellipse and circle : prove that one pair of straight lines through the four points of contact with the circle will envelope the hyperbola la - ^) a' -2b' b confocal with the ellipse. 1079. From a fixed point {X, Y) are drawn tangents OF, OQ to a conic whose foci are given : prove that the locus of the centre of the circle OPQ is the straight line 2xX 2yY _ X^H^T^-T? ^ XU Y' - c^ ~ ' and the lotus of the centre of perpendiculars is a rectangular hyperbola CONIC SECTIONS, ANALYTICAL. 177 of -wliicli one asymptote is parallel to CO, reducing to two straight lines if lie on the lemniscate of which the given foci are vertices. [The equation of the rectangular hyperbola is {Xx + Yy) {Xu - Yx) - c- [Xy + Fa? - 2XY) ; and if 0' be its centre, CO'.CO- CS', and the angle SCO' is three times the angle SCO.] 1080. Two tangents OP, OQ being draA\Ti to a given conic, prove that two other conies can be drawn confocal with the given conic and having for their polars of the normals at F, Q, 1081. Two conies have common foci aS*, S', a point is taken such that the rectangle under its focal distances is equal to that iinder the tangents to the director circles : px-ove that the polars of will be normals to a third confocal conic at points lying on the polar of with respect to that conic. 1082. A diameter PP of a given ellipse being taken, the normal at P intersects the ordinate at P in (^ : prove that the locus of Q is the ellipse and that the tangents from Q meet the tangent at P in points on the auxiliary cii'cle. 1083. A chord PQ of an ellipse is normal to the ellipse at P, and •p, q are perpendiculars from the centre on the tangents sX P, Q : prove that iq' _ (a' + b'-p y 108-1. The locus of the centre of an equilateral triangle inscribed in a given ellipse is the ellipse ^] (a' + wy + ^^ {?>a' + by = («-" - by. 1085. From two points on the polar of a point are drawn two pairs of tangents at right angles to each other to a given ellipse : prove that the four other points of intersection of these tangents lie u]ion the tangents at to the confocals through : and the tangents drawn from a pair of these points to the corresponding confocal will be parallel to each other. [The latter proposition is more readily proved geometrically.] 1086. An ellipse is described passing through the foci of a given ellipse and having the tangents at the end of the major axis for direc- trices : prove that it will have double contact with the given ellipse, and that its foci will lie on two circles touching the given ellipse at the ends of its major axis and having diameters equal to half the latus rectum. w. p. 12 178 CONIC SECTIONS, ANALYTICAL. 1087. The least distance between two points lying respectively on the fixed ellipses a a is / '{^"b' - a'b"-) {a" - a" - b' + b") V {a'-b'){a"-b") Explain how it comes to pass that this vanishes for confocal and for similar ellipses. 1088. Prove that if the ellipse -^ — r + ,f—z = 1 touch a parallel ^ a'-\ b^-\ IX of I/' to the ellipse -i + tj = 1, ^^he distance between the ellipse and its parallel will be ^/x, and the ratio of the curvatures at the point of contact will be Xy. {a' - X) (6- - X) : {a-¥ fx - X') {X - /x). lY. Ili/perhola, re/erred to its axes or asymjytoles. [The eqviation of the hyperbola, referred to its axes, only differing from that of the ellipse by ha\'ing — b' instead of b', many theorems which have been stated for the ellipse are obviously also true for the hyperbola. It js convenient still to use the notation of the excentric angle and denote any point on the hyperbola by a cos a, hi sin a, and all the corresponding equations, but the excentric angle is imaginary. A point on the hyperbola may be denoted by a sec a, b tan a, but the resulting equations ai'e not nearly so symmetrical as the corre- sponding equations in terms of the excentric angle are for the ellipse. The angle a so used is sometimes called the excentric angle in the case of the hyperbola. When referred to its asj-mptotes the equation of the hyperbola is ixy = a^ + b', but the axes are not generally rectangular, and questions involving perpendicularity should not be referred to such axes. The equation is often ^vl'itten xy = are the angles which the central radii to F, Q make with either asymptote : prove that tan^ ^ tan <^ = 1, tan FCQ + 2 tan CFQ = 0, and that the least value of the angle CQP is sin~'l. Also if the diameter FF be drawn, QP will subtend a right angle at F'. 1109. In a rectangular hyperbola the rectangle under the distances of any point of the curve from two fixed tangents is to the square on the distance from their chord of contact as cos <^ : 1, where is the angle between the tangents. 1110. A cii'cle is described on a chord of an ellipse as diameter which X "?/ is parallel to the straight line - cos a + V ^i^^ a = : prove that the locus of the pole with respect to the circle of the straight line joining the two other common points is the hyperbola cos^a sin" a X y [If the diameter be the straight line - cos a+y sin a = cos ^, tlie pole of the other common chord is the point (A", Y), where = jr + («- + 6") cos o, cos a cos p -. — = ^ + (a' + I)-) cos B.\ sm a cos p / I J 1111. An ellipse and an hyperbola are so related that the asymptotes of the hyperbola are conjugate diameters of the ellipse : prove that by a j^roper choice of axes their equations may be expressed in the forms a' b'~ ' a' b'~ 182 CONIC SECTIONS, ANALYTICAL, 1112. An hyperbola is described with a pair of conjugate diameters of a given ellipse as asymptotes : prove that the angle at which the curves cut each other at any common point is t,,,-. ( 29c^ and an hyperbola when XY <^c^, but degenerates when XY = 9cl If OOf be bisected by the centre of the given hyperbola, the centre of the envelope divides 00' in the ratio 3:1. CONIC SECTIONS, ANALYTICAL. 183 1117. A triangle is inscribed in the hyperbola xy = c* so that its centroid is the fixed point (ca,-j, (a point on the hyjjerbola) : prove that its sides will touch the ellipse f 3 - + 3ay - 8c j = ixT/, which touches the asymptotes and the hyperbola (at the fixed point), the curvatures of the two curves at the point of contact are as 4 : 1, and the tangents to the ellipse where it again meets the hyperbola are parallel to the asymptotes. 1118. The circle of curvature of the rectangular hyj^erbola at the point (a cosec 6, a cot 6) meets the curve again in the point (a cosec ST, SQ : prove that SF and SQ' will be of constant length. 1127. Two circles are described touching a parabola at the ends of a focal chord and passing through the focxis : prove that they intersect at right angles and that their second point of intersection lies on a fixed circle. Also prove that the straight line joining the centres of the cii'cles touches an ellipse whose excentricity is ~ and w^hich has the same focus and directrix as the parabola. [The equation of the parabola being 2a = r (1 + cos 6), those of the circles may be taken to be r cos^ a = a cos {6 - 3a), r sin^ a = a sin {6 - 3a).] 1128. A conic is described having a common focus with the conic c = r (I +e cos 6), similar to it, and touching it where 6 = a: prove that 2c(l — e^) its latus rectum is ^, — — ^^ — , , and that the angle between the axes 1 +2ecosa + e^ ° of the two conies is 2 tan~* (- — -. | . If e> 1, the conies will inter- \ sm a . / sect again in two points lying on the straight line - (1 + e cos a) + c sin a {e sin 6 + sin ($ - a)} = 0. 1129. Two chords QP, PR of a conic subtend equal angles at the focus : prove that the chord QR and the tangent at P intersect on the directrix. 1130. Two conjugate points Q, Q' are taken on a straight line through the focus S of a conic, and the straight line meets the conic in P : prove that the latus rectum is equal to 2SP . SQ 2SP . SQ' SQ-SP ^ SQ' -SP' CONIC SECTIONS, ANALYTICAL. 185 1131. Tlu'ougli a point on the axis of an ellipse at a distance / h* . / a* — ^ from the centre is ckawn a straight line YOP meeting the ellipse in P and a tangent at right angles in Y : prove that the rectangle PO . OY is equal to the square on the semi latus rectum. 1132. The points of an ellipse at which the circle of curvature passes through the other ends of the respective focal chords are given by the equation 2r' - r (3a + c) + 2ac = 0, where 2a is the major axis, r the focal distance, and 2c the latus rectum. 1133. Tlie two circles which are touched by any circle whose diameter is a focal chord of a given conic have the directrix for their radical axis and the focus for one of their point cii'cles. [The equations of the two circles are r* (1 ± e) + cer cos 9 = c', that of the conic being c = r (\ +e cos 0).] 1134. The radii of two circles are a, b and the distance between theii' centres is c, where c (a + b + c) — 2 (a — b)~ ; the centre of a circle which always touches them both traces out an ellipse whose vertex (the nearer to the centre of the smaller circle) is A : jjrove that the ends of the diameter of the moving circle drawn through A lie on a fixed ellijise with its focus at A. 1 1 35. Prove that any chord of the conic c = r (1 + e cos 0) which is normal at a point where the conic is met by the straight lines -fe + -\ = j= sin 6+ {e-- 1) cos 6 will subtend a right angle at the pole. 1136. A conic with given excentricity and direction of axes is described with its focus at the centre of a given circle : prove that the tangents to this conic at the points where it meets the cii'cle touch a fixed conic of which the given cii'cle is auxiliary cii'cle. 1137. Two parabolas have a common focus and axes opposite, a circle is drawn through the focus touching both parabolas : prove that 3r^ = J-Jb^ + b^, a, b being the latera recta and r the radius of the circle. 1138. Four tangents to the parabola 2a = r (1 + cos 6) are drawn at the points 29^, 20^, 26^, 26^ : prove that the centres of the circles cii'cum- scribing the four triangles formed by them lie on the circle 2r cos 6^ cos 6^ cos 6.^ cos $^ = a cos (^, + ^3 + ^3 + ^4 - ^)- LS6 CONIC SECTIONS, ANALYTICAL. 1139. Through a fixed jtoint is clrawu any straiglit line, and on it are taken two points snch that tlieir distances from the fixed point are in a constant ratio and the line joining them subtends a constant angle at another fixed point : pi'ove that their loci are circles. 1 1 40. Two circles intersect, a straight line is drawn through one of their common points, and tangents are draAvn to the circles at the points where this line again meets them : prove that the locus of the point of intersection of these tangents is the cardioid cr = 2ab {I + cos {6 + a- (3)}; the second common point of the cii'cles being the pole, the common chord (c) the initial Hue, a, b the radii, and a, (3 the angles siibtended by c in the segments of the two cix'cles which lie each without the other circle. 1141. Tlie equation of the circle which touches the conic c =r (1 +ecos^) at the point where 6 = a, and passes through the pole, is T - (l +e cos a)^ = cos (a-6) +e cos (2a - 6) ; and the equation of the chord joining their points of intersection is - (1 + 2e cos a + e") = e^ cos 6 + e" cos {0 — a). 1142. Two ellipses have a common focus S, a common excentricity e, axes in the same straight line, and the axis of the outer (t^) is to that of the inner ( F) as 2 - e^ : 1 — e^ ; on a chord of U, which touches V, as diameter is described a cii'cle meeting U again in the points P, Q : prove that the circle FS'Q will touch U and that FQ will toiich a fixed similar ellijjse having the same focus S and its centre at tlie foot of the directrix oHU. 114.3. Two similar ellipses U, Fhave one focus aS' common, and the centre of V is at the foot of that directrix of U which is the polar of S; a tangent drawn to F at a point F meets U in two points : prove that the circle through these points and S will touch (J a,t a point Q such that iSF, IIQ ai'e equally inclined to the axis, // being the second focus of U. 1144. A conic is described having the focus of a given conic for its focus, any tangent for directrix, and touching the minor axis : prove that it will be similar to the given conic. [Also easily proved by reciprocation.] 1 1 45. Any point F is taken on a given conic, A is the vertex, aS' the nearer focus, and on A F is taken a point Q such that FQ exceeds SF by the sum of the distances of A^ S from the directrix : prove that the locus of ^ is a conic whose focus is A, similar to the given conic and having its centre at the farther vertex. f)r r,. I. ^ fJ' r COXIC SECTIONS, ANALYTICAl^. ' 187 1 1 -40. The point *S' is a focus of an ellipse c - r (1 + e cos 6), 0, 0' are two points on any tangent such that SO = SO' = mc, and SO, SO' meet the ellipse ia. P, Q : prove that PQ touches the conic c - mr \\ + e (1 + m) cos 6] ; and the tangents at P, Q intersect on the conic i" (1 — e') ( — e cos ^ J + 2em cos 6 i- — e cos j = 1. [The latter conic is a circle \\ii\\ its centi-e at the second focus and radius equal to the major axis, when m (1 — e^) = 2.] 11 -i?. With the vertex of a given conic as focus and any tangent as directrix is described a conic passing through the nearer focus : prove that its major axis is of constant length equal to the distance between the focus and directrix of the given conic, and that the second directiix envelopes a conic similar to the given conic and having a focus in common with it. 1148. Two straight lines bisect each other at right angles: prove that the locus of the points at which they subtend equal angles is r^ a cos 6 — h sin 6 ah b cos 6 — a sin 6 ' 2a, 26 being the lengths of the lines, theii- point of intersection the pole and the initial line along the length 2a. 1149. The focal distances of three points on a conic being r^, r,, r^ and the angles between them a, (3, y, prove that the latus rectum (2^) is given by the equation 4.a.^.y 1. l.-l. ^ sm - sm ^ sm K = - sm a + — sm o + — sm y : t 2 2 2 r, r,, r^ the angles a, j3, y being always taken so that their sum is 2ir. 1150. An ellipse circumscribes a triangle ABC and the centre of perpendiculars of the triangle is a focus : prove that the latus rectum will be 2E cos A cos B cos C . A . B . C ' sin^- sm^sm^ 7? being the radius of the circle ABC. 1151. Two ellipses have a common focus and axes inclined at an angle a, and triangles can be inscribed in one whose sides touch the other : prove that c," ± 2CjC„ = e^'c^ + e^c^ — 2e^e^c^c^ cos a, c^, c^ being the latera recta, and e,, e^ the excentricities. Also i£ $, /- + C+ 2fu + 2gx + 21ixy = 0, tlie equations giving its centre are ax dij The equation determining its excentricity may be found at once from the consideration that a + 6 — 2/i cos (£) ah -I? sin* CD ' sin* w are unchanged by transformation of co-ordinates ; and therefore that ((1 + 5-2^ cos (o)* _ (a* + ^y {ah — K') sin" w a"^' ' where to is the angle between the co-ordinate axes, and 2a, 2^ the axes. The excentricity e is thus given by the equation (a 4- 6 — 2/i cos 0))^ + 4 = ^ -. , 1 - e* (ah - h^) sin* w The area of the conic u = is ttA sin {ah - /r)^ ' A being used to denote the discriminant a, h, g I ^> *, / . or ahc + 2fgh - of- - hg- - ch^. The foci may be determined from the condition that the rectangle under the perpendiculars from them on any tangent is constant. Thus, taking the simple case when the origin is the centre and the axes rectangular, if the equation of the conic be aa^ + hf + 2hxy + c = 0, CONIC SECTIONS, ANALYTICAL. 189 and (A', Y), (- X, - Y) two conjugate foci, we must have in order that the straight line px + qy = \ may be a tangent il-pX-qY)ilyX.qY) ^ ^ ^^^^^^^ ^ V + q or jf {f, + X^-) + q'- (fx+ Y') + 2pqXY -I =0 (J). But the sti-aight line will be a tangent if the quadratic equation aaf + b>^ + c (px + qyf + llixy = 0, found by combining the equations, have equal roots ; that is, if (« + cp-) {h + c -2 + 2X-^-'|5 + (l+X»)-^— r3 = 0, a' b ci' ah b ^ ' a -b will subtend a right angle at the centre. 1 184. Two common tangents to the cii-cle x-' -f- y' = 2ax and the conic x'+{y-\xy+2ax = subtend each a right angle at the origin : also the tangents are parallel to each other, and the straight lines joining the origin to the points of contact with either curve are parallel to the axes of the conic. Hence prove that, if at a point on an ellipse where the rectangle under the focal distances is equal to that under the semi-axes a circle equal to the circle of curvature be dra\\Ti touching the ellipse externally, and FF", QQ' be 13—2 lOG CONIC SECTIONS, ANALYTICAL. the other common tangents, PQ', FQ will jiass through the point of contact and be parallel to the axes. 1185. Two paral)olas are so situated that a circle can be described through their four common points : prove that the distance of the centre of this circle from the axis of one parabola is equal to half the latus rectum of the other. 1186. An hyi)erbola is drawn touching the axes of an ellipse and the asymptotes of the hyj^erbola touch the ellipse : prove that the centre of the hyperbola lies on one of the equal conjugate diameters of the ellipse. 1187. On two fixed straight lines are taken fixed points A, B; C, D : prove that the parabola which touches the two fixed straight lines and the asymptotes of any conic through A, B, C, B will also touch the straight line which bisects AB and CD. 1188. With two conjugate semi-diameters CP, CD of an ellipse as asymptotes is described an hyperbola, and j^d is a common chord parallel to PD and bearing to it the ratio n : \: the curvatures of the two curves at any common point will be as 1 : 1 - n^. 1189. Five fixed points are taken, no three of which are in one straight line, and five conies are described each bisecting all the lines joining four of the points, two and two : prove that these conies will have one common point. 1190. A conic is drawn touching a given conic at P and passing through its foci S, S' : prove that the pole of SS' with respect to this conic will lie on the common normal at P, and will coincide with the common centre of curvature when the conies osculate. 1191. A parabola is drawn having its axis parallel to a given straight line and having double contact with a given ellipse : pi'ove that the locus of its focus is an hyperbola confocal with the ellipse and having one asymptote in the given direction. [If the given direction be that of the diameter of the ellipse through the point P (a cos a, b sin a) and the latus rectum of the parabola be 2ka^b'--i-CP^, the co-ordinates of the focus are i a cos a -2 ^ ' . + T > i 6 sm a -^ — „' j . '. , + y / > ^ V« COS* a + 6 sin* a A:/ ' ^ \a* cos* a + 6* sin* a kj and the equation of the directrix is , . ft* cos* a 4- 6* sin* a h (a" + ¥) , ax cos a + by sin a - ^, \- —^ — .J 1191*. An hyperbola is drawn touching a given ellipse, passing through its centre, and having its asymptotes parallel to the axes ; prove that the centre of curvature of the ellipse at the point of contact lies on the hyperbola, and that the chord of intersection of the two curves touches the locus of the centre of the hyperbola at a point whose distance from the centre of the hyperbola is bisected by the centre of the ellipse. At the point of contact the curvature of the hyperbola is two-thirds of that of the ellipse. CONIC SECTIONS, ANALYTICAL. 107 1192. Taking the equation of a conic to be u = 0, ii \ be so deter- luiiied that the equation u + \ (x- + y^ + 2xi/ cos w) = represent two straight lines, the part of any tangent to the conic intercepted l)etween tliese sti'aight lines will be divided by the point of contact into two parts subtending equal (or supplementary) angles at the origin. If the co- ordinates be rectangular f w = - j and px + qy =\ be one of the two straight lines, then will ^P1 -^fP + ffQ + ^>'=^, « + ^9P + (^P^ = b + 2/q + cq-. 1193. The part of any tangent to the ellipse a^i/ + b^x^ = a'b' intercepted between two fixed straight lines at right angles to each other is divided by the point of contact into two parts subtending equal angles at the point (X, Y) : prove that .Y" _r_ a'-b' a' ~ b' ~a'+b'* and that the two straight Hues intersect in the point (A"', Y'), where X^_ Y' _a' + b' X~ Y~ ^?^^ * 1194. The tangent at a point P to the parabola y' = 4:ax meets the tangent at the vertex (A) in Q and the straight line a; + 4ft = in (?' : prove that the angles QAF, Q'AF are supplementary; and, generally, that the two straight lines y = mx + — , y + m (x + 4a) -I — = 0, have a similar . . / a 2a\ property with respect to the point ( — ; , - — 1 . 1195. An ordinate MF is drawn to the ellipse -^ + r3 = 1 and the tangent at F meets the axis of a; in ; from are drawn two tangents x' ■?/ 1 to the ellipse ^ +4^ = -2', prove that the parts of any tangent to the first ellipse intercepted between these two wUl be divided by the point of contact into two parts subtending equal angles at M. For the two lines to be real, F must lie between the latera recta. 1196. The straight lines A A', BB', CC are let fall from A, B, C per- pendicular to the opposite sides of the triangle ABC, and conies are described touching the sides CA, AB and the perpendiculars on them : prove that the locus of the foci is (x^ + y-){{b^ + ac)x + b{a-c)7/- {b^ + ac) {a - c)} = ac {{b" + ac) x — b {a - c) y}, reducing to X (.«* + y* — «') = 0, when c = a. (The origin is A', A'A the axis of y, and the lengths A'B, A'A, CA' are denoted by a, b, c.) In the last case trace the positions of the foci for all different positions of the centre on A'A. 198 CONIC SECTIONS, ANALYTICAL. 1197. Two conies have four-point contact at 0, their foci are S, H, S', ir respectively, and the circles OSH, OS'W are drawn : prove that the poles of SII, S'W with resjiect to the corresponding circles lie on the common chord of the two cii'cles. 1198. Two conies osculate at a and intersect in 0, the tangents at meet the curves again in b, c, the tangents at h, c meet the tangent at a in C, B and each other in A : prove that Aa, £b, Cc, meet in a point and that A, 0, a lie on one straight line. 1199. Two conies osculate at and intersect in P, any straight line drawn through P meets the conies again in Q, Q' : prove that the tangents at Q, Q' intersect in a point whose locus is a conic touching the other two at and also touching them again, and the ciu'vature at of this locus is three-foui-ths of the curvature of either of the former, and that the straight lines joining with the other two points of contact form with OP and the tangent at a hai'monic pencil. If one conic be a cii-cle and the angle POQ a right angle, OP, OQ will be parallel to the axes of the other. [The equations of the two conies being a? -f h'lf + hxy = ax, X' + h'lf + h'xy = ax, that of the locus is {(/i - h') X + hyY = 4:& {x" + hif + hxy - ax).'\ 1200. A given conic turns in one plane about (1) its centre, (2) a focus : prove that the locus of the pole of a fixed straight line with respect to the conic is (1) a circle, (2) a conic, which is a parabola when the minor axis can coincide wdth the fixed straight line. [The locus in general is to be found from the equations - . - «- cos^ 6 + lf sin^ 9 x = p cos a — q sm d + -, -^ -. — -^ , h—j} cos 6 ■¥ q sm . . . id' — 6") sin 6 cos Q •^ ^ ^ 7t - p cos ^ + g sm ^ ' the fixed point about which the conic turns being origin, the fixed straight line being x = h ; a, b the semi-axes, and p, q the co-ordinates of the centre when its axis (26) is parallel to the fixed straight line.] 1201. A conic has double contact with a given conic : prove that its real foci lie on a conic confocal with the given conic, and its excentricity is given by the equation 2-e- { a'-c b--c-) ( cr-r b--c-} where —2+-jj=^ is the given conic, (A', Y) the pole of the chord of contact, and a^ — c^, ¥ — c* the sqviares on the semi-axes of the confocal through the foci. The foci are given by the eqiiations CONIC SECTIONS, ANALYTICAL. 199 the equation of tlie conic of double contact being 1202. Tangents are drawTi to the conic ax- + by' + 2/txy= 2x from two points on tlie axis of x equidistant from the origin: prove that their four points of intei'section lie on the conic 6y* + ^I'^y = ^• 1203. On any diameter of a given ellipse is taken a point such that the tangents from it intercept on the tangent at one end of the diameter a length equal to the diameter : prove that the locus of the point is the curve \a? 67 \ce - by V by " 1204. On the diameter through any point P of a pai"abola is taken a point Q such that the tangents from Q intercept on the tangent at P a length equal to the focal chord parallel to the tangent at F : prove that the locus of ^ is the parabola ?>if + 4a (ic + 4a) = 0. 1205. Tangents are drawn to the conic ax^ + btf + lUxy = 2x from two points on the axis of x, dividing harmonically the segment whose extremities are at distances />, q fi-om the origin : the locus of their points of intersection will be the conic {p (ax + hi/ -l)-x}{q (ax + hy-l) — x} — i'^PQ ~P ~l) (^'^' + ^y' + 21ixy — 2x]. 1206. From P, P' ends of a diameter of a given conic ai'e di-awn tangents to another given concentric conic : prove that their other points of intersection lie on a fixed conic touching the four common tangents of the given conies ; so that if the two given conies be confocal the locus is a third confocal and the tangents form a parallelogram of constant perimeter. In this last case, if Q, Q' be the points of intersec- tion and tangents be drawn at P, P', Q, Q', their points of intersection will lie on a fixed circle. JC* ?/' flj' v* [The equations in the latter case are —, - r+ i^ — c = 1> -a+ ra- 1. '- ^ a" + \ b' + \ a b^ ' as* , 2/' , , . 2, „ a , (a^ + X) (¥ + X) , ^, -5 + -j-^ — = 1, where Au = ab- x + y = - . ^ ' , and the a^ + ix b- + fji ' '^ ' '^ \ ' perimeter is twice the diameter of the circle. If X be negative, one of the conies must be an hyperbola for real tangents, in which case the locus will be an hy})erbola and the difference of the sides of the paral- lelogram will be constant.] 1207. From two points 0, 0' are drawn tangents to a given conic whose centre is C : prove that if the conic drawn through the four points of contact and through 0, 0' be a circle, CO, CO' will be equally inclined to the axes and 0, 0' will be conjugate with respect to the rectangular hyperbola whose vertices are the foci of the given conic. 200 CONIC SECTIONS, ANALYTICAL. 1208. The area of the ellipse of nnuimum excentricity which can be clra-\\ni touching two given straight lines at distances h, k from their point of intersection is 77,70 7 "\ (li'' + ^" - 2M cos 0))* . irhk (h- + «■) —. sin w : (/t* + ^*+2/<^•coso))« and, if e be the minimum excentricity, e* (h'-ky 1 — 6* /i'k:' sin'oi' r„. a+b — 2h cos = 0, so ■^ V l + «ec6»' ^ that either axis of the ellipse of least excentricity is equally inclined to the axes of the two parabolas. 1211. Three points A, B, G are taken on an ellipse, the circle about ABC meets the ellipse again in P, and PP' is a diameter : prove that of all ellipses through A, B, 0, P' the given ellipse is that of least excen- tricity. CONIC SECTIONS, ANALYTICAL. 201 1212. Of all ellipses circumscribing a parallelogram, the one of least excentricity has its equal conjugate diameters parallel to the sides. 1213. The ellipse of least excentricity which can be inscribed in a given parallelogram is such that any point of contact divides a side into segments which are as the squares on the respective adjacent diagonals. 1214. Four points are such that ellipses can be drawn through them, and e is the least excentricity of any such ellipse, e the excentri- city of the hyperbola on which the centres of the ellipses lie : prove that e - 1 e - 1 also that the equal conjugate diameters of the ellipse are parallel to the asymptotes of the hyperbola. 1215. The equation of the conic of least excentricity through the four points («, 0), [a', 0), (0, h), (0, h') is X^ ^XIJ cos (J i/ cm' aa + bb' bb' a4)-KM>-°' and its axes are parallel to the asymptotes of the rectangular hyperbola through the four points. 1216. The axes of the conic which is the locus of the centres of all conies through four given points ai-e parallel to the asymptotes of the rectangular hyperbola through the four points. 1217. The equation of the director circle of the conic is (\^ — 1) (of + 2/^ + 2xy cos w) + Aa; + ky + (kx + hy — hk) cos u = 0. 1218. The equation of a conic, ha\'ing the centre of the ellipse a'y^ + b^x' = d'¥ for focus and osculating the ellipse at the point 6, is (»' + f) («' cos'^ + ¥ shi'6)' = {(rt^ - 6-") {ax cos' 6 - by sin'O) + a'by. 1219. A rectangular hyperbola has double contact with a parabola : prove that the centre of the hyperbola and the pole of the chord of con- tact will be equidistant fi*om the directrix of the parabola. 1220. A conic is drawn to touch four given straight lines, two of which are parallel : prove that its asymptotes will touch a fixed hyper- bola and that this hyperbola touches the diagonals of the quadrilateral, formed by the given lines, at their middle points. 1221. A parabola has four-pointic contact with a conic : prove that the axis of the parabola is parallel to the diameter of the conic through 2a*6* the point of contact, and that the latus rectum of the parabola is — 3- , where a, b are the semi-axes of the conic and r the central distance of the point of contact. If the conic be a rectangular hyperbola, the envelope of the dii-ectrix of the parabola is 202 CONIC SECTIONS, ANALYTICAL. 1222. The locus of the centre of a rectangular hyperbola having foui'-pointic contact with th^ ellipse a*y" + b^x' = crb' is the curve ?/ b'' \a- + 6"/ a- 1223. Tlie locus of the foci of all conies which have four-pointic contact with a given curve at a given point is a curve whose equation, referred to the normal and tangent at the given point, is of the form {mx + y) (x" + y') = axij. 1224. The excentricity e of the conic whose equation, referred to axes inclined at an angle w, is w = 0, satisfies the equation e* {a — by sin*, =, or < « - b, according as the conic is an ellipse, parabola, or hyperbola. If the contacts are on different branches, the excentricity < ~ , and the asymptotes always touch a fixed parabola. 1230. A triangle ABC circumscribes an ellipse whose foci are *S', *S" and SA -- SB = iSC : prove that S'A.S'B .S'C , SA.SB~rSG~ ' and that each angle of the triangle ABC lies between the acute angles cos -^ . [When the conic is an hyj^erbola, e* - 1 replaces 1 - e*, and one angle g \_ of the triangle will be obtuse and > tt — cos"' — — .] 1231. On every straight line can be found two real points conjugate to each other with i-esjiect to a given conic and the distance between which subtends a i-ight angle at a given point not on the straight line. 1232. Prove that the axis of a parabola, which passes through the feet of the four normals drawn to a given ellipse from a given point, will be parallel to one of the equal conjugate diameters of the ellipse. [If (A", Y) be the given point and a' if + 6V = a"h- the given ellipse, the equation of the axis of the parabola will be 1/ «^r \ 1/ h'Y \ ^^ 1233. A conic is drawn through four given points lying on two parallel straight lines : jirove that the asymptotes touch the parabola which touches the other four joining straight lines. [The equation of the conic being taken to be that of the asymptotes will be found by adding ^^ - m to the A sinister so as to give for the asymptotes the equation and for the envelope the equation {(M)'-(i"')(M)f=4^'(i.„,).. The student should observe, and account for, the factor ( — r ) -1 204. CONIC SECTIONS, ANALYTICAL. 1231. An ellipse of constant area ttp* is described liaving foui'- pointic contact with a given parabola wlioso latiis rectum is 2??^ : ])i-ove that the locus of the centre of the ellipse is an equal parabola whose — j from the vei-tex of the given parabola; also that when c = m the axes of the ellipse make with the axis of the para- bola angles I tan"' (2 tan^), where ^ is the angle which the tangent at the point of contact makes with the axis. 1235. An ellipse of constant area irc^ is described having four- pointic contact with a given ellipse whose axes are 2a, 26 : prove that the locus of its centre is an ellipse, concentric similar and similarly situate with the given ellipse, the linear ratio of the two being ) : 1. Also the desciibed ellipse will be similar to the given ellipse when the point of contact P is such that CP' -.CD'^d^ : {ab)^. \ab YII. Envelopes (of the second class). [The equation of the tangent to a parabola, in the form a gives as the condition of equal roots in m, if = iax ; and the equation of the tangent to an ellipse X V • - cos a + f sin a = 1, a written in the form gives as the condition of equal roots in z a" b' ' So in general if the equation of a line in one plane, straight or curved, involve a parameter in the second degree, it follows that througli any proposed point can be drawn two lines of the series represented by the equation. These two lines will be the tangents (rectilinear or curvilinear) from the proposed point to the curve which is the envelope of the system. If the proposed point lie on this envelope the two tan- gents will coincide, hence the equation of the envelope may be found as the condition of equal roots. Thus, "To find the envelope of a system of circles each having, for its diameter a focal chord of a given conic." CONIC SECTIONS, ANALYTICAL. 205 li LSL' be a focal chord, ASL-a, tlie mid point of LL\ and P any point on the circle, we shall have 1+ecosa' 1 — ecosa' 1— e'cos'a' 1— e"cos"a' and OF' = SO' + ST' -2S0 . SF coH FSO, whence the equation of the circle c' = r' (1 - e' cos" a) - 2cer cos a cos {6 — a), or, if tan a = A, (?'- - c") (1 + X-) - e-r" - 2ce?' (cos ^ + X sin ^) = 0, whence, as the condition for equal roots in A, {r^ (1 - e-) - c- - 2cer cos 6} (r' - c') = c'e'r^ sin* d, equivalent to (?•■ — c'- cer cos $)' = eV*, or r' (1 ± e) — cer cos 6 = c", so that the envelope is two circles, one of which degenerates into a straight line when e= 1, being the directrix of the parabola; and one degenerates into a point when e~2, being the farther vertex of the hyperbola. In general, if ^, A' be the nearer and farther vertices, the two cii-cles will have for diameters the segments ASM, A'SM', where J/'aS'= aS'J/= c the semi latus rectum. In this case every one of the system of curves has real contact in two points with the envelope, but it frequently happens that the contact becomes impossible for a jiart of the series. A method which is often the best is exemplified in the following : " To find the envelope of a chord of a conic which subtends a right angle at a given point." Move the origin to the given j^oint, and let the equation of the conic be u= ax' + hy' ■\-c + 2fij+ Igx + 2hxy = 0, and let px + qy=\ be the equation of a chord. The equation of the straight lines joining to the origin the ends of this chord will then be ax' + hf + Ihxy + 2 {gx +fy) {px + qy) + c (j)x + qyf = 0, which will be at right angles if a + b + 2 {2)g + q/) + c {}f + q') = 0. Hence the equation (« + 6) {px + qyf + 2 {]yg + qf) {px +qy) + c {p' +q')^0 represents two parallel chords of the series and involves the parameter p : q in the second degree ; whence, for the envelope, {a + bcc^ + 2gx + c) {a + by" + 2/y + c)={a + b xy +/x + gyf, or {r-hc^f- ca) {a? + if) = {gx +fy + c)^ or the envelope is a conic having a focus at the given point, directrix the polar of the given point with respect to the given conic, and excentricity y f[+f r+g^-c{a'+b)' 20G CONIC SECTIONS, ANALYTICAL. Tlie envelope of any sei'ies of linos is to he found from the condition that the equation shall give two equal A-alues of the parametei", but in all the following examples it will be found that the equation of the line can be wTitten in the form U+2XV+\' W=0, where X is the pai'ameter, so that the envelope is UV= IP. A common form is JF= U cos$ + Fsin 6 when the envelope ia, as ali'eady seen, 1236. A conic has a given focus and given length and direction of major axis : the envelope is two parabolas whose common focus is the given focus and whose common latus rectum is in the given direction and of t\vice the given length. [This is obvious geometrically.] 1237. Tlie envelope of the circles where jj, q are connected by the equation ^jg ± a (^ — ^) = 0, is the two circles a- + ^z* db Icix = 0. [It will be found best to make the ratio p : q the parameter, so that the equation will be pq {«- + y^) ± ay {p' - q') + a' (j) - qf = 0.] 1238. The envelope of the cii'cles x' + {y-p)ijif-q) = 0, where p, q are connected by the equation {p + a) {q-a) + b^ = 0, is the two circles 1239. The ellipse q^x" + p^y^ = p'q^ has its axes connected by the equation a'p^ = 9^ {p^ ~ 9^) '• pi'ove that the envelope is the two circles x^ -^y^^ 2ax = ; and, if the relation between the axes, be a'p^ = g^ (nv^p^ - n^q^), the envelope will be mV + Tiy ± Inax = 0. 1240. The envelope of the ellipse x^ + y^ - 2 {ax cos a + by sin a) ( - cos a + ^ sin a j + (a* + b^- c') (- cos a + J- sin a\ = a^ sin'^ a+b cos a — c CONIC SECTIONS, ANALYTICAL. 207 is the two confocal ellipses ^+^=1 -^+--^-=1. [The parameter taa a is involved in the second degree.] 1241. The director circle of a conic and one point of the conic ai-e given : prove tliat the envelope is a conic whose major axis is a diameter of the given cii'cle, [The equation may be taken to be '—^ + 2X -y + 72 = !> ^> ^ being parametei-s connected by the equation and a, c, given. The envelope is ^^ + -^ — j == 1 •] c c — ct 1242. Through a fixed point is drawn a straight line meeting two fixed straight lines parallel to each other in L, J/: the envelope of the circle whose diameter is LJI is a conic whose focus is and whose transverse axis has its ends on the two fixed straight Hnes. 1243. A variable tangent to a given parabola meets two fixed tangents, and another parabola is dra-woi touching the fixed tangents in these points : prove that the dii'ectrix of this last envelopes a third parabola touching straight lines, drawn at right angles to the two fixed tangents through their common point, in the points where they are met by the directrix of the given parabola. 1244. A variable tangent to a given parabola meets two fixed tangents, and on the intercepted segment as diameter a circle is de- scribed : the envelope is a conic touching the two fixed tangents in the points where they are met by the directrix of the given parabola. [The given parabola being (-] +(^j"=l, the envelope is {x (a + b cos (a) + 1/ (b + a cos w) - ab cos w}" = iahxy sin' w.] 1245. Through a point P are drawTi two cii'cles each touching two fixed equal cu-cles which touch each other at A : prove that the angle at which the two cu'cles intersect at P is 2 sec"' («), where n is the ratio of the I'adius of the cii'cle drawn through P to touch the two at A to the radius of either. If P lie on BB\ a common tangent to the two fixed cii'cles, the cii'cle through P touching the two will make with BB' an angle ^PAB. 1246. Through each point of the straight line h ~ = 1 is di-awn a chord of the ellipse a' if + b^af = a'b' bisected in the point : prove that the envelope is the parabola whose focus is the point X - y _ a' — b' al ~ bm ^"a" + m'i* ' 208 CONIC SECTIONS, ANALYTICAL. and dii'ectrix the straight liue lax + mhy =a^ + 6*. 1247. An hyperbola has a focus at the centre of a given cii'cle and its asymptotes in given directions ; prove that tangents drawn to it at the points where it meets the given circle envelope an hyperbola to which the given cii'cle is the auxiliary circle. /• [The equation of a tangent will be - = e cos 6 + cos {9 - a) where - = 1 + e cos a, a or we may write the equation - (1 + e cos a) =e cos 9 + cos {9 - a), involving only the parameter a, and giving the envelope ( ecos^j =fcos^ — ^ j +sin^^, 1248. A circle subtends the same given angle at each of two given points : prove that its envelope is an hyperbola whose foci are the two given points and whose asymptotes include an angle sujiplementary to the given one. 1249. A circle has its centre on a fixed sti'aight line and intercepts on another straight line a segment of constant length : prove that its envelope is an hyperbola of which the first straight line is the conjugate axis and the second straight line is an asymptote. 1250. A conic is drawn having its focus at A, the vertex of a given cordc, passing through a focus aS' of the given conic, and having for dii'ectrix a tangent to the given conic : prove that its envelope is a conic having its focus at the given vertex and excentricity . Also the envelopes of the minor axis, the second latus rectum, and the second, directrix are each conies similar to the given conic, and bearing to it the linear ratios 1 ± e : 2e, 1 ± e : e ; and 1 + e ± e : e, the upper or lower signs being taken accordiiTg as S is the nearer or farther focus. 1251. A triangle is inscribed in the hyj^erbola xy = c^ whose centroid is the fixed point (cm, cm~^): prove that its sides envelope the ellipse ^3m2/ + -|-8cj =ixij, which touches the asymptotes of the hyperbola; and also touches the hyperbola at the point (cm, cm~^), the curvatures at the point being as 4:1. Where the ellipse again meets the hyperbola, its tangents are parallel to the asymptotes. CONIC SECTIONS, ANALYTICAL. 209 1252. The centre and directrix of an ellipse are given : the envelope is two parabolas having their common focus at the given centre. [The ellipse may be taken '■ — h — = 1, where jp* = c' (^j — (?) J or (qx^ +py')p = c'g (p — g), involving only the parameter p : q.] 1253. One extremity of the minor axis, and the directrix, of a conic are given : the envelo]>e is a circle with centre at the given point and touching the given line, 2 2 n 4 CC ■?/ Ij?/ ft [The equation is —^ + j^ =—-, where -^ — j^ = c^ or may be written (6V + a'y^y = 4cV (a^"^ - 6*), involving only the parameter a^ : b^. This theorem is easily proved geometrically.] 1254. Find the envelope of the circle (a; - 7i)* + y* = r*, when A, r are connected by the equation h^ = 2 (r^ + a'), and a is given. 1255. A circle rolls with internal contact upon a circle of half the radius : prove that the envelope of any chord of the rolling cii'cle is a circle which reduces to a point when the chord is a diameter. 1256. A parabola rolls on an equal jiarabola, similar points being always in contact : prove that the envelojje of any straight line perpen- dicular to the axis of the moving parabola is a circle. [Also obvious geometrically.] 1257. A parabola has a given focus and intercepts on a given straight line a segment subtending a constant angle (2a) at the focus : prove that the envelope of its directrix is an ellii^se having the given point for focus, the given straight line for minor axis, and excentricity cos a. 1258. The envelope of the straight line X y sin are parameters connected by the equation , « k sin $ a h cos d sin cos <^ ' is the parabola i^ix + hj — a^y = ihkxy. [By combining the two equations we may obtain the equation in the form Jix + hj — a' - \kx - - hy = 0.] A 1259. The envelope of the conic a {a — kb) b {ka — b) is the four straight lines {x ± yY = a* - i'. w. p. 14 210 CONIC SECTIONS, ANALYTICAL. 12G0. A parabola is drawn touching a given straight line at a given point 0, also the point on the normal at 0, chords through which subtend a right angle at 0, is given : the envelope is a circle in which the two given points are ends of a diameter, and each parabola touches the envelope at the point opposite to in the parabola. [The equation of the parabola may be taken Jl—Xx+Jl+ky^ V **•] 12G1. Through each point P of a given circle is drawn a straight line PQ of given length and direction (a given vector), and a circle is described on PQ as diameter : prove that tlie envelope of the common chord of the two circles is a parabola. Tlie envelope of the circle is obviously two circles. [If a;* + y' = a* be the given cii-cle, 2h the given length, axis of y the given direction, the envelope is af + (y — Jif ={a — y\ .] 1262. Two points are taken on a given ellipse such that the normals intersect in a point lying on a fixed normal : prove that the envelope of the chord joining the two points is a parabola whose directrix passes through the centre and whose focus is the foot of the perpendicular from the centre on the tangent which is perpendicular to the given normal. [If a be the excentric angle of the foot of the given normal, the equation of one of the chords will be X II p - cos a + q , sin a = 1 , •^ a 6 where p, q are connected by the equation ^j^' + p + 5 = 0.] 1263. A chord PQ. is drawn through a fixed point (X, Y) to the X^ IT ellipse -2 + ?g = 1, the normals at P, Q meet in and from are drawn OP', OQ' also normals : prove that the envelope of P'Q' is the parabola ixyXY (xX vY ,\^ ixy. whose focus is the foot of the perpendicular from the origin on the line xX vY -^ + ^ + 1=0, and directrix the straight line a^xY+ b^yX=0. 1264. Three points are taken on a given ellipse so that their centroid is a fixed point, the straight lines joining them two and two will touch a fixed conic. [Refer the ellipse to conjugate diameters so that the fixed point is (-n-ci, 0), then taking the three points (acosa, 6 sin a), &c., we may X ?/ 1 take the chord joining two to be p- + 5'r = 9 > where {m' -l){p' + q"-) - 2mp + 1=0. The envelope is CM J 'I = 1.1 b'{l-m'j J CONIC SECTIONS, ANALYTICAL. 211 1265. A triangle inscribed in the ellipse ay+ 6V = a*i' has its centroid at the point (^, q-j : prove that its sides touch the conic = ^^,(2.r2/-xr-yX)V [The poles of the sides lie on the conio 1266. Three points are taken on a given ellipse such that the centre of perpendiculars of the triangle is a fixed point : the envelope of the chords will be a fixed conic whose asymptotes are perpendicular to the tangents from the fixed point to the given conic. [If (A', Y) be the given point, a, fi, y excentric angles of the corners of one of the triangles, then (1026) (a' + b') cos a -{a"- ¥) cos (a + /? + y) = 2aX - (a' + ¥) (cos /? + cos y), (a- + 6-) sin a - (a' - ¥) sin (a + /3 + y) = 26 F- (a' + b') (sin /? + sin y) ; square and add, and we obtain the relation (a'X' + b' Y'){p' + q') - aY - * V - 2 {a' + b') (paX +qbY) + {a' + b'Y = 0, X ?/ connecting the pai'ameters in the equation /»- + 5't = 1 of one of the sides. The equation of the envelope is ■ +x'{X'-a') + f{Y'-b') = 2xyXY.] x^ V* 1267. The envelope of a chord of the conic -3 + ^^ = 1 which sub- tends a right angle at the point (X, Z) is \{x - xy 4- {1/ - Yy-} {a^ + ¥-x^- Y^) = a^v (^ + |r_ ly ; and thence that if F be the point (A", Y), F' the second focus of this envelope, FF' is divided by either axis of the given ellipse into segments in the ratio a* + b^ : a^ - b', and that the major axis of the envelope bears to the minimum chord of the director circle through F the ratio 2ab : a* + ¥. 1268. A parabola has its focus at the focus of a given conic and touches the conic : prove that its directrix and the tangent at its vertex both envelope circles, the former one of radius equal to the major axis and with its centre at the second focus of the given conic, the latter the auxiliaiy circle of the given conic. 14—2 212 CONIC SECTIONS, ANALYTICAL. 1 2G9. Each diameter of a given parabola meets a fixed straight line, and from their common point is drawn a straight line making a given angle -with the tangent corrcsjjonding to the diameter : the envelope is a parabola, degenerating when the given angle is equal to the angle which the fixed straight line makes with the axis, 1270. From the i)oint where a diameter of a given conic meets a fixed straight line is drawn a straight line inclined at a given angle to the conjugate diameter : the envelope is a parabola. If the constant angle be made with the first diameter, the envelope is another parabola. [The equations of the conic and sti*aight line being - +■^=1, ;/- + 9f =1, a ^ a and t the tangent of the given angle, the envelopes are respectively {pa {y - tx) + qh {x + ty) + t (a^ - b^) Y = 4ab {p (x + ty) -a]{q{y- tx) -h], ■ and {hp {x + ty) -aq{y- tx)f + 4a& [q {x + ty) - ht] {;? {y - tx) + at] = 0.] 1271. A fixed point A is taken on a given circle, and a chord of the circle PQ is such that PQ = e{AP^-AQ): prove that the envelope of PQ is a circle of radius a{\~ e") touching the given circle at A, a negative value of the radius meaning external contact. 1272. A fixed point A is taken within a given circle, and a chord of the circle PQ is such that PQ = e (AP + AQ) : prove that the envelope of PQ is a circle coaxial with the given circle and the point A and whose radius is J (a- - e^c") (1 — e^), where a is the radius of the given circle and c the distance of A from its centre. 1273. One given circle U lies within another V, and PQ is a chord of V touching U, S the interior point circle coaxial with Uand V, PSP' a chord of V : prove that the envelope of QP' is a third coaxial circle such that the tangents drawn from any point of it to U, V are in the ratio a^ - c^ : a^, where a is the radius of TJ and c the distance between their centres. 1274. A circle passes through two fixed points A, B, and a tangent is drawn to it at the second point where it meets a fixed straight line through A : the envelope of this tangent is a parabola, whose focus is B, whose directrix passes through A, and whose axis makes with BA an angle double that which the fixed straight line makes with BA . [If the two points be (± a, 0), and y cos a={x — a) sin a the given straight line, the envelope is {x + ay ■\-y^={{x- a) cos 2a + y sin 2a}^] 1275. Through any point (9 on a fixed tangent to a given parabola is drawn a straight line OPTF meeting the parabola in P, P' and a given straight line in T, and 02' is a mean proportional between OP, OP' : prove that PP' either passes through a fixed point or en- velopes a parabola. CONIC SECTIOXS, ANALYTICAIi '^ - 213 127G. Tho envelope of the polar of the origin with respect to any circle circumscribing a maximum triangle inscribed in the ellipse ay + b^x- = a%^ is a \a — u / [If $ be the fourth point in which the circumscribing ciixle meets the ellipse, the equation of the polar will be COS + ^ sin = 2 — — p .] 1277. A chord of a given conic is drawn through a giA^en point, another chord is drawn conjugate to the former and equally inclined to a given dii-ection : prove that the envelope of this latter chord is a parabola. 1278. A triangle is self -conjugate to the circle {x — cf-\-'>f = b^, and two of its sides touch the circle x"^ + y' = a' : prove that if the equation of the third side be p(x—c) +Q>/ = ^, p, Q will be connected by the equation p- {a^ + ¥) + q" (a^ + b^- c") + 2^:>c + c" - 2«- = ; and find the Cartesian equation of the envelope, 1279. A triangle ABC is inscribed in the cii'cle x'+y* = a', and A is the pole of £C with respect to the circle (x — c)" +y^ =^b^ : prove that AB, AG envelope the conic (2a; -cr ^ 2/ 2a' + 26- - c' a- + b' = 1. 1280. From a fixed point are drawn tangents OP, OP' to one of a series of conies whose foci are given points S, S' : prove that (I) the envelope of the normals at P, P' is the same as the envelope of PP", (2) the cii'cle OPP' will pass throiigh another fixed point, (3) the conic OPP'SS' will pass through another fixed point. 1281. A chord of a parabola is drawn through a fixed point and on it as diameter a circle is described : prove that the envelope of the polar of the vertex with respect to this circle is a conic which degenerates when the fixed point is on the tangent at the vertex. [This conic will be a circle for the point f — ^, Oj , and a rectangular hyperbola when the point lies on the parabola 7/* = 4rt (2.t; — rt).] 1282. The centre of a given circle is C and a diameter is AB, chords AP, PB are drawn and perpendiculars let fall on these choi'd.s from a fixed point : prove that the envelope of the straight line join- ing the feet of these peri)endiculars is a conic whose directrices are AB and a parallel through 0, whose excentricity is CP : CO, and whose focus corresponding to the directrix through lies on CO, 214 CONIC SECTIONS, ANALYTICAL. 1283. From the centres A, B oi two given circles are drawn radii AP, BQ whose directions inchide a constant angle 2a : prove that the envehi])e of PQ is a conic whose excentricity is Ja^ + h*-2abcos2a where a, h are the radii and c the distance A B. [The conic is always an ellipse when one circle lies within the other, and always an hyperbola when each lies entirely without the other ; when the circles intersect, the conic is an ellijjse if 2a be greater than the angle subtended hy AB at a common point, reduces to the two com- mon points when 2a has that critical value, and is an hyperbola for any smaller numerical value. When 2a = or tt, the envelope degenerates to a point. For diflerent values of a, the foci of the envelope lie on a fixed circle of I'adius -, — ^ „ , and whose centre divides AB externallv in the ratio a' : h'J\ 1284. Two conjugate chords AB, CD of a conic are taken, P is any point on the conic, PA, PB meet CD in «, 6, is another fixed point, and Oa, Oh meet PB, PA in Q, P : prove that QP envelopes a conic which degenerates if lie on ^J5 or on the conic. 1285. The point circles coaxial with two given equal circles are S, S", a straight line parallel to SS' meets the circles in JI, W so that SH = S'H', and with foci H, H' is described a conic passing through ^S*, S' : proA^e that its directrices are fixed and that its envelope is a conic having IS, S' for foci. 1286. Two conies U, V osculate in and PP' is the remaining common tangent, PQ, P Q' are drawn tangents to F, U respectively : pi-ove that PP', QQ' and the tangent at meet in a point, and that, if from any point on PP' be drawn other tangents to U, V, the straight line joining their points of contact envelopes a conic touching both curves at and touching U, V again in Q', Q respectively. 1287. Normals PQ, PQ' are drawn to the parabola if^^iax from a point P on the cui-ve : prove that the envelope of the circle PQQ' is the curve y' (^ + f ) = ^' l^'* ~ ^)» which is the pedal of the parabola y" ■■=a{x — 2a) with respect to the origin A. [The chord QQ' always passes through a fixed point C (— 2a, 0), and if aS' be the focus of the given parabola, C, S are single foci and A a double focus of the envelope : for a point P on the loop, CP = AP + 2SP, and for a point on the sinuous branch, C-P= 2SP - AP, so that AP may be regarded as changing sign in vanishing.] CONIC SECTIONS, ANALYTICAL. 215 VIII. Areal Co-ordinates. [In this system the position of a point P with respect to three fixed points A, Jj, C not in one straight line is determined by the values of the ratios of the three triangles PBC, PC A, PAB to the triangle ABC, any one of them PBC being esteemed positive or negative accord- ing as P and A are on the same or on opposite sides of BC. These ratios being denoted hj x, y, z will always satisfy the equation x + 7j + z -\. A point is completely determined by the ratios of its areal co-ordinates {X : Y : Z) or by two equations, as Ix = mt/ = nz. It is sometimes convenient to use trilinear co-ordinates, x, y, z being then the distances of the point from the sides of the triangle of reference ABC and con- nected by the equation ax + by + cz= 2K, where a, b, c are the sides and K the area of the triangle ABC. A point would obviously be equally well determined by x, y, z being any fixed multiples of its areal or trilinear co-ordinates, a relation of the form Ax -i- By ■¥ Cz—\ always existing. In the questions under this head areal co-ordinates will generally be taken foi- granted. The general equation of a straight line is px + qy + rz=0, aid p, q, r are proportional to the perpendiculars from A, B, C on the straight line, sign of course being always regarded. When p, q, r are the actual perpendiculars, px + qy -i- rz is the perpendicular distance from the line of the point whose areal co-ordinates are x, y, z. The condition that the straight lines p^x+q^y-i-r^z=0, p^x+q^y + r s; =0 shall be pai-allel is I i, 1, 1 ! and that they may be at right angles is p^p^ sin^ A + ... + ...--^ (q^r^ + q^r^) sin ^ sin C cos .-f + ... + . . . If p, = 9', = »", , or if p^^q^ = r„ , both these equations are ti-ue. The straight line x + y + z = 'd, the line at infinity, may then be regarded as both parallel and perpendicular to every finite straight line, the fact being that the direction of the line at infinity is really indeterminate. In questions relating to four points it is convenient to take the points to be (A', ± J", ± iT), or given by the equations Ix = ± my = =ir nz ; and similarly to take the equations of four given straight lines to be p.c ^ qy^ rz — 0, The general equation of a conic is V. = ax- + by' -t- cz° + 2/yz + 2gzx -t- 2hxy = 0; 21 G CONIC SECTIONS, ANALYTICAL, and the polar of any point (X : T : Z) in ^dii ,-rdit „dit « X-r-+ Y-f- + Z-j- = 0, ax ay az or (the same thing) dU dU dU ^ dX ^ dY dZ The special fonns of this equation most useful are (1) circumscribing the triangle of reference (a, h, c = 0) fyz + gzx + lixy = ; (2) inscribed in the triangle of reference (?.r)^ + (m2/)U(ns)i=0j (3) touching the sides AB, ^C at the points B, C ]fX^ = yz ; but "when this form is used it is often better to take such a multiple of the ratio ^PBC : /^ABC for x as to reduce the equation to the form x^ = yz; (4r) to which the triangle is self-conjugate (f, g, h — 0) Ix^ + my^ + ws^ = ; here again it is often convenient to use such multiples of the triangle ratios as to give us the equation cc' + T/' + s'-O. As a general rule, when metrical results are wanted, it will be found simpler to keej) to the true areal or trilinear co-ordinates. The form (4) is probably the most generally useful. We may denote any point on such a conic by a single variable, as with the excentric angle in the case of a conic referred to its axes, which is indeed a particular case of this fonn. Thus any point on the conic ic^ + 3/^ -t- »^ = may be represented by the equations X y cos^ sin^"~ ' and we may call this the point 6. The equation of the tangent at 6 is X cos 6 + y sin 6 = iz, that of the chord through 6, cfy is xcoa^ {6+ (fi)+y sin |(^ + ^) = iz cos |(^ - ^) ; CONIC SECTIONS, ANALYTICAL. 217 and the intersection of the tangents at 6, cf> is ?/ cos ^{d + ff>) sin ^{0 + ) cos ^{6 -<}>)' The equations of any two conies may be taken to be of + y^ + :^ = 0, ax^ + hif + cs" = 0, ar ■?/* ^ their common points being ^ = = — -, . The multipliers of the areal co-oi'dinates will not all be real when the triangle of reference is real, and when the conies have two real and two impossible common points the triangle will be imaginary. Any point on the conic x' = yz may be denoted by the co-or- dinates (A : 1 : A') and called the point A. The tangent at this point is and the chord through X, /a is Xfxy - (\ + /a) a? + s = 0. Any point on the conic fyz + gzx + hxy = may be taken to be X cos^ _y sin^ ^ _ ^ the tangent at the point being 5cos'^ + ^sin*5 + J = 0: and any point on the conic (Ix)- + {my)- + {nz)- = to be Ix my cos*" Q sin' with the corresponding tangent - — 5-7; + -7— ^ — 7i~ = : but these equa- cos" Q sin^ Q tions are not often requii-ed.] 1288. The equations of the straight lines each bisecting two of the sides of the ti-iangle formed by joining the feet of the perpendiculars of the triangle ABC are x = y cot" B -vz cot' C, kc, and the perpendiculars from (x, y, z) on them are 1R sin' B sin' C {y cot^ B -k-z cot^ C - x), kc. 1289. The sides of the triangle of reference are bisected in the points -4,, B^, G^; the triangle A^B^C^ is treated in the same way and so on n times : prove that the equation of BjC^ ia 2"+' + (_])" 218 CONIC SECTIONS, ANALYTICAL. 1290. The equation of the straight line passing througli the centres of the inscribed and circumscribed cii-cles is X bin —. (cos 7? - cos C) + . - -„ (cos C-cosA)+ — — -^ (cos ^ - cos -B) = A ^ 'auxB^ ' sin 6^ and the point sin A (m + n cos A ) -. sin ^(w + n cosB) : sin C {m + ?icos C) lies on this straight line for all values of )/i : n. 1291. If X, 2/, z be perpendiculars froni any poijit on three straight lines which meet in a j>oint and make with each other angles A, B, 0, the equation Ix" + mif + nz' — will represent two straight lines whicli will be real, coincident, or imaginary, according as 711)1 sin* A + nl sin^ B + ha sin* C is negative, zero, or positive. 1292. The perpendiculai's from A, B, C on a straight line are p, q, r, and the areal co-ordinates of any point on the line are x, y, z : prove that px + qy + rz = 0, and the perjjendicular distance of any point (x, y, z) from the line is j*^ + ^2/ + *'^' 1293. The perpendiculars from A, B, C on the straight line joining the centres of the inscribed aud cii'cumscribed circles are p, q, r : prove that , _ 2 /e- (1 - co s ^ - C) (1 - cos B ) ( 1 - cos C) ^~ 3^Tcos^-2cos^-2cos6' ' and two similar equations for q, r. 1294. The straight lines bisecting the external angles at A, B, C meet the opposite sides in A', B', C, and p, q, r are the perpendicular from A, B, upon the straight line A'B'C : prove that . . _ . ^ ^R sin A sin B sin C pfiinA=q%viiB^ r sm C ~ J'5 - 2 cos A- 2 cos B-2 cos C 1295. Within a triangle ABC are taken two points 0, 0' ; AO, BO, CO meet the opposite sides in A', B', C, and the points of intersection oiO'A,B'C'; O'B, C'A'; O'C, A'B' are respectively Z>, E, F : prove that A'D, B'E, C'F will meet in a point which remains the same if 0, 0' be interchanged in the construction. . [If ,(^, : 2/i : «,) and (x^ : y^ : s^) be the points 0, 0' the point is detei*mined by the equations 1296, The perpendiculars p, q, r are let fall from A, B, C on any tangent (1) to the inscribed circle, (2) to the circumscribed circle, (3) to the nine-points' circle, and (4) to the polar circle : prove that CONIC SECTIONS, ANALYTICAL. 211) (1) p sin A 4- 5'sin^ + r sin G = 2ii sin A sin 5 sin C ; (2) p sin 2A + (? sin 25 + r sin 2C = 4^? sin A sin 5 sin C ; (3) /y sin ^ cos (i) - C) +...+ ... = 27? sin J sin /? sin C ; ,,, , , . n , ^ 2A* sin -4 sin 5 sin C (4) p tan ^4 + y tan ZJ + ?• tan 6 = -y ; J- cos -4 cos B cos C and also that in (4) p^ tan A +q^ tan 5 + r° tan C = 0. [The general relation for the tangent to any circle, whose centre is (x^ : 2/, : zj and radius p, is px^ + q>/,+rz^=(x^+!/, + z^)p.] 1297. The feet of the perpendiculars let fall from (x^ : y, : z^) on the sides of the tj-iangle of reference are A', B', C": prove that straight lines drawn through A,B,C perpendicular to B'C, C A\ A'B' respectively meet in the point a' h' C-' • 1298. A triangle LMX has its angular points on the sides of the triangle ABC, and AL, BM, CX meet in a })oiut {x^ : y^ : sj ; a straight line px + qi/ + 7-z = is drawn meeting the sides of LMN in three points which ai'e joined to the corresponding angvilar points of ABC : prove that the joining lines meet the sides of ABC in points lying on the straight line X y z {(l + r)x^ {r + p)y^ {p + q) z^ 0. 1299. The two points at which the escribed circles of the triangle of reference subtend equal angles lie on the straight line (6 - c) X cot A + (c — a) v/cot B + {a-h)z cot (7 = 0. 1300. Four straight lines form a quadrilatei'al, and from the middle points of the sides of the triangle formed by three of them })er[)endieulars are let fall on the sti-aight line which bisects the diagonals : prove that these perpendiculars are inversely pro])ortional to the perpendiculars from the angular points of the triangle on the fourth straight line. [The three being taken to form the triangle of reference, and the fourth being px + qy + rz = 0, the equation of the bisector of the diagonals will be found to be 2 {qrx + rpy + pqz) = {qr + rp+pq) {x + y + z).] 1301. A straight line meets the sides of the triangle ABC in A', B', C", the straight line joining A to the point (BB', CC) meets BC in a, and 6, c are similarly determined : prove that if any point be taken the sti'aight lines joining a, 6, c to the intei-sections of OA, OB, OG with A'B'G' will pass through a point 0' ; and that 00' will pass through a point whose position is independent of 0. 220 CONIC SECTIONS, ANALYTICAL. [If jix + qj/ + rz = be the line A'B'C, and {x^ : y, : z^ the point 0, the straight line 00' is 2)x {qy^ - re,) + qij {rz^ -;).x-,) + rz (px^ - q>/J = 0, passing through the point ^^x = q>/= rz.] 1302. The two points whose distances from A, B, C are as BC, CA, AB respectively both lie on the straiglit line joining the centroid and the centre of perpendiculars L of the triangle. [The two points are given by the equations S — b'z — c^y S — c'x — ii?z S — ary — Vx a* " b' " ? ~' where S = a'yz + Irzx + c^xy.] 1303. The distances of L from A, B, C are as cos A : cos^ : cos C : prove that the other point P whose distances from A, B, C are as cos A : cos B : cos C also lies on the straight line GL and is reciprocal to L with respect to the circumscribed circle. Also AP' BP' CP' OP 1 AL' BU CL' OL l-8cos^cos^cosC" where is the centre of the circle ABC. 1301:. Each of the straight lines X sin^ A+y sin^ B + z sin" C =0, x cos* A+y COS" B + z cos^ C = 0, is perpendicular to the straight line joining the centroid and the centre of perpendiculars. 1305. The equation of the straight line bisecting the diagonals of the quadrilateral, whose four sides are ^?a; ^qy± rz = 0, is ifx + /)^ + (nz)^ = 0, _, IW + m^b^ + n^c' + 2mnbc cos A + ... + ... a- + /3- = 4 (^-1- 111 +n)- aB = I' be sin A . / — r^ . 1343. Of all ellipses inscribed in a given cii'cle, that has the least director circle whose centre is the centre of perpendiculars, the radius of its director circle being 2B ^cos A cos Ji cos C and the area of the ellipse 27r^° ^(cos A — cos £ cos C) (cos B — cos G cos A) (cos C — cos A cos B) ; provided that of the segments into Avhich each perpendicular is divided by the centre of perpendiculars the segment next the base is less than the one to the vertex. {R denotes the radius of the circle ABC.) 1344. A conic is inscribed in the triangle J^C with its centre at the centre of perpendiculars, and 6 is the angle which its axis makes with the side BG : prove that tan 2^ cos ^4 — cos -5 cos C tan B — tan G cos^ — 2 cos -5 cos G ' also that, if A', B\ G' be its points of contact with the sides, the centre of perpendiculars of the triangle AB'G' will lie on the conic and the tangent there will be parallel to BG. 1345. If {x :y :z) be the centre of an inscribed conic, the sum of the squares on its semi-axes is ^R' cos A cos B cos G +;/ ; where p is the distance of its centre from the centre of perpendiculars. • 1346. The centre of a conic is the point (;« •.y.z), its excentricity is e, the radius of its dii-ector cii'cle p, and K denotes twice the area of the triangle of reference ABG : prove that (1) for a conic inscribed in the triangle ABG (e' - 2)' _ {x- cot ^ + / cot i? + z" cot Gf 1 - e" ~ {x + y-^-z){-x + y + z) {x-y + z){x-\-y-z)' 2 TT ^ ^0* A-^y^ cot B + s? cot G ^ {x + y + zy CONIC SECTIONS, ANALYTICAL. 227 (2) for a cu'cumscribed conic {e'-2y _ {a'yz+ ... + ... -x'hc cos A -...- ...y 1 - e* ~ K" {x + ij + z){- X ■\- y + z) (x - y + z) (x ■{■ y - z)^ 3 _ ^xijz {u?yz + . . . - x^hc cos ^ -...-... ) ^ ~ {x^y + z)-{-x+y-Vz)i(x-yvz) [x + y-z)' (3) for a conic to which the triangle is self-conjugate, (e* - 2)^ _ {a^yz + Vzx + c^xyf 1 — e" K'xyz (^x + y + z) ' 2 a'yz + b'zx + c'xy ^ ~~ i^ + y + ^T 1347. Two similar conies have a common centre, one is inscribed and the other circumscribed to the triangle of reference : prove that their common centre lies either on the circumscribed cii'cle or on the circle of which the centroid and centre of perpendiculars are ends of a diameter. 1348. The equation of an asymptote of the conic yz = kx^ is 2fjLkx - ky — ^'z = 0, where /x is given by the equation f^ +fx + k-0 ; and the asymptotes, for different values of k, envelope the parabola (y - zf + ix (x + y + z)=(y + z + 2a;)- - iyz = 0. 1349. A conic is inscribed in the triangle ABC and its centre lies on a fixed straight line parallel to £C (y + z = kx) : prove that its asymp- totes enveloi^e the conic {k-\){x + y + zY=Uyz. 1 350. The radius of curvature of the conic x^ = kyz at the point B kR sin" C IS -. 7—; fj. sui A sm B 1351. Prove that the equation x' = iyz represents a parabola ; and that the tangential equation of the same parabola is qr = p'. 1352. The tangents to a given conic at two fixed points A, B meet in C, and the tangent at any point P meets CA, CB in B\ A' respec- tively : find the locus of tlie point of intersection of AA', BU ; and, if AB, BP meet CB, CA in a, b respectively, find the envelope of ab. [Taking the original conic to be i? = kxy, the locus and envelope are respectively iz^^kxy, z^=\kxy.^ 1.3—2 228 CONIC SECTIONS, ANALYTICAL. 1353. Tlie tangents to a given conic at A, B meet in C and a, b are two other fixed jmints on the conic ; a tangent to the conic meets CA, CB in B' , A' : prove that the locus of the intei'section of aB', bA' is a conic passing through a, b and the intersections of Ca, Bb, and of Cb, Aa. [Taking the given conic to be z" = xy, and {x^ : y^ : z^), {x^ : y^ : zj) the points rt, b, the locus is y^i yj\x^ yj \x, ^./Vy, ^J ■' 1354. The tangents to a given conic at ^, ^ meet in C ; /* is any other point on the conic and AF, BP meet CB, CA ux a,b : prove that the triangle abP is self-conjugate to another fixed conic touching the former at A, B. [Taking the given conic to be z^ = xy and (x^ : ?/, : zj the point P, the fixed conic is z' = '2xy, which is equivalent to U, «,/ \y, zj ~ U, ^ y, zj •-■ 1355. Prove that the locus of the foci of the conic x° = kyz, for different values of k, is the circular cubic X {if — z~) + 2yz {y cos B-z cos C) = 0, trilinear co-ordinates being used. 1356. Three tangents touch a conic va. A, B, C and form a triangle abc; BC, CA, AB meet a fourth tangent in a, P, y, and Aa, B(3, Cy meet the conic again in A', B', C : prove that B'C, be ; CA', ca ; A'B', ab intersect in points on the fourth tangent, and aA', bB' , cC meet in a point. If a family of conies be inscribed in the quadrilateral forme 1 by the four tangents, the centre of homology of the two triangles abc, A' B'C' lies on the cun^e {2oxf + {qyf + {rzf = 0, X, y, z and i^x + qy + rz being the four tangents. 1357. Thi'ee points A, B, C are taken on a conic and the tangents- form a triangle abc, a fourth point {X : Y : Z) is taken on the conic and Oa, Ob, Oc meet the tangents at A, B, C in a, P, y from which points other tangents are drawn forming a triangle A' B'C : prove that A A', BB', CC will meet in and that the axis of the two triangles ABC, A' B'C envelopes the curve [If the points be fixed and the conic variable, the straight lines B'C , CA', A'B' each envelope a fixed tricusp, two of the cusps (for B'C) being B, C and the tangents OB, OC, and the thii-d cusp, lying on OA and having OA for tangent, being at a point 0' such that, '\i AO meet BC'va.I),{AaOI)\=^\ CONIC SECTIONS, ANALYTICAL. 229 1358. In the la^t question, if ABC and the conic be fixed and the point vary, the axis envelopes the ciu-ve (^x)~^ + (»?y)~^ + (ns)"^ = 0, the conic being {lx)~^ + (^y)"' + (*^^)~' = ^• 1359. Prove that the general equation of a conic, with respect to which the conic Ix' + my' + nz' = is its own polar reciprocal, is (?«' + my' + nz') [p'mn + q-nl + r^lm) = 2lmn {jpx + qy + ?*z)*. 1360. Every hj-perbola is its own polar reciprocal with respect to a parabola having double contact with it at the ends of a chord which touches the conjugate hyperbola. 1361. An ellipse is its own reciprocal polar with respect to a rectangular hyperbola which has double contact with it at the ends of a chord touching the hyperbola which is confocal with the ellipse and has its asjTiiptotes along the equal conjugate diameter of the ellipse. 1362. A pai-abola is its own recijoi-ocal with respect to any rect- angular hj-perbola which has double contact with it at the ends of a chord touching the other parabola wliich has the same latus rectum. 1363. An hyperbola is its owti reciprocal with respect to either circle which touches both branches of the hyperbola and intercepts on the transverse axis a length equal to the conjugate axis. 1364. Each of two conies U, V is its own i-eciprocal with respect to the other, prove that they mvist have double contact and that each is its own reciprocal with respect to any conic which has double contact with both U and V provided the contacts are different. 1365. Through the fourth common point of the two conies lyz + mzx + nxy = 0, I'yz + m'zx + n'xy = 0, is drawn a straight line meeting the conies again m P, Q : prove that the locus of the intersection of the tangents at P, Q is the curve (tricusp quartic) [W {inn — m'n) yz^ + {mm {nV — n'l) sx-}- + [nn {Im' — I'm) xy]^ = 0. 1366. From any point on the fourth common tangent to the two conies X- + y'' + z'^ = 0, (Ix)''' + (myY + (nz)- = are drawn two other tan- gents to the conies : prove that the envelope of the straight line joining their points of contact is the cm-ye {I (m - ?i) x}^ + {m {n - I) y)^ 4- {n (l - m) z]^ = 0. 1367. The sides of the triangle ABC touch a conic U; 0, (9,, 0^, 0^ are the centres of the inscribed and escribed cucles of ABC, a conic J' is described through B, C, 0, 0,, and one focus of U, and a conic ]V through B, C, 0„, 0^ and the same focus of U : prove that the fourth common point of T, H' Avill be the conjugate focus of U ; also that, if the conic W be fixed, the major axis of the conic U will always pass through a fixed point on the internal bisector of the angle A, and if the conic r be fixed, through a fixed point on the external bisector of the angle A. 230 CONIC SECTIONS, ANALYTICAL. 1368. Four conies are described with respect to each of which three of the four straight lines px^qij ^rz = ^ form a self-conjugate triangle and the fo\irth is the polar of a 6xed ])oint (A' -.Y-.Z): prove that all four will have two common tangents, meeting in {X : Y : Z), whose equation is yZ-zY zX-xZ xY-yX 1369. The triangle ABC is self-conjugate to a given conic and on the tangent to the conic at any point F is taken a point Q such that the pencil Q [ABCP] is constant : prove that the locus of ^ is a quartic having nodes Sit A, £, G and touching the conic in four points. [If the conic be Ix' + my^ + nz" = 0, and k the given anharmonic ratio, the locus of Q is , , + — ., + — „ = 0.1 Ix" my nz -" 1370. Two given conies intersect in A, B, C, JD and from any point on AB are drawn tangents OP, OQ to one conic, Ojj, Oq to the other : prove that Fp, Qq intersect in one fixed jioint and Fq, Qp in another ; that these points remain the same if A, B be interchanged with C, D ; and that the six such points corresponding to all the common chords lie on four straight lines. [Taking the conies to be x" ■{■ y^ -\- z^ = 0, ax" + hf -t- cz^ = 0, the six points are a; = 0, h{c- a)y^ = c[a — h) x", Arc] 1371. A triangle A'B'C is drawn similar to the triangle of reference ABC and with its sides passing respectively through A, B, C ; another simDar triangle abc is drawn with its sides ])arallel to those of the former and its angular points upon the sides BC, CA, AB respectively : prove that the triangle ABC is a mean proportional between the triangles abc, A'B'C ; and that the straight lines A'a, B'b, C'c meet in the point X y sin 2^ + sin 2 (^ - 6) + sin 2 (C + 0) sin 2B + sin 2(C- 6) + sin 2(^ + 6) z "sin2C + sin2 (^-e)+sin2 (5 + ^)' where 6 is the angle between the directions of BC, B'C. Prove also that the locus of this point is a conic having an axis along the straight line joining the centroid and the centre of perpendiculars. [The equation of the conic is u° + v^ = lo^, where rt = a; cos 2A+y cos 2B + z cos 2C + 4 (a; + y + 0) cos A cos B cos C, v=x (sin 2B - sin 2C) + y (sin 2(7 - sin 1A) + z (sin 2 A - sin IB), w=2{xco%2A +y cos 25 + s cos 1C) + x + y-^z, of which u and w are parallel to each other and perpendicular to v.] CONIC SECTIONS, ANALYTICAL. 231 1372. Foiir fixed tangents are drawn to a given conic forming a qiiadriliiteral whose diagonals are aa\ hh\ cc' ; tliree other conies are drawn osculating the given conic at the same point F and passing through a, a; b, b'; and c, c' respectively : prove that the tangents at a, a, b, h', c, c' all meet in one point ; that the locus of this point as F moves is the envelope of tlie straight line joining it to F and is a fourth class sextic having two cusps on each diagonal and touching the given conic at the points of contact of the four tangents. [If the four tangents be px' ±qy ±rz-0 and the conic Im" + my- + nz" = 0, the locus is {2i'vinx'f + {q* nhf)^ + {rHmz')^ = 0.] 1373. A conic is drawn through B, C osculating in F the conic (Ix)^- + {myY + {nz)^ = : prove that the locus of the pole of BC with respect to this conic is the cubic {Ix + imy + inzY = 27lx [my - nz)' : also if A' be the point of contact of BC and another conic be drawn also oscidating the given conic in F but passing through A, A', the tangents Sit B, C, A, A' will meet in a point. 1374. A parabola touches the sides of the triangle ABC and the straight line B'C joining the feet of the perpendiculars from B, C on the opposite sides : prove that its focus lies on the straight line joining A to the intersection of BC, B'C. 1375. A triangle is self-conjugate to a parabola: prove that the straight lines each bisecting two of the sides are tangents to the parabola, and thence that the focus lies on the nine-points' ciixle and the directrix passes through the centre of perpendiculars. 137G. A triangle is self-conjugate to a pai'abola and the focus of the parabola lies on the circle circumscribing the triangle : jiroA'e that the poles of the sides of the triangle with respect to the cii'cle lie on the I)arabola. 1377. A rectangular hyperbola is inscribed in the triangle ^^C : jn-ove that the locus of the pole of the straight line which bisects the two sides AB, AG is the circle X- {a- + b- + C-) + {y- + 2xy) {cc + 1'- c") + (;:■ -I- 2zx) (a' -b^ + c'') = 0; that tliis circle is equal to the polar circle of the triangle and its centre is the point of the circle ABC opposite to A. 232 CONIC SECTIONS, ANALYTICAL. 1378. A conic is clra-svn touching the four straight lines prove that its equation is Ix^ + m]f + nz'^ = 0, where I, tn, n are connected by the equation p'mn + q^nl + rl7n = 0, and investigate the species of this conic with respect to the position of its centre on its rectilinear locus. [If the roiddle points of the internal diagonals of the convex quad- rangle be L, 2[, and that of the external diagonal be N, L, M, N being in order, the conic is an hyperbola when the centre lies between - oo and L, an ellipse from L to M, an hyperbola from M to iY, and an ellipse from N to + co . Hence there are two true minimum excen- tricities.] 1379. A conic is drawn touching the four straight lines 2)X d=qy :A= rz = 0, ]>rove that any two straight lines p^x + q^i/.+ r^z = 0, p^jCc + q,^T/ + r_^z = 0, will be conjiigate with respect to this conic if p' (f r^ 1380. The straight lines 2'>i^ + \^ = 0, ^;.,.r + qjj + r^s = will be conjugate with respect to all parabolas inscribed in the triangle of reference if [A particular case of the last with different notation.] 1381. The two points {x^ : y, : s,), {x^ : y^ : z^^ will be conjugate with respect to any conic through the four jjoints {X : ± F : ± ^), if ^1^2 _ yjl2 _ £1^2 1382. A triangle is self-conjugate to a rectangular hyperbola : pi-ove that the foci of any conic inscribed in the triangle will be conjugate with respect to the hyperbola. [A particular case of the last.] 1383. The locus of the foci of all conies touching the four straight lines px ^qy^rz = ^ is the cubic whose equation is, if (?, m, n) E V' sin^ A + m^ sin^ h + n' %\vl C — '2mn sin 5 sin (7 cos A — 2nl sin C sin A cos B — 2lm sin A sin B cos C, (I, m, n) (- I, m, n) {I, — ni, n) (l, ???., - n) Ix + my + nz — Ix + my + nz Ix — my + nz Ix + my — nz ' and this equation may be reduced to the form {x + y + z) {rx^ cot A + m^y- cot B + n'z^ cot C) he sin A = {I'x + m'y + n^z) {a-yz + Vzx + c"xy). CONIC SECTIONS, ANALYTICAL. 23'? 1384. Of all the conies inscribed in a given quadrilateral there are only two which have an axis along the straight line which is the locus of the centres of the conies, and the two conies will be real and the axis the major axis when the centre of perpendiculars of the triangle formed by the diagonals of the quadrilateral is on the opposite sides of the locus of centres to the three corners of the triangle. 1385. The equations determining the foci of the conic Ix' + my^ + nsr = are a'X I m nj b'\ m n I) c' { n I vi) ' 1386. One directrix of the conic Im^ + my^ + nz' = passes througii A : prove that mn = I (m cot" C + n cot' B) ; and that the conjugate focus lies on the straight line joining the feet of the perpendiculars from B, C on the opposite sides. 1387. A conic is described to which the ti-iangle ABC is self-conju- gate and its centre lies on the straight line bisecting two of the sides of the triangle formed by joining the feet of the perpendiculars of the tri- angle ABC, prove that one of its foci is a fixed point. [It is at the foot of the perpendicular from A on BC] 1388. Given a point on a conic and a triangle ^l^C'self-conjuga to to the conic; AO, BO, CO meet the opjjosite sides in three points and the straight lines joining these two and two meet the coiTesponding sides in A', B', C": prove that the intersections of BB', CC" ; CC", AA' j and A A', BB' also lie on the conic. 1389. Any tangent to a conic meets the sides of the triangle ABC •which is self-conjugate to the conic in a, 6, c; the straight line joining A to the intersection of Bb, Cc meets BC in A', and B', Care similarly determined : prove that B'C, C'A', A'B' are also tangents to the conic. 1390. Two conies U, Fhave double contact and from a point on the chord of contact are drawn tangents OP, OQ ; Op, Oq ; another conic W is drawn througii ]), q touching OP, OQ : prove that the tangent to W at any point where it meets U will touch V. 1391. A conic passes through four given points: prove tliat the locus of tangents drawn to it from a given point is in general a cubic, which degenerates into a conic if the point be in the same straight line ■with two of the former and in that case the locus passes through the other two points and tlie tangents to it at them pass through the fifth point. 234 CONIC SECTIONS, ANALYTICAL. 1392. A conic is inseribod in a given quadrilateral and tangents are drawn to it from a given jioint : prove tliat the locus of their points of contact is a cubic passing tlirough the ends of the diagonals of the quadrilateral, through the given ]»()int, and through any point where the straight line joining the given point to the intersection of two diagonals meets the third : there is a node at the given point the tangents at which form a harmonic pencil with the straight Hues to the ends of any diagonal. [The node is a crunode when the given point lies within the convex quadrangle or in any of the portions of space vertically opposite any angle of the convex quadrangle.] 1393. Prove that, if ;5.x- + qy+rz = be the equation of the axis of a parabola inscribed in the triangle AJBC, or the asymptote of a rect- angular hyperbola to which the triangle is self-conjugate, a^p }fq c~r q -r r —p p- q 1394. A parabola is inscribed in the triangle ABC and S is its focus (a point on the circle ABC), the axis meets the circle ABC again in 0: prove that, if with centre a rectangular hyperbola be described to which the tx-iangle is self-conjugate, one of its asymptotes will coincide with OS. 1395. The conies passing throvigh two given points and touching three given straight lines are either all four real or all four impossible. [If the three given straight lines form the triangle of reference and (x^ : ?/j : z^) (x^ : y^ : s^) be the two given points, the conies will be where (Ix)-^ + {myY + {nzY - 0, Til yi^2 + 2/2-. =^ 2 Jy^y^z^z„_ z^x^ + z^x^ ± 2 Jz^z^x^x,^ x^y^ + x^j^ ± 2 Jx^x^y^y^ ' in which ambiguities an odd number of negative signs must be taken. If the points of contact with BG he A^, A^, A^, A^ these can always be taken so that BA^ . BA^ :CA^.CA^ = BA^. BA^ : CA^ . CA^J] 139G. The locus of the foci of a rectangular hyperbola, to which the triangle ABC is self-conjugate, is the tricyclic sextic x{U-x{x+y + z)bcco%A\ where U E a^yz + h^zx + c~xy. + ... = 0, CONIC SECTIONS, ANALYTICAL. 2^)0 1397. A triangle circumscribes the conic x' + y^ + z^ = and two of its angular points lie on the conic Ix^ + ?»y^ + nz' = : pi-ove that the locus of the third angular point is the conic {- I + m + y?)" (l-m + nY {1+ m- n) that this will coincide with the second if Z- 4- m^ + n- = ; and that the three conies have always four common tangents, 1398. The angular points of a triangle lie on the conic Ix' + mif + nz' = and two of its sides touch the conic x' + 'if + z^ = ', prove that the cn\e- lope of the third side is the conic l{-l+m + n)- X- + m{l- m + n)* y^ + n {I + m - n)' s* = ; that this will coincide with the second if l^ + 7n'^ + iv^ = 0, and that the three conies have always four common points. 1399. A triangle is self-conjugate to the conic x" + ?/^ + c^ = and two of its angular points lie on the conic Ix^ + inif + w:^ = ; prove that the locus of the thii-d angular point is the conic {y)i + 11) X- + (rt + y" + (^ + wi) s° = ; that this will coincide with the second if ^ + «i + ?i = ; and that the three have always four common points. Also prove that the straight line joining the two angular points will touch the conic I"" ?/^ i^ m + 7i n+ 1 1 + m 1400. A triangle is self-conjugate to the conic Ix^ + 7n)f + nz' = and two of its sides touch the conic x^ + y' + z'' = ; prove that the enve- lope of the third side is the conic m + n n+ 1 l + m that this will coincide with the second \i I + m-\-n = Q ; and that the three have always four common tangents. Also prove that the locus of the intersection of the two sides is the conic I (m + 7i) a;* + m{n + I) if + 7i (^ -f- m) 'J = 0. SoG CONIC SECTIONS, ANALYTICAL. IX. Anharmonic Ratio. llomogroj)hic Pencils and Ranges. Invo- lution. [The anharmonic ratio of four points A, B, C, D in one straight line, deuoted by \ABCD], means the ratio -ttt. : 77T^^ or -7—; — ^^: the order ■^ * '' BB CD' AC.BD' of the letters marking the direction of measui-ement of the segments and segments measured in opposite directions being aflected with opposite signs. So, if A, B, C, JJ be any four points in a plane and F any other point in the same plane, P \ABCD\ denotes —. — rv.^-^ — 7. i, 7^> J ^ 1 5 I J am A PC am BPJy the same rules being observed as to direction of measurement and sign for the angles in this expression as for the segments in the other. Either of these ratios is called harmonic when its value is — 1 ; in which ease ^Z? is the harmonic mean between AB and AC, and BA is the harmonic mean between BB and BC. The anharmonic ratio of four points or four straight lines can never be equal to 1 ; as that value leads immediately to the result AD . BC = or sin ^PZ> sin ^PC — making two of the points or two of the lines coincident. A series of points on a straight line is called a range, and a series of straight lines through a point is called a pencil, the straight line or point being the axis or vertex of the range or pencil respectively. If two ranges abed ... , a'h'c'd' ... be so connected that each point a of the first determines one point a' of the second and each point a of the second determines one point a of the first, the ranges are homographic. So also two pencils, or a range and a pencil, may be homographic ; and in all such cases the anharmonic ratio of any range or pencil is equal to the anharmonic ratio of the corresponding range or pencil in any homographic system. If four fixed points A, B, C, D be taken on any conic and P be any other 2)oint on the same conic, P {A BCD} is constant for all positions of P and is harmonic when BC, AD are conjugates with respect to the conic. Also, if the tangent at P meet the tangents at A., B, C, D in the points a, b, c, d, the range [abed] is constant and equal to the former pencil. A range of points on any straight line is homographic with the pencil formed by their polars with respect to any conic. If the equations of four straight lines can be put in the form u = fx^v, u = fx^v, u = ix^v, n = fi^v, the anharmonic ratio of the pencil formed by them or of the range in which any straight line meets them is (/^ l-/^2)(/^3-/^. ) _ (Mi-/^3)(/*2-/^4)' A very great number of loci and envelopes can be determined imme- diately from the following theorems : (1) The locus of the intersection of corresponding rays of two homographic pencils is a conic passing through the vertices of the pencils (0, 0') and the tangents at 0, 0' ai-e the C'(JNIC SECTIONS, analyticaK^ 237 rays correspoutling to O'O, 00' respectively : (2) The envelo[)c of a straight line whicli joins corresjionding points of two homographic ranges is a conic touching the axes of the two ranges in the i:)oints which cor- res])ond to the common point of the axes. A series of pairs of points on a straight line is said to be in involu- tion when there exist two fixed points {/, f) on the line such that, a, a heing any pair, {ftjf'a'] = — 1 . The points f, f are called the foci or double points of the range, since when a is at f, a! Avill also be at f. The middle point (C) oi Jf is called the centre and Ga . Ca' — Cf^. The foci may be either both real or both impossible, but the centre is always real ; and when two corresponding points are on the same side of the centre the foci are real. Similarly a series of pairs of straight lines, or rays, drawn from a point is in involution when there exist two fixed rays forming with any pair of corresponding rays a harmonic pencil in which the two fixed rays are Conjugate. This paii* of fixed straight lines is called the focal lines or double rays. Any straight line is divided in involution by the six straight lines joining the points of a quadrangle, and any two corresponding points of the involution will lie on a conic round the quadrangle. The pencil formed by joining any point to the six points of intersec- tion of the sides of a quadrilateral is in involution and any pair of cor- responding rays touch a conic inscribed in the quadrilateral. The locus of the intersection of two tangents to a given conic drawn from corresponding points of an involution is the conic which passes through the double points of the involution and through the points of contact of tangents to the given conic drawn from the double points. The envelope of a chord of a given conic whose ends lie on corresponding rays of a pencil in involution is a conic touching the double rays of the involution and also touching the tangents drawn to the given conic at the points where the double rays meet it. These two theorems will be found to include as particular cases many well-known loci and envelopes. It may be mentioned that a large proportion of the questions which are given under this head might equally well have appeared in the next division : Reciprocal Polars and Pi'ojection.] 1401. Two fixed straight lines meet in A ; B, C, D are three fixed points on another straight line through A ; any straight line through D meets the two former straight lines in h, c and Bb, Cc meet in P, Be, Cb in Q : prove that the loci of P, Q are straight lines through A which make with the two former a pencil whose ratio is {{ABCD\y. 1402. On a .straight line are taken points 0, A, B, C, A', B', C such that {OABC} = {OAB'C] = {OA'BC} = {OA'B'C} ; prove that each = {OA'B'C'\, and that the ranges {OBCA'}, {OCAB'}, and {OABO'\ will each be harmonic. 238 CONIC SECTIONS, ANALYTICAL. 1403. Two fixed straight lines intersect in a point on the side BC of a triangle ABC \ any point P being taken on AO the straight lines BBy FC meet the two fixed straight lines in B^, B^, C, , G^ respec- tively : prove that Bfi^ and B^C\ pass each tlu'ough a fixed point on BC. 1404:. From a fixed point are let fall perpendiculars on conjugate i-ays of a pencil in involution : prove that the straight Ime through the feet of these perpendiculars passes through a fixed point. 1405. Two conjugate points a, a of a range in involution being joined to a fixed point 0, straight lines drawn through a, a' at right angles to aO, a'O meet in a point which lies on a fixed straight line. 140G, Chords are drawn thi'ough a fixed point of a conic equally inclined to a given direction ; prove that the straight line joining theii- extremities passes through a fixed point. 1407. Through a given point are drawn chords PP', QQ' of a given conic so as both to touch a confocal conic : prove that the points of intersection of PQ, P'Q', and of PQ', P'Q are fixed. 1408. A cii'cle is described having for ends of a diameter two conju- gate points of a pencil in involution : prove that this cii'cle will be cut orthogonally by any circle through the two double points of the range. 1409. Two triangles are formed each by two tangents to a conic and their chord of contact; prove that their angvilar points lie on one conic. 1410. Four points A, B, C, D being taken on a conic, any straight line through D meets the conic again in D' and the sides of the triangle ABC in A', B', C : prove that the range {A'B'C'JD'} is equal to the pencil {A BCD] at any point on the conic. 1411. The sides of a triangle ABO touch a conic in the points A', B', C and the tangent at any point meets the sides of the two triangles in a, b, c, a', b', c' respectively : prove that {Oabc} = {Oa'b'c']. 1412. Four chords of a conic are drawn through a point, and two other conies are drawn through the point, one passing through four extremities of the four chords and the other through the other four extremities : prove that these conies Avill touch each other at the point of concourse. [Also very easily proved by projection.] 1413. Through a given point is drawn any straight line meeting a given conic in Q, Q', and a point P is taken on this line such that the range {OQQ'P} is constant: prove that the locus of P is an arc of a conic having double contact with the given conic. 1414. Given two points ^, ^ of a given conic; the envelope of a chord PQ such that the pencil {APQB} at any point of the conic has a given value is a conic touching the given conic at A , B. CONIC SECTIONS, ANALYTICAL. 230 1415. Through a fixed point is drawn any straight line meeting two fixed straight lines in Q, li respectively ; E, F are two other fixed points : prove tliat the locus of the point of intersection of QE^ RF is a conic passing through E^ F, and the common point of the two fixed straight lines. 1416. Three fixed j^oints A, B, C being taken on a given conic, two other points P, F' are taken on the conic such that the ])eucils [PABC], {P'ABC] are equal at any point on the conic : prove that PJ'', CA, and the tangent at B, meet in a point. 1417. Six fixed points A, B, C, A', B', C" are taken on a given conic such that, at any point on the conic, {B'ABC} = {BA'B'C} ; and P, P' arc two otlier ])oints on the conic such that, at any point on the conic, {PABC]^{FA'BC']: prove that PF, AA', BB', CC all intersect in one point. [Of course the six points subtend a pencil in involution at any point of the conic, and conjugate rays pass through F^ F.] 1418. A conic passes through two given points A, A', and touches a given conic at a given point ; prove that their other common chord will pass through a fixed point B on A A', and that if the straight line through A, A' meet the given conic in C, C" and the tangent at in F, the points A, A' ; B, B' ; C, C will be in involution. 1419. Two chords AB, CD of a conic being conjugate, the angle ACB is a right angle, and any chord DP through D meets AB in Q ; prove that the angle PCQ is bisected by CA or CB, 1420. Three fixed points A, B, C being taken on a conic, and P being any other point on the conic, through P is dra^v^l a straight line meeting the sides of the triangle ABC in points a, h, c such that {Pabc} has a given value : prove that the straight Hue passes through a fixed point on the conic such that the pencil [OABC] at any point of the conic has the same given value. 1421. Pi'ove that the two points in which, a given straight line meets any conic through four given points are conjugate with respect to the conic which is the locus of the pole of the given straight line with respect to the system of conies. 1422. Three fixed tangents to a given conic form a triangle ABC, and on the tangent at any point P is taken a point such that the pencil {PABC] has a given value : prove that the locus of is a straight line which touches the conic. 1423. Two conies cii-cumscribe a triangle ABC, any straight line through -.1 meets them again in F, Q : pi'ove that the tangents at P, Q divide BC in a constant anharmouic ratio. 1424. Conies are described touching four given straight lines, of which two meet in A, and the other two in B ; on the two meeting 240 CONIC SECTIONS, ANALYTICAL. ill A are taken two lixecl points C, D, and the tangents drawn from them to one of tlie conies meet in F : ])rove tliat the locus of F is a straight line through B which forms with BC, BD and one of the tangents through B a pencil equal to that formed by BA^ BC, BD and the other tangent through B. 1425. The diagonals of a given quadrilateral are A A', BB', CC, and on them are taken points a, a ; h, b' ; c, c , so tliat each diagonal is divided harmonically : prove that if a, 6, c be collinear, so also will a , U, c', and their common point will be the point where either of them is touched by a conic inscribed in the quadrilatex'al. [This is also a good example of the use of Projection.] 1426. Two fixed tangents OA, OB are drawn to a given conic and a fixed point C taken on AB ; through C is drawn a straight line meeting the fixed tangents in A', B' : prove that the remaining tangents from A', B' intersect in a point whose locus is a fixed straight line through 0. 1427. Two fixed points A, B are taken on a given conic and a fixed straight line drawn conjugate to AB \ any point F being taken on this last straight line chords AFQ, BFQ' are drawn ; prove that QQ' passes through a fixed point on AB. 1428. A conic is inscribed in a triangle ABC, the polar of A meets BC in a, and aF is drawn to touch the conic ; prove that if fi'om any point Q on aF another tangent be drawn, this tangent and QA will form with QB, QC a harmonic pencil. 1429. Two chords AO, BC of a conic are conjugate, any chord OP meets the sides of the triangle ABC in a, h, c : prove that the range {abcF] is harmonic. 1430. Two fixed points A, B are taken on a given conic, F is any other point on the conic : prove that the envelope of the straight line joining the points where FA, FB meet two fixed tangents to the conic is a conic which touches at ^, ^ the straight lines joining these points to the points of contact of the corresponding fixed tangents, and which also touches the two fixed tangents. 1431. One diagonal of a quach-ilateral circumscribing a conic is AA' : prove that another conic can be described touching two of the sides of the quadrilateral in A, A' and passing through the points of contact of the other two. 1432. On the normal to an ellipse at a point P are taken two points 0, 0' such that the rectangle FO . FO' is equal to that under the focal distances of F, and from these points tangents are drawn to the ellipse : prove that their points of intersection lie on the circle whose diameter is QQ', where Q, Q' are the points in which the tangent at F meets the director circle. CONIC SECTIONS, ANALYTICAL. 241 1 433. A range of points in involution lie on a fixed straight line and a homographic system on another fixed straight line ; a, a' are con- jugate points of the former and ^, -4' the corresponding points on the latter: prove that the locus of the intersection of aA, a' A' or of aA', a' A is a sti-aight line. 1434. A pencil in involution has a point for its vertex, and a homographic pencil is drawn from another point 0', corresponding rays of the two intersect in F and the conjugate rays in F' : prove that FF" passes through a fixed point. 1 435. In any conic the tangent at A meets the tangents at C, -S in b, c which are joined to a point by straight lines meeting BC in 6', c' : prove that AC, cb' intersect on the polar of 0, as also AB, be'. 1436. The triangle ABC is self-conjugate to a given conic U, a conic V is inscribed in the triangle and its points of contact are A', B', C: prove that, if B'C" touch (/, so also will C'A\ A'B', and the straight line in which lie the points (BC, B'C), (CA, C'A'), and {AB, A'B'). 1437. A variable tangent to a conic meets two fixed tangents in F, Q ; A, B are two fixed points : prove that the locus of the intersection oi AF, BQ is a conic, passing through A, B, and the intersections of (OA, Bb) and (OB, Aa); Oa, Ob being the fixed tangents. 1438. Parallel tangents are drawn to a given conic and the point where one meets a given tangent is joined to the point where the other meets another given tangent : prove that the envelope of the joining line is a conic to which the two given tangents are asymptotes. 1439. Through a fixed point of an hyperbola is di*awn a straight line parallel to an asymptote, and on it are taken two points F, F" such that the rectangle OF . OF is constant ; the locus of the intersection of tangents drawn from F, F' is two fixed straight lines passing through the common point of the tangent at and the asymptote, and forming with them an harmonic pencil. 1440. Four fixed points A, B, C, D are taken on a given conic; thi'ough D is drawn any straight line meeting the conic again in F and the sides of the triangle ABC in A', B', C' : prove that the range [FA'B'C] is constant. 1441. The tangent to a parabola at any point F meets two fixed tangents CA , CB in a, b, the diameters through the points of contact A, B in a, b', and the chord of contact AB in c : prove that Fa . Fa : Fb . Fb' = ac : be'. 1442. A tangent to an hyperbola at F meets the asymptotes in a, b, the tangent at a point Q in c, and the straight lines drawn through Q parallel to the asymptotes in o', 6': prove that Fa : Fb' - ca : be. w. p. 16 242 CONIC SECTIONS, ANALYTICAL. 1443. Tlie anliarinonic ratio of the pencil subtended by the four points whoso excentric aiigles are a,, a.,, a^, a^ at any point of an ellipse is sin I (g, - g^) sin 1 (a3 - a J sin 1 (a, - aj sin 1 (a^ - a J * 1444. Tangents are drawn to a conic at four points A, B, C, D, and form a quadrilateral whose diagonals are aa', hh', cc', the tangents at A, B, C forming the triangle ahc, and being met by the tangent at D in a', b', c' ; the middle points of tlie diagonals are A', B', C and the centre of the conic is 0: prove that the range {A'B'C'O) is equal to the pencil \ABCD\ at any point of the conic. 1445. A conic is drawn through* four given points ^1, B^ C, D; BC, AD meet in A'; CA, BD in B' ; AB, CD in C" ; and is the centre of the conic : prove that the pencil {ABCD} on the conic is equal to the pencil {A'B'C'O} on the conic which is the locus of 0. 1446. The anharmonic ratio of the four common points of the two conies OCT + 1/- + z~ = 0, ax^ + hy" + c^ = 0, at any point on the former is one of the three a-h h — G c — a a - c' b—a' c — b^ or the reciprocal of one of them, according to the order of taking the four points; also these are the values of the range formed on any tangent to the second conic by their four common tangents. 1447. Two fixed tangents are drawn to a given conic intersecting each other in and a fixed straight line in Z, M ; from any point on LM are drawn two tangents to the conic meeting the two fixed tangents in A, B ; A', B', respectively : prove that a conic dra^vn to touch the two fixed tangents at points where they are met by LJI, and touching one of the straight lines AB', A'B, will also touch the other. 1448. A quadrilateral circumscribes a conic and A A', BB' are two of its diagonals ; any point F being taken on the conic, BP, B'P and the tangent at P meet AA' in the points t', t, p respectively : prove that Af : A'f = At . At' : A't . A't'. , [Also easily proved by projecting A, A' into foci.] 1449. Four tangents TP, TQ, T'P', T'Q' are drawn to a parabola : prove that the conic TPQT'P'Q' will be a circle if TT' be bisected by the focus. [A parabola can be drawn with its focus at T' touching PQ and the normals at P, Q ; and another with its focus at T touching FQ' and the normals at F, Q' ; and the axis of the given parabola will be the tangent at the vertex of either of these.] CONIC SECTIONS, ANALYTICAL. 243 U50. Four tangeuts TP, TQ, TP\ T'Q' are drawn to a given ellipse : prove that the conic TPQT'P'Q' will be a circle when CT, CT' being equally inclined to the major axis and T, T' on the same side of the minor axis, CT . CT' = CS^, where C is the centre and 8 a focus of the given ellipse. [A parabola can be drawn with T' for focus and CT for directrix touching PQ and tlie noimals at P, Q ; and another parabola with T ibr focus and 6' 7'' for directrix will touch P'Q' and the normals at P', Q' ; and these parabolas are tlie same for a series of conies confocal with the given ellipse.] 1451. The locus of the intersection of tangents to the ellipse fix' + hy^ + 2hKy = 1 drawn parallel to conjugate diameters of the ellipse a'x' + h'y^ + llixy = 1 is {cih — h') {a'x~ + h'y' + Ih'xy) = ah' + a'b — Ihh'. 1452. Through each point of the conic aof + by' + Ihxy— 1 is drawn a pencil in involution whose doiible I'ays are parallel to the co-ordinate axes : prove that the chord cut oif by a pair of conjugate rays passes through a fixed point whose locus is the conic 2 7 " 07 ^^ ax + by + zhxy = j^ . 1453. Two conjugate rays of a pencil in involution meet the conic u = ax' + by- + c +2/y + 2(jx + llixy = in the points /*, F \ Q, Q', the double rays of the pencil being the axes of co-ordinates : prove that the conic enveloped by PQ, PQ', P'Q, P'Q' is 4 (fg - ch) xy = {fy + gx + cy. [li fff = ch, the double rays are conjugate with respect to the conic ?«, and the chords pass through the two fixed points where tlio double rays meet the polar of the vertex : if c = 0, the vertex is on the curve and the chord determined by conjugate rays passes through the poiut (-{•-!)•] 1454. Four fixed tangents to a conic fonn a quadrilateral of which AA', BB' are two diagonals, any other tangent meets AA' in P and the range {APFA'\ is harmonic : prove that the locus of the intersection of BP', or of B'P', with the last tangent is a conic passing through AA' and touching the given conic where BB' meets it. [Taking the given conic to be a? = yz, and the straight line A A' to be px + qy + rz = 0, the locus is P^xr-yz) = {qy-rzy, degenerating to the straight line qy + rz = px when 7/ = 47?-; that is, when AA' is a tangent to the given conic] 1 ( »— 2 2i^ CONIC SECTIONS, ANALYTICAL. 1455. Tlircc fixed tangents are dra\sni to a conic and their points of intersection joined to a focus; any other tangent meets these six lines in an invohition such that the distance between the double points subtends a right angle .at the focus. Also the locus of the double points for dilierent positions of the hitst-named tangent is the curve - =e cos 6 + cos (a + /3 + y - 3^), where c = r{l +e cos 6) is the equation of the given conic, and a, /8, y are the values of at the points of contact of the fixed tangents. X. Beciprocal Pulars and Projections. [If there be a system of points, and straight lines, lying in the same jilane and we take the j^olars of the points and the poles of the straight lines with respect to any conic in that plane, we obtain a system of straight lines and points reciprocal to the former ; so that to a series of points lying on any curve in the first system correspond a series of straight lines touching a cei-tain other curve in the second system, and vice versd : and, in particular, to any number of points lying on a straight line or a conic, correspond a number of straight lines passing through a point or touching a conic. Thvis from any general theorem of position may be deduced a reciprocal theorem. It is in nearly all cases ad\ isable to take a circle for the auxiliary conic with respect to which the system is reciprocated ; the point (^j) corresponding to any proposed straight line being then found by di-awing through 0, the centre of the cii'cle, OP perpendicular to the proposed straight line and taking on OP a point jy such that OP . Op = k', k being the radius of the circle ; and similarly the straight line through p at right angles to Op is the straight line corresponding to the point P. To draw the figure reciprocal to a triangle ABC, with respect to a circle whose centre is or more shortly loith respect to the point 0, draw Oa perpendicular to BC and on it take any .point a ; through a, draw straight lines perpendicular to OC, CA, meeting in b ; and through b, draw sti'aight lines perpendicular to OA, AB meeting in c ; then the points a, b, c will be the poles of the sides of the triangle ABC and the straight lines be, ca, ab the polars of the points A, B, C, wdth respect to some circle with centre 0. Now suppose we want to find the point corresponding to the perpendicular from A on BC ; it must lie on be and on the straight line through at right angles to Oa since Oa is parallel to the straight line whose reciprocal is required ; it is therefore determined. Hence to the theorem that the three perpendicidars of a triangle meet in a point corresponds the following : if through any point (0) in the plane of a triangle (abc) be drawn straight lines at right angles to Oa, Ob, Oc to meet the respectively opposite sides, the three points so determined will lie on one straight line, or be coUinear. CONIC SECTIONS, ANALYTICAL. 245 So from the theorem that the bisectors of the angles meet in a point we get tlie following : the straight lines drawn through bisecting the external angles (or one external and two internal angles) between Oh, Ch ; Oc, Oa; Oa, Ob, respectively, will meet the opposite sides in three col linear points. If a circle with centre A and radius 7? be reciprocated with respect to 0, the reciprocal curve is a conic whose focus is 0, major axis along OA, excentricity OA -4- Ji, and latus rectum 2k' -=- R or 2 -i- Ji if we take the radius of the auxiliary circle to be unity. The centre A is reciprocated into the directrix. Focal properties of conies are thus deduced from theorems relating to the circle. For instance, if (9 be a point on the circle and OP, OQ chords at right angles, PQ will pass through the centre. Reciprocating with respect to 0, to the cix-cle corresponds a . parabola and to the points P, Q two tangents to the })arabola at right angles to each other ; jjcrpendicular tangents to a parabola therefore intersect on the directiix. Again, to find the condition that two conies which have one focus common should be such that triangles can be inscribed in one whose sides touch the other. Take two circles which have this propei-ty, and let R, r be theii- radii, S the distance between theii* centres ; then 8" = i?^ ± 2Rr. Reciprocate the system with respect to a point at distances X, y from the centres, and let a be the angle between these distances. Then a will be the angle between the axes of the two conies, and, if 2Cj, 2c^ be the latei-a recta, e^, e^ the exceutricities, \ \ V cc 1 2 e: e' „e,e„ whence -^± — = -%+ -\~2 -^—- cos a, or c- ± 2c,C3 = e\^ + e, V - 2e,e,c^c^ cos a ; the requii-ed relation. If a system of confocal conies be reciprocated with respect to one of the foci, the reciprocal system will consist of circles having a common radical axis ; the radical axis being the reciprocal of the second focus, and the first focus being a ])oint-cii-cle of the system. The reciprocal of a. conic with resjject to any point in its plane is another conic which is an ellipse, parabola, or hyperbola ac- cording as the point lies within, upon, or without the conic. To the points of contact of tangents from the point correspond the asymptote-s, and to the polar of the point the centre of the reciprocah So also to the asymi^totes and centre of the original conic correspond the points of contact and polar with respect to the reciprocal. As an example we may reciprocate the elementary property that the tangent at any point of a conic makes equal angles with the focal distances. The theorem so obtained is that if we tiike any point 24G CONIC SECTIONS, ANALYTICAL. in the jilano of tlie couic there exist two fixed straiglit Hues (recipro- cals of till' foci) such that if a tangent to the conic at P meet them in Q, Q\ OP makes eijual angles with OQ, OQ'. (More correctly there are two such pairs of straight lines, one [tair only being real.) If however the i)oint lie on the curve the original curve was a jiarabola ; and one of the straight lines being the reciprocal of the point at infinity on the jiarabola will be the tangent at 0. Another jn-operty of the focus, that any two straight lines thi-ough it at right angles to each other are conjugate, shews us that if on cither of the two straight lines we take two points L, L' such that LOL' is a right angle, Z, L' will be conjugate. Since the anhannonic ratio of the pencil formed by any four rays is ecfual to that of the range formed l)y their poles with respect to any conic it follows that, in any reciprocation whatever, a pencil or range is replaced by a range or pencil havijig the same anharmonic ratio. The method of Projections enables ns to make the proof of any general theorem of position depend upon that of a more simjjle par- ticular case of that theorem. Given any figure in a ])lane we have five constants disposable to enable us to simplify the projected figure, three depending on the position of the vertex and two on the direction of the j>lane of Projection. It is clear that relations of tangency, of pole and ])olai-, and anharmonic ratio, are the same in the original and jirojected figure. As a good example of the use of this method, we will by means of it prove the theorem that if two triangles be each self-conjugate to the same conic theii* angular points lie on one conic. Let the two triangles be ABC, DEF, and ahc, def their pro- jections; project the conic into a circle with its centre at d, then e,f will be at infinity, and de, df at right angles. Draw a conic thi'ough (ihcde, then since abc is self-conjugate to a circle whose centre is d, d is the centre of perpendiculars of the tiiangle abc, the conic is therefore a rectangular hyperbola, and e being one of its points at infinity, f must be the other. Thus abcdef lie on one conic, and therefore ABCDEF also lie on one conic. Again, retaining the centre at d, take any other conic instead of a circle ; de, df will still be conjugate diameters, and therefore if any conic pass through a, b, c, d, its asymptotes will be parallel to a })air of conjugate diameters of the conic whose centre is d and to which abc is self-conjugate. The same must tlierefore.be the case with respect to the four conies each having its centre at one of the four j)oints a, b, c, d, and the other three points corners of a self-conjugate triangle. These four conies must therefore be similar and similarly situated. Moreover if we di-aw the two parabolas which can be drawn through a, b, c, d their axes must be parallel i-espectively to coincident conjugate diameters of any one of the four conies ; that is to the asymjv totes. But the axes of these parabolas must be parallel to the asymp- totes of the conic which is the locus of the centres of all conies through a, b, c, d, since the centre is at infinity for a parabola. Hence, finally, if we have four points in a plane, the four conies each of which has one of the four i^oints for its centre and the other three at the comers of a CONIC SECTIONS, ANALYTICAL. 247 self -con jugate triangle are all similar and similarly situated to each othvA- and to the conic which is the locus of centres of all conies through the four jjoints. (The same results might also be proved l)y orthogonal projection, making d the centre of perpendiculars of the triangle abc, in which case the five conies are all cii'cles.) Let A, B he any two fixed points on a circle, co , oo ' the two impossible circular ]>oints at infinity, P any other point on the circle ; then F {Aao co'JJ] is constant. Hence FA, PB divide the segment terminated by the two circular points in a constant anharmonic ratio. Hence two straiglit lines including a given angle may be projected into two straight lines dividing a given segment in a constant aidiarmouic ratio. In particular, if APB be a right angle, AB passes through the centre of the circle (the pole of o) co '), and the ratio becomes har- monic. Thus, projecting properties of the director circle of a conic, we obtain the following important theorem : the locus of the intersec- tion of tangents to a conic which divide a given segment harmonically is a conic passing through the ends of the segment and through the points of contact of tangents to the conic dra^vn from the ends. If the straight line on which the segment lies touch the conic, the locus degenerates to a straight line joining the points of contact of the other tangents drawn from the ends of the segment. Recipi'ocating, we get the equally important theorem : if a chord of a given conic be divided harmonically by the conic and by two given straight lines its envelope will be a conic touching the two given straight lines and also the tangents to the given conic at the points where the given straight lines meet it ; but when the two given straight lines intersect on the given conic the chord which is divided harmoni- cally will pass through a fixed point, the intersection of the tangents to the given conic at the points where the given straight lines again meet it. If tangents be drawn to any conic through oo , co ' their four other points of intersection are the real and impossible foci of the conic. When the conic is a parabola the line joining 00,00' is a tangent, and one of the real foci is at infinity, while the two impossible foci are the circular points. Many focal jiroperties, especially of the parabola, may thus be generalizei.l by i)rojcction. Thus since the locus of intersection of tangents to a parabola including a constant angle is a conic having the same focus and directrix, it follows that if a conic be inscribeS"; proA^e that the corresponding directrices coincide and pass through the j^oint of contact of the parabola which osculates the given circle at P and touches the given straight line. CONIC SECTIONS, ANALYTICAL. 251 1482. An ellipse is drawn osculating a given circle at P and having one focus at a ]>oint O of the cii-cle; a parabola is also drawn osculating at P and touching the tangent at : ])rove that the directrix of the ellijtse is jiarallel to the axis of the parabola and passes through the point of contact of the parabola with the tangent at 0. 1483. A point is taken within a circle, and with as focus is described a parabola touching the radical axis of the circle and the point-circle ; ADA' is a chox'd of the circle bisected in 0: prove that tangents from A, ^I'to the parabola touch it in points lying on the cii'cle. 1484. A chord LL' of a given cii'cle is bisected in and P is its pole ; a parabola is drawn with its focus at and directrix passing through L : prove that the tangents drawn to this parabola at points where it meets the circle pass thro\igh L or L' ; and, if two such parabolas intersect the circle in any the same point, the angle between theii* axes is constant. 1485. Two fixed points are taken on a given conic and joined to any point on a given straight line : prove that the envelope of the straight line joining the points in which these joining lines again meet the conic is a conic having double contact with the given conic at the points where the given straight line meets it and also touching the straight line joining the two fixed points. 1486. Any straight line drawn through a given point meets two fixed tangents to a given conic in two points from which are drawn other tangents to the given conic : the locus of the common point of these last tangents is a conic which touches the given conic at the points of contact of tangents from the fixed point and passes thi'ough the common point of the fixed tangents. 1487. Four fixed points 0, A, B, C being taken, OB, CA meet in B', 00, AB in C", and from a fixed point on OA two tangents are drawn to any conic through 0, A, B, 0: prove that the points of contact and the points B, 0, B', C lie on a fixed conic. 1488. With the centre of perpendiculars of a triangle as focus are described two conies, one touching the sides and the other passing through the feet of the perpendiculars; prove that these conies will touch each other and that their point of contact will lie on the conic which touches the sides of the triangle at the feet of the perpendiciilars. 1489. A conic is inscribed in a triangle and one focus lies on the polar circle of the triangle : prove that the corresponding directrix passes through the centre of perpendiculars. 1490. With the centre of the circuuiscribed cii-cle of a triangle as focus are described two ellipses, one touching the sides and the other passing through the middle points of the sitles : prove that they will touch each othex". 252 CONIC SECTIONS, ANALYTICAL. 1491. Four lixcd straight lines form a quadrilateral wliose diagon;il.s are A A', BIi\ CO': prove that the ouv(dope of tangents drawn to any conic inscribed in the quadrihiteral at the points wliere it meets a fixed sti-aight line through A is a conic Avhich touches Bli', CC and the two sides of the quadrilateral whicli do not jiass through A ; and if BB\ CC meet AA' in o, b and tlie fixed straiglit line through A in h', c, that bU, CC are also tangents to this envelope. 1492. Five points are taken no three lying in one straight line, and with one of the points as focus are described four conies each touching the sides of a triangk? formed by joining two and two three of the remaining four points : prove that these four conies have a common tangent. [If A, B,C, D, E be the five points, A the one taken for focus, AP,AQ two choi'ds at right angles of the conic ABODE, then the common tangent is the locus of the intersection of the tangents at F, Q.^ 1493. Through a fixed point are drawn two straight lines meeting a given conic in P, i*'; Q, Q' ; and a given straight line in R, li\ and RE! subtends a right angle at another fixed point : prove that RQ^ FQ', P'Q, RQ' all touch a certain fixed conic. 1494. Given a conic and a point in its plane 0: prove that there exist two real points L, such that if any straight line through L meet the polar of Z in P and F be the pole of this straight line, RF will subtend a right angle at 0. 1495. Any conic drawn through four fixed points meets two fixed straight lines drawn through one of the points again in R, Q: prove that the envelope of RQ is a conic touching the straight lines joining the other three given points. 1496. Two equal circles U, V touch at a point *S', a tangent to V meets U in. R, Q, and is its pole with respect to U: prove that the directrices of two of the conies described with focus S and circumscribing the triangle ORQ will touch the circle U. 1497. A conic touches the sides of a triangle ABC in a, b, c and Aa, Bb, Cc meet in S; three conies are drawn with AS'for focus osculating the former at a, b, c; prove that all four conies have one common tangent which also touches the conic having one focus at S and touching the sides of the triangle ABC. . 1498. Given four straight lines, i:>rove that two conies can be constructed so that an assigned straight line of the four is directrix and the other three form a self-conjugate triangle ; and that, whichever straight line be taken for directrix, the corresponding focus is one of two fixed points. 1499. A quadrilateral can be projected into a rhombus on any plane parallel to one of its diagonals, and the vertex will be any point on a certain circle in a certain pai'allel jilane. CONIC SECTIONS, ANALYTICAL. 253 1500. A conic inscribed in a triangle -4 5(7 touches BC in a and Aa again meets tlie conic in A' ; the tangent at any point P meets tho tangent at A' in T: prove that tlie pencil T[ABCP] is harmonic. 1501. A conic is inscribed in a triangle ABC and OP, OQ are two other tangents; another conic is drawn through OPQBC and T is the l)ole of BG with respect to it: pi-ove that A [OBCT] is harmonic. Also prove that if lie on the straight line joining A to the point of contact uf BC, T will coincide with A, 1502. A conic is inscribed in a given triangle ABC and touches BC in a fixed point a; h, c are two other fixed points on BC : prove that tangents drawn from h, c to the conic intersect in a point lying on a fixed straight line through A. 1503. A triangle is self-conjugate to a rectangular hyperbola ?7and its sides touch a parabola V; a diameter of U is drawn through the ioown of V: prove that the conjugate diameter is parallel to the axis of V. 1504. Two tangents OP, OQ are drawn to a parabola; an hyperbola drawn through 0, P, Q with one asymptote parallel to the axis of the parabola meets the parabola again in R: prove that its other asymptote is parallel to the tangent at R to the parabola. 1505. Two tangents OP, OQ are dra-wn to an hyperbola; another hyperbola is di-awn through 0, P, Q with asymptotes parallel to those of the former : prove that it will pass through tlie centre C of the former and that CO will be a diameter. 1506. A triangle is self-conjugate to a conic f/" and from any other two points conjugate to U tangents ai"e drawn to a conic V inscribed in the triangle : prove that the other four points of intersection of these tangents are two pairs of conjugate points to U. 1507. A conic drawn through four fixed points A, B, C, D meets a fixed straight line L in P, Q : prove that the conic which touches the straight lines AB, CD, L and the tangents at P and Q will have a foiu-th fixed tangent which with L divides AB and CD harmonically. 1508. Through two fixed points 0, 0' are drawn two straight lines which are conjugate to each other with resjiect to a given conic U: prove that the locus of their common point is a conic V passing through 0, 0' and the ])oints of contact of the tangents from 0, 0' to the given conic. Also, if two points be taken on the polars of 0, 0' which are conjugates with respect to U, the envelope of the straight line joining them is a conic V which touches the polars of 00' and the tangents from 0, 0' to U. 1509. From two points 0, 0' are drawn tangents OP, OQ ; O'P', OQ' to a given conic U, and a conic V is drawn through OPQO'P'Q' ; a triangle is inscribed in V, two of whose sides touch U : prove that the 254 CONIC SECTIONS, ANALYTICAL. third sido passes through the common ])oint of PQ, P'Q'. Also the tanuonts to U at the points wlioro tlie straiglit line 00' meets it meet V in the i>oints of contact of the conimon tangents to U, V. [ )'' is the locus of the intersection of tangents to U which divide 00' harmonically, and U is the envelope of straight lines divided har- monically by V and by the tangents to U at the points where 00' meets it.] 1510. From two points 0, 0' are drawn tangents OP, OQ ; O'P', O'Q' to a given conic U ; a conic V is drawn thi'ough OPQO'P'Q' , and another conic V touches the sides of the triangles OPQ, O'P'Q' : prove that V, V are polar reciprocals of each other with respect to U. Also PQ, P'Q' and the tangents to V at 0, 0' intersect in one point. 1511. Any conic is drawn touching four fixed straight lines and fi'oni a fixed point on one of the lines a second tangent is drawn to the conic: prove that the locus of its point of contact is a conic circum- scribing the triangle formed by the other three given lines. [If the four be the sides of a triangle ABC and a straight line meeting the sides in A', B', C and the fixed point be on the last, the locus passes through A, B, C and through the point of concourse of Aa, Bb, Cc, where a is the point (BB', CC) ; also if any other straight line through meet the sides of the triangle ABC in A", B", C", and BB", CC" meet in a, kc, then Aa, B/3, Cy intersect in a point on the locus.] 1512. A conic is inscribed in a given quadrilateral and from two fixed points on one of the sides are drawn other tangents to the conic : prove that the locus of their common point is a conic passing through the two given points and the points of intersection of the other thi'ee straight lines. 1513. Two common tangents to two conies meet in A, the other two in A' ; from a point on AA' tangents OP, OQ, OP, OS are drawn to the two conies, and the conic through OPQPS meets AA' again in 0' and the conies again in P', Q', R', S' : prove that O'P', O'Q', O'P', O'S' will be the tangents to the two conies at P', Q', P', S', and that the conic OPQPS will pass through the other four points of intersection of the four common tangents. 1514. A tangent OP i^ drawn from a given point to a conic inscribed in a given quadrilateral of which A A', BB', CC are diagonals, and a straight line drawn through P which with PO divides AA' harmonically : prove that the envelope of this line is "also the envelope of the polar of and is a conic which touches the three diagonals. Also, if OP, OP' be the two tangents from the conic through OAA'PP' will pass thi-ough a fourth fixed point. 1515. A conic is inscribed in a given quadrilateral and from two given points on one of the diagonals tangents are drawn : prove that their points of intersection lie on a fixed conic which passes through the ends of the other two diagonals and divides harmonically the segment terminated by the two given points ; also if tangents be drawn to the CONIC SECTIONS, ANALYTICAL. 2;'35 former conic at points where the second conic meets it four of their points of intersection will lie on a conic which passes through the points of contact of the given quadi'ilateral and thi-oiigh the ends of the given diagonal. 1516. A conic U is inscribed in a given quadrilateral and another conic V is drawn through the ends of two of the diagonals : i)rove that the tangents to U at the points where it meets V jjass through the points of intersection of V with the third diagonal ; and the points of contact with V of the common tangents to U, V lie on the tangents to (/ at the points where it meets the third diagonal. 1517. A conic is dra^vn through four given points : prove that the envelope of the straight line joining the points where this conic again meets two fixed straight lines through one of the ))oints is a conic which touches the two fixed straight lines and the straight lines joining two and two the other three given points. 1518. Find the locus of a point such that one double ray of the involution detennined by the tangents from the point to two given conies may pass through a fixed point ; and prove that the other double ray will envelope a conic, which touches the diagonals of the quadiilateral formed by the common tangents to the two given conies. 1519. Three conies U, V, W have two and two double contact, not at the same points : prove that the chords of contact of F, W with U will pass through the intersection of the common tangents to V, W and form with the common tangents an harmonic pencil. 1520. Two ellipses have the same (impossible) asymptotes: prove that any ellipse which has double contact with both will touch them so that the chords of contact will lie along conjugate diameters. 1521. A point Q is taken on the dii-ectrix of a parabola whose focus is ~' •••^ P~ P — * Explain these results when ^^- < 4:q. 1574. Prove that, when the equation x^- j^x' + qx — r = Q has two equal roots, the thii-d root must satisfy either of the equations x [x -2^)' = 4r, {x —2>) {^x + p) + 45- = 0. 1575. Find the relation between p, q, r in order that the roots of the equation a;' -j9x^ + 5'aj-r = may be (1) the tangents, (2) the cosines, (3) the sines, of the angles of a triangle. [The results are {I) p-^ r, (2) 2r -2q + 2r = 1, (3) y - ijfq + 8^>r + 4r = 0.] 1576. Prove that the roots of the equations {!) x^-5x'+%x-\=Q; (2) a;'-6x'+10aj-4 = 0j (3) a;^- 7x^4- 15a;' -10a;+ 1 = 0; (4) x' - 1 \x' + 45a;* - 84a;' + 70a;' - 42a; + 1 1 = ; (1) 4cos'y A "2^ 4 cos -zr , 7 ' A 2^77 4 cos' y ; (2) 4cos'|, A 2^^ 4cos'-g-, A 2^^ 4 cos'-g ; (3) 4cos'^, A '277 4 cos- — , , Stt 4 cos- — , A 2^'" 4 cos'- (4) 4cos'^, A •'27r 4cos-^. 4 cos 3-:r . 1 THEORY OF EQUATIONS. 207 1577. Determine the relation between q and r necessary in order that the equation x^ — qx + r = may be put into the form (re* + mx + ny = x* ; and solve in this manner the equation Sx-'' - 36x- +27 = 0. 1578. Find the condition necessary in order that tlie equation ax^ + hx' + CX + d=Q may be put under the form [xr -^ jyx + qY = x* ; and solve in this manner the equation x^ + 3a;^ + -Ix- + 4 = 0. [The condition is c' — Abed + 8ad' = ; and the })roposed equation may be written (a;* + 2x + 4)^ = as''.] 1579. Prove tliat, if the roots of the equation x^ -px^ + qx-r = are in h.p., those of the equation {j/ (I -n) + n^ {pq - w>-)} 0^ - {if - 2npq + Zn^r) x^ + (pq - 3wr) x-r = are also in h.p. 1580. Reduce the equation x^ -j)^ + ^x — r -0 to the form tf ±Zy + m = Q by assuming x = ay + h; and solve this equation by assuming y — z^^p-. Hence prove the condition for equal roots to be 4 (/ - Zqf = (2/ - ^pq + 27r)^ 1581. Prove that the roots of the auxiliary quadratic, used in solving a cubic equation by Cardan's (or Tartaglia's) rvde, are (2a-^-y)(2;8-y-a)(2y-a-^)^37-3(^-y)(y-a)(a-^) . 54 wliere a, ^, y are the roots of the cubic. 1582. Prove that any cubic equation in x can be reduced to the form {ay + hf = cy^ by putting x = y + z, and the roots of the quadi*atic for z aWII be ^l3-yy+{y-ay + {a-(3y 1583. Prove that, if the cubic (p^, p^, p,, l^a^^) 1)' = be put bx the form A [x + ay + B[x + (Sy = 0, a, ^ will be the roots of the quad- ratic ip' -i\p,) -' - {P^V,-PJh) - + (p/ -P,P,) ^ i and thence deduce the condition for equal roots. [The true condition for two equal roots is given by making this quadratic have equal roots ; yet, if a = /3, the equation reduces to (.1 + B) (,r + ay = 0. The student should explain tliis result.] 2GS THEORY OF EQUATIONS. 1584:. A cubic e(|uatiou is solved by putting it in the form (x+]>y* = z {x + (/Y : prove that the roots of the quadratic for z are /a+ fiui'+ywY /a + ySto + ywV . e .^ -^ — — ' ., I , ^T-T. — ' — , where a, o, y are tlie roots oi the \a + I3w + yu}-j ' \a + /3(i}- + yo)/ ^ r-j i cubic, and co an impossible cube root of 1. Solve the equation x^ + 'dx^ — 33ic + 27 = in this manner. . [5(a;-lr=4(a;-2)^] 1585. Prove that the equation {x — rt) {x - h) (x - c) -f'x- {x — a) - (fx' {x — h)— h^x- (x-c) + 2/(jhx^ = 0, when rt, h, c are all of the same sign, will have two equal roots only when ft/ _ ^U _ ^^* f-gh' g-hf h-fg' [The equation may be reduced to the form gj!: ¥ Jg_ /« , g^ . Jic 1 X a X (3 X y where a, (i, y are the thi-ee f—j > «^c.] 1586. The equation x^ - ix^ + 5.-*;^ - 3 = can be solved as follows : {x* + 5a;- - 3)" = 1 Gx-", therefore {x* - 3a;' + 5)' = 1 6, or a;' - 3a;' + 5 ± 4 = : prove that the equation a;" - 2rta;^ + (a' + 1) a;' = rt" - 1 can be solved in the same way ; solve it, and select the roots which belong to the original equation. 1587. Prove that the equation x^ + (rt + 6 + c) tc' + 2 (6c + ca + ah — or — b^-c')x — iahc — has all its roots real for all real values of a, h, c, and that the roots are separated by the three J be y ca T ah a - ~ c , — c — rt— -,-, c— rt — 6 . rt c [If these three expressions be denoted by a, ^, y, the equation may be written X2 2 2 2 27 2 , be c a rt b T abC = + ;; + . X- a X - p x — y-' 1588. Investigate whether the general cubic eqiiation can be reduced by assuming it to coincide with either of the forms (1) (2a;' + (rt + rt') a; + 6 + h'f = (2a;' + {a- a') x + (b- b'f ; (2) {x- + rta; + 6 + c)' = (a;' - rta; + 6 - c)'. THEORY OF EQUATIONS. 2G9 1589. Prove that, if a, /3, y, 8 be tlie roots of the equation X* + qx'^ + ')'X + s = 0, the roots of the equation sV + qs{l- s)V + r(l-syx + {l-sy = will be /? + y + 8 + 7r ,, &c. fdyb 1590. Prove that the equation x* - 2x^ + m [Ix - 1) = has two re.al and two impossible roots, for all real finite values of m, except when m— 1. [The equation may be written {x- -x + zf = (22 + 1) a;- - 2 {m + z)x + in + z\ and the dexter is a square when 2,~^ = m (m — 1)] 1591. Prove that the equation x* + 2px^ + 2rx + rp = has in general two real and two impossible roots : the only exception being when three roots are equal. 1592. Prove that the roots of the equation x^ — 6x' + 9x- i sin^ a = are all real and positive, and that the difference between the greatest and least lies between 3 and 2 ^3. [/(O) is negative, /(I) positive, / (3) negative, and/ (4) positive. The actual roots are readily found, by putting a; = 4 sin" 6, to be 4 sin^ — ^— and 4 sin" ^ , and a may be supposed to lie between and ^ .] 1593. In the equation a;" -piX^ + p^x" - p^x + p^ - 0, prove that the s\;m of two of the roots will be equal to the sum of the other two, if Sp _ 4p p +j)^^ z= ; and the product of two equal to the product of the other two, ^ 2\^J>^=pf- 1594. The roots of a biquadratic are a, /3, y, 8, and it is solved by putting it in the fomi (ic^ + ax+ by = {ex - d)' ; prove that the values of 26 are fiy + aB, ya + ^8, a^ + y8 ; those of ± 2c are j8 + y-a-8, y-f-a-;8-8, a + ^-y-8; and those of ± 2d are I3y — n8. ya - fto, aft - yf^. 270 THEORY OF EQUATIONS. 1595. A biquadratic in x may be solved by putting x=m}/ + n. and making the equation in y reci}>rocal : prove that the three values of n are /3y — aS ya — /3S a^ — y8 /3 + y - a - 8 ' ^oT^yS^' oTT ^ - y - 8 ' and those of wi" are • (a-y)(a-8)(^-y)(^-8) , itc. 1596. Prove that the equation 3x* + Sx" - Gx' - 24:X + o- = will have four real roots, if r<— 8> — 13; two real roots if r>— 8<19; and no real roots, if ?• ;> 1 9. 1597. Prove that, if -- be the ii^^ convergent to the infinite con- tinued fraction 1_1_ 1 a — a — a — , ^"*' ~ 9,,^ "^Pn ""^^^ ^® divisible by x' — ax + 1, and conversely. p 1598. Prove that, if — be the n^^ convergent (unreduced) to the infinite continued fraction b_ b_b_ a— a— a— , x"'^^ — qx + p^^ will be divisible by x^ — ax + b, and the quotient will be a;""' + g^x"~^ + qjif~^ + . . . + q^_„ x + q^^_^, 1599. Prove that, if — be the n^^ convergent to the smaller root % of the equation x^ - ax + h, which has real roots, the convergents to the other root will be 1> > } J ^•- > J • • 9'. ?2 9n-X 1\ V, 1\ 1600. The n roots of the equation ,^ Ax-h){x-c) {x-c)(x-a) {x-a){x- b) _ {a-b}{a-c) "^ {b-c){b-a) {c-a){c-b)~ ' •difierent from o, h, c are given by the equation x" + H^x"-' + E^x"-'' + . . . + ZT. = 0, where 11^ is the sum of the homogeneous products of powers of a, b, c of 2J dimensions. THEORY OF EQUATIONS. 271 1601. The n roots of the equation 1(0;- a,)/' (a.) -^ (.c - «:)/ (a,) "^ - •" (,« - g )/ (.oK^'"-'^ " "= ' different from g^, g^, ... g^, the roots of /(./;) = 0, arc given by tlie equation x" + ir^x"-' + Il^x"-' +...+//= 0, •where 11^, is the sum of the homogeneous products of powers of g , g ...g of p dimensions. 1602. Prove that, if {1 +x -i-x" + ... +x^~^)'' = a^ + (c^x + a^o^ + and iS^ = a^ + a^^p + ct^^^f+ ... , where r may have any of the 2^ values 0, 1, 2, ... 7?- 1, then of the;; quantities aS\, iS\, /S'^, ... Sp_^,p- 1 are equal to each other, and differ from the jp"* by 1. [If «EO(mod.;.), ^,=^^=...=^^_,=^^-(-l)"; and if n E r (mod. p), S^ = S^= ... =S^_^= S^_^ -(-!)"•] 1603. Prove that the equation x" - 7'x''~'' + s = will have two equal roots if {l(n-p)] -{'-(n-p) 1601. Prove that, if (x) have two roots equal to a, and the corre- .sponding partial fractions in ., / be -, r, + , J{x) {x-af (sc-g)' 2^^ 2 3^>)/>)-^(aXr'(a) /"(«) '3 [TW • 1605. The coefficients g^ g^ ay..a^ can be so determined as to make the expression g, (x'"*' + 1) - a^x (a:^+ 1) + g,a;^ (x'"-' + 1)... + (- lyax' (x^ + 1) equal to {x - ly {{n + 1) (a^ + 1) + 2nx}. [The necessary value of a , , is '^ C ,^ 1606. Prove that, if x^, x^,...x^ be determined by the n simple equations a;,-2X + 3%-... + (-l)'-VX=(-ir-(« + ir, r having successively the values 1, 2, ... 7i, x-2n + ^ X _ (!!L±l)12n+l) ^!LLg 272 THEORY OF EQUATIONS. 1G07. Provo the identity (for integral values of «) _,,,(„ (2n4l)2n {2n+\)2n{2n-\){2n-2) \ = {n+\){x ^ X + J X -...J, the n\imber of terms being oi in the dexter, and — ^ or - + 1 in the sinister. 1G08. Prove that the expi-ession ,n o, „-,, (3^^-l)(2^-2) „-. {2n-2){2n-S){2n-A) is unchanged, or changed in sign only, if 4 — a; be substituted for x; and deduce the identity 4-_2„-4'-.i ^"-^>f"-^ Jri!lz2)r-'-... n \2 n(ii — \) ^{2n-r+ 1) {2n -r)... (2n - 2r + 2) ~ n{n—l)...{u — 7-+l) [The roots of the expression are 4 cos^ -— , 4 cos^ -^ > • • ■ where a^, a^, . . . . sin (n + 1) 6 » -, are the roots ot ^^^ — y. — = U. sm^ -■ 1609. Prove that the roots of the equations (1) x^-i2u-l)x-^.^^^'^K-^ {2n-S){2n-i){2n-5) ^ X +...-U, (2) x' -nx- +— S^ — x"" 5^ r^ 'x"" +... = 0, are respectively „ TT . , 27r , „ mr (1) 4cos^_ r-, 4cos^jr — -,,..4 cos-- r-; '^ '' 2n+\ 2n + \ 2n+ 1 (2) 4sm--, 4sin^ — ,...4sur-^ — tt^— ■ ^ ' n n In 1610. Prove that, if a^, a^,...a^ be the roots (all unequal) of /(.'), and the coefficient of x" in/(.r) be 1, + -I /K) /K>) ■■■ /'(O THEORY OF EQUATIONS. 273 will be eqiial to the sum of the homogeneous products of r dimensions of powers of the n quantities a,, a,^,.,. a^. Prove also that 11 1/11 IV —^+-^+... + — 2 -(-+- + . ..+ — ). = —.+ 1611. Prove that the equation in x + — *— + . . . + — V = will be an identical equation if 2 {a) = 0, S («6) = 0, 2 {ah') = 0, ... 2 (aJ""') = ; but that these conditions are equivalent to % (a) = 0, b^=bg- ... = b^. 1612. If four quantities a, h, c, d be such that he + ad + ca + hd +ah + cd - 0, J 111,. and y ^+- n+-7 }=0; be + ad ca + o» a6 + ca 1111 while a+h + c + d, and - + -j- + - + -. are real and finite, two of the four abed vnM be real and two impossible. 1613. Prove that, if all the roots of the cubic x^ - S^^x" + Z qx-r = be real, the dijfFerence between any two roots cannot exceed 2^3(7 ?'— g ), and the difference between the greatest and least must exceed 3 Jp* - q. Also, if /3 be the mean root, 1614. Prove that the sum of the ninth powers of the roots of the equation x^ + 3a; + 9 = is 0. 1615. The system of equations of which the type is a^^x^ + ajx^ + • . . + ajx^ = c' is true for integi-al values of r from r=l tor=^?i+l: i>rove that they are true for all values of r. 1616. Having given the two equations cosna+^jj cos(yt — l)a +p^cos(n — 2) a -t- ... +p„-^0, sin wa + 7?j sin (» - 1) a+;^^sin (/t-2) a + ...;>,_, sin a = 0; prove that 1 + J)^ cos a 4- p^ cos 2a + ... + p^^ COS na = 0, and p^ sin a -^-p^ sin 2a -•-...+ ^)„ sin 7ia = 0. W. P. 18 27-4 THEORY OF EQUATIONS. 1617. The sum of two roots of the equation a;*-8a:='+ 21.x' -20a;+ 5 = is equal to 4 : explain wliy, on attempting to solve the equation from the knowledge of this fact, the method fails. 1618. The equation a;' - 209cc + 56 = has two roots whose pi-oduct is 1, determine them : also determine the roots of the equation x' - 3S7a; + 285 = whose sum is 5. 1619. Prove that all the roots of the equation /i sm /I \m-i rii (m—\) n(n—V\ „,, ,__, _ (1 -xY -mnx{l-xY +-^ ' -~i — 'a;''(l-a;)'"^-...=0, the number of tei-ms being ??i+ 1, are all real, and that none lie beyond the limits 0, 1 ; m^ n being whole numbers and ni > n. 1620. Find the sum of the n^^ powers of the roots of the equation X* — x' +\ =0; and form the equation whose roots are the squares of the differences of the roots of the proj^osed equation. [If S^ denote the sum of the ?-^^ powers, aS^j^.j = 0, aS'^^^ = 4 cos -^ ; and the required equation is {x' + 4x + 2>){x' + 2a; V3 + 4)(a;V 7 - 4 ^3) = 0.] 1621. The sum of the r**^ powers of the roots of the equation x" + ^jO;""^ -^Vi^'^ + • • • + i^„ = 0) is denoted by s , and S = s, + 5„ + s, + . . . + s ; prove that, if S^ have a finite limit when m is indefinitely incx-eased, that limit is jj, + 2/7^ + 3;?3+... + ?iy^ l+i'i+^. + '-'+P, 1622. The roots of the equation a" - j^jfc""' +7'2-'«"~^- ... = are a, /?, y, S,... : prove that 2(2a-/3-y)(2^-y-a)(2y-a-^) = (n - 1 ) (?i - 2) p/ - 3n (w - 2) f^f^ + 3?i>3 ; and determine what symmetrical functions of the differences of the roots are equal to (1) (,i _ 2) {n - 3)^/ - 2 (n - l)(7i - 3)i?,;j3 + 2^ {n - \)p^, (2) - (n - 1) {n - 2) (n - 3) p^ + 4n (ti - 2) {n - 3) f^p^ - Sn' (n - 3) p^p^ + 8n'p^. THEORY OF EQUATIONS. 275 1623. Tlic roots of the equation x^-2\x*-¥ p^x' - i^x' +PkX-p^ = exceed those of the equation x' - q^x* + q,x' - q.a? + q,X-q, = respectively, each by the same quantity : prove that 2p,'-5p,^2q;-5q^, ^J\' - ^'^1\V. + 25/^3 = 4?/ - 15?,?, + 25^3, 3;^/ - Sp,p, + 20p, = 3?/ - 8q^q^ + 20?,, S/'xX - 3;^P/ - 50p,p, + 5pj^^ + 250;., = 8?;?3 - 3y,7/ - 50?,?, + 5?,?3 + 250?,. 1624. Prove that if the n roots of an algebraical equation be a> ^> 7> S, . . . , 62(a-/3)(a-y)(a-S) = (ji-3)2(2a-i3-y)(2;8-y-a)(2y-a-^). 1625. Prove that, if // denote the sum of the homogeneous products of r dimensions of the powers of the roots of the equation x" +^^33""' + pjio"~' + ...+p^ = 0, 1626. Two homogeneous functions of x, y of n dimensions are denoted by u^, v^\ prove that the equation found by eliminating y between the two equations «t , = a, v^ = b, will be a rational equation of the ?i"^ degree in a;". 1 627. Prove that — 2a, a + b, a + c b + a, -2b, b + c c + a, c+b, —2c (b + c)', c% b- c*, (c + ay, a^ b"^, a^, {a + by \a — b — c, 2a, 2a 2b, b-c-a, 2b 2c, 2c, c-a-b (1) (2) (3) (1) E 4 (6 + c) (c + a) (a + 6) J = 2(bc + ca^ aby ; = (a+b + cy; — be , b + c' — ca c+ a a, b. -ah a + b _ {be + ca + aby ~ {b + c) {c + a) {a + b) ' 18—2 276 (5) (6) (7) THEORY OF EQUATIONS. {b + c)', h\ c' =2abc{a + b + cY; a', (c + a)*, & a^ b', {a + by d% b\ {a + by a-nb- nc, {n + 1) a, {)i + l)a (71 + 1)6, b-nc-na, {n + l)b {n+l)c, (n+l)c, c-na-nb -a{b' + c'-a'), 26^ 2c«|e I'ia + b+cf; {n+l)c, (n+l)c, c-na-nb -a{b' + c'-a'), 26^ 2c''\= abc{a' + b' + c')'. 2a^ -b{c' + a'-b'), 2cH 2< 2^)^ _c(a'+6^-c*)| 1628. Prove that (1) 1, 1, cos COS (/? + y), sin^ ^— ^ -0 . ,y-^ T- ^^^ 2 (y + '^)> sin' i^-sin'-^- 1, COS (a + ^), sm' —2— sm^ i-^— = 2 sin ^- — ^ Rin :- — —X sin ^ sin ^ {sm2^ + 2 sin (a + ;S + y - e) J J J (2) ^r— sm — ^ (Sin zy + z sun^a + p + y - p; - sin (^ + y) - sin (y + a) - sin (a + ft)} ', 1, COS (a + 0), Bin' '^ ^ sin' ' sin' = 2 sin - sin '—^r— si 1, COS (a + 6), sin' '^ sin' -^-^ — 1, COS (^+ ^), sin' i-n— sin'— 2— / m . 2^-^ • 2/5-^ , COS (y + 6), sin' —^ sin ■^— ^ 2 ^sin^^{2sin20 + sin(a + /? + y-e) ~ sin (a + e) -sin (/3 + 6) -sin (y + 0)}. 1629. Prove that the determinant 1, cos^, cos2^, cos(w-l)^ COS0, cos 2^, cos 30, COSW0 cos 26, COS 30, cos 40, cos {n+\)6 cos(w-l)0, COS710, cos(2w-2)0 andaU its first 2nd, ... minors, to the w-3^^ =0, if ?i be any integer >3. THEORY OF EQUATIONS. 277 1C30. Prove that the value of the determinant 1, 1, 1, 1 1, 2, 3, n n{n+l) 1, 3, 6, . 1, 4, 10, n{n+l){n + 2) 11 1, n, (w + 1) n{7i + l) ...(271-1) \n-l is 1, and that of its first, second, &c. principal minors are w(?i+ 1) /I 1 1 ^ ^\ E 2sx,xxjx. {- + -+—+ — ~ zjt ' ^ 3 *\aj, x^ x^ x^ 8/ 1631. Prove that the determinant s — xj, x^ , ajg , x^ ajj , X^ , X^ , {S — XJ where 8 = x^+x^ + x^ + x^. 1G32. The determinant of the {n + If^ order "iJ "'2» "■3» a n' »3> «4J is equal to (— 1) ^ ''»> **'«> aj^ are the roots of the equation (l-a;;+')(l-a;;^'), (1 -«;^'); where a„a" + a^_^x" ' + +a^x + z= 0. 1G33. Prove that the determinant 1, cos a, cos (a + /S), cos (a + ^ + y) cos(^ + y + 8), 1, cos^, cos(/8 + y) cos(y + 8), cos(y + 8 + a), 1, cosy COS 8, cos (8 + a), cos (8 + a + )8), 1 and all its fii*st minors, will vanish when a + ft + y + S=2ir. 278 THEORY OF EQUATIONS. lG3-i. Prove tliat, if ?t, denote the determinant of the n*^ oa-der, a, "l, 0, 0, 1, a, 1, 0, 0, 1, a, 1, 0, 0, 0, 1, a, I, ...... 0, 0, 0, 1, a, 1 0, 0, 0, 0, 1, a u —au+u_=0; and thence (or otherwise) obtain its value iii one of the three equivalent forms (1) a''-(n-l)a''-'+ .^ (2) sin (?i + 1) a -r- sin a, where 2 cos a = a, (3) (p"^' - q""^^) -^{p- 9)} where 2^, 9' are the roots of x^ -ax + 1 = 0. 1635. Prove that (n-2){n-3) ^„_, (1) (2) (3) 1-n, 1, 1, 1, 1-M, ^5 1, 1, I -01, = 0, n being the order of the determinant ; 1, 1, 1, 1-^ X, 1, 1, 1, 1 ={x-lY-'{x + n-l); 1, 00, 1, 1, 1 1, 1, so, 1, 1 1, 1, OS,, Ij 1> 1, 1, oc^> 1, 1, 1, (4) X, x% fl3^ X. where x^, x^, ... cc^, are roots of the equation x" - p^x"'^ + p^x'*'^ - ... +{-l)"p^=0; ■ X" tC • *fyy ■ *^ J (n — 1) n = (-1) ' x^ix"-!)"-'. THEORY OF EQUATIONS 1636. Prove that COS^, C0S2^, C0S3^, COSM^ cos 2^, cos 3^, cos 4^, ... cos?i^, cos^ cos 3^, cos4^, ... cos«^, cos^, cos 2^ cos w^, cos $, cos 20, ... cos (n~ 1) 6 1637. Prove that 279 _ {cos^-cos(?i+l)^}''-(l-cosn^)' 2(-l) ■' {I- cos n6) „0 „3 „2 ^ 1 a , a , a , a, 1 iS", /s^ /3^ (3, 1 6 3 2 1 y> 7> y> y> i 8^ h\ B', s, 1 £^ €', Cl €. 1 5 4 2 1 a , a , a , a, l iSS /8^ (S% (3, 1 /, y, y% y, 1 , , o", o, 1 €\ c\ e\ €, 1 E(a-)8)(a-y)(a-S)(a-e)(/3-y)(^-8) S (a - ^)^ 1638. Prove that the determinants 0, 0, 0, a, h, c\] = 0. 0, 0, ;2, a, h, 0, 2/, 0, a, 0, c a;, 0, 0, 0, h, c \x, y, z, 0, 0, 1639. Prove that, if m_, denote the determinant of the n^^ order, a, 1, 0, 0, 0, a, a, 1, 0, 0, 1, a, a, 1, 0, 0, 1, a, a, 0, 0, 0, 0, 0, 1, a, a, 1 0, 0, 0, 0, 0, 1, a, a w,^, - au^ + au^_^ - w^_j= ; and express the developed determinant in the forms (1) ■V -1 a— 3 , where v. E (a - 1)" - (n - 1) (a - 1)""' 280 THEORY OF EQUATIONS. (2) {2f''-p'^'-p-q'"°-+q"'' + q]-^{p-q){p + q-% wliere /?, q are the roots of the equation a;* — (a - 1 ) x + 1 = ; -1 f, , .sin(?i + l)^ + sin{«+2)^) (3) 2(l+cos^) {i + (-ir 3in^ ')' where 2 cos 5 = 1 - a. 1G40. Prove that, if u^ denote the determinant of the n^^ order, rt,, 1, 0, 0, 0, 0, «2» «1' 1' ^' ^' •• %> »2> f*l> 1> 0> <^' 0, «,, «_i, ^2, «,, 1, 0, 0, 0, 0, a,, 0, 0, 0, «, a^_^, a^, a^, 1 .... ^3, «„ a, and that, if aJj , »2» ^3 , a?, be the roots of the equation f{x) E cc' - ftjCc''"' + a^cc''"^ - + (- lya^ - 0, u.= + I . . + Also prove that, when a^ — a^_^ = ... = a^ = a^ = l, w_^ = except when w = or 1 (mod, r), and is then equal to (- 1) *■ or (- 1) »• . 1G41. Prove that l-x^, xjl-x.;), x^x^{l-x.;), x^x.^ ... a,_^(l-a;„), aj^a;^ -1, l-o;^, x^{l-x^), x^ ...x^_^{l-xj, x^x^ 0, -1, 1 —X. ^u-l (1 - ^n)y 0, 0, 0, -1, 1 is equal to 1; the second row being formed by differentiating the first with respect to x^ , the third by differentiating the second with respect to x^, and so on. DIFFERENTIAL CALCULUS. 1642. Having given sin a; sin (a + x) sin (2a + x) ...sm{{n — l)a + x} = 2'"" sin 7ix, where w is a whole number and wa = tt : prove that (1) cot X + cot (a + x) + cot (2a + x) + ... + cot {{n—l)a + x} = n cot iix, (2) cot*cc+cot^(a+a;) + cot-(2a + a3) + ... +cot^ {{n-l)a + x} = n{n- 1) + V? cot' wx. 1643. Prove that the limit of (cos a;)^*'^% as x tends to zero, is c~'. 1644. Prove that the equation (1 -a;^) -^-a;y + 1 = is satisfied either by yj\-x^ = cos"' x, or by y Jx' - 1 = log (a; + Jx^- 1). 1645. Prove that the equation ^^^''> dx'^'^dx~*'^ is satisfied by any one of the four functions and therefore by the sum of the four functions each with an arbitrary multiplier ; and account for the apparent anomaly. 1 G 4 6. Prove that, if y = cot" ' x, ^ = (- 1)- |n-l sin ny sin> ; and, if y = tan ( ;j ) , ' '' \1 + a; cos a/ ^ = (- 1)-' 1|:^ sin n (a - y) sin" (a - y). 282 DIFFERENTIAL CALCULUS. 1G47. Prove that tlie function 4a; log a; - a;* - 2a; + 3 is positive for all values of x lying between and 2. lG-i8. Prove that, if n be a positive integer, the expression (a; - n) ^ + , -. + , ^ + -j ^ + ... + (?i - 1) « + »* ^ ' \n—\ m—Z \n— 6 will be positive for all positive values of x ; and will be positive or nega- tive for negative values of x accoi'ding as n is odd or even. d^V dv 1 G49. Having given (t? ^ + x-^ + y = 0, prove that 1050. Having given y= {x + Jl + a;')"* + {x + Jl+ x')'"', prove that 1651. Assuming the expansion of sin [m tan"' x) to be x" X" a.x + a -^+ ... +a, |- + ... , prove that a„,.^ + (27i^ + w'^) a, + {n- \f {n - 1 ,'- 1) a„_^ = 0. 1652. Assuming the expansion of {log (1 + a;)}^ to be x^ X* x" prove that «„.. + (^+l)«^„.. = 6(-l)-•|l^{i + l + |+... + l|; and thence that <'.„=5(-ir'ti+l(i + 5(l + i) + J(l + i + 5) + T(l + * + 5 + l) + - 1653. Prove that, if y = x" (log a;)', whenr^l, x^,=\n,', whenr=2, a;^ -^,f + a; ^-^ = 2 [n ; •u o 3(^"'^^y o od''^-y d"'^\ - , whenr = 3, a;^ -^ + 3x- ^J + a; ^,f = 1 3 |^ ; • ■ . 1 1 DIFFERENTIAL CALCULU and generally that \r dx*' j^-l dx"^' ' |2 dx"^' dx"^' ~- [It is a singular pi-operty, but easy to pi'ove, that the sum of the coefficients of the sinister is equal to the limit of the product of the two infinite series 1654. Prove that, if y = x^~^\og{l+x), where r is a positive integer, dy_, / 1 1 1 \ ^'~^\i+x'^ Jpn^''^ '" "^ {i +xy) ' and, if w > r, — ^ = (- 1)" ' ' ' Ax'-' +7ix"-'+ — ^T — ^a;'-^+ ... to r terms } dx" ^ ' (l+a;)" I ^ i (- 1)"-M?i fa;'-' r-1 ,_, ,. , (r- 1) (r-2) ,.3 ,, .^ ^ '^ w-r+ IJ 1655. Prove that, if y = {\+xy log x, =--^ = (- 1) - [X -nx ' + — ^-^r — - X - - ... to 7* + 1 terms ) ; dx"^' V / ^™+r \ [2 /' and deduce the identity 9Z [71/ ■"* 1 ) (1 + a;)' - n (1 + a;)'-' + —^ — - (1 + x)"'- - ... to r + 1 terms = X - (n - r) X + '^ '-^ X - ... to r + 1 terms, [n > r). li 1656. Prove that, in the expansion of (1 +a;)" log(l +a;), the co- efficient of 1 is (- l)'"' In \t-\ ', and that of , is \n + r ^ ' ■— \n-r l^i/1 1 1 \ 7=(-+ 1 + ... + =-), n, r, and n-r being all positive integei-s. ^ , , . /. log (1 + ^O • 1657. Prove that the expansion of -°-^ — r^ is (1 + xy iix n --?i(?i+ 1)(- + ^— r) To + 7i{n + l)(7i + 2)(-+ -i-+ -i-V:^-... ^ ^ ^ '^\» n+l ?i + 2/ 3 284 DIFFERENTIAL CALCULUS. and deduce the identity 1 n + l (n+l)(n + 2) 1 ^ ^ - + :r + ^ TK ——h + • • • to ** terms r r-1 [2 r-2 \n + r (-1 1 1 -J \n\r[n + r n + r-l ?i + lj [When n is not a whole number, the last identity should be corrected by writing {it + I) {n + 2) ... {71 + r) for \n + r -h \n.] n— J 1658. Prove that in the expansion of (1 +x) log (1 + x), when 91 is a positive integer, the coefficient of x"" is 0. 1659. Prove that, in the equation /(aj + h) =f{x) + hf (x + 6h), the limiting value of 0, when h tends to zero, is -J ; and that, if 6 be constant, y(.'v) E ^ + ^aj+ Ca;^, where A, B, C are independent of x. Also prove that, if be independent of x, f{x) = A + Bx + Cnc', where A, B, G, on are independent of x, and find the value of 6 when f {x) has this form. 1 Ce?ilogm_ 11 [The value of 6 is , , log { -^-^ > . | "- hio^m {_ II log m ) -" 1660. In the equation /(ic + li) =f{x) + hf {x + 6h), prove that the first three tei"ms of the expansion of 6 in ascending powers of h are , hf"'{x) 1c f"{x)r' { x)-{f"'{x)Y and calculate them when f{x) = sin a;. \h + — tt: ,„ . ^ , provided cot x be finite.] L- 24 48 sm" a; ^ -■ 1661. In the equation /(« + A) =/(a) + hf («) + 1/" («) + ...+ ^/" (a + ^A), the limiting value of 6, when h tends to zero, is r ; and, if /(«) be a rational algebraical expression of n + l dimensions in x, the value of - 6 is always ^ . Also prove that, if /(a;) = e""", ^ is independent of x ; and that the general form of /(a;) in order that 6 may be independent of X IS J„ + A^x + A^x' +...+ A^^x" + B^'\ 1662. Prove that, in the equation fix) =/(0) + xf (0) +^/" (0) + ... +|> (^a;). if /(a;) E (1 -a)"* where ?/i is any positive quantity, the limiting value of ^ as a: tends to 1 will be 1 - ( — )'" ". DIFFEllENTIAL CALCULUS. 285 1663. Prove that, m the equation F{x + h)-F{x) _ F'{x + eh) f{x+h)-f{x) ~ f'{x+eh) ' the limiting value of 9 when h tends to zero is ^ ; and that when F(x) = sin X and /(x) — cos x, the value of 6 is always ^. 1664. Prove that the expansion of (vers"' x)' is „ / la;' 1.2 a;" 1.2.3a;- \ 1665. Prove that ,,, TTC'^-l 1 1 1 (2) l^^,.g^...tooo^(l4.)(l.i.)(l + |-3)...tocc. 1666. Prove that the limit of the fraction _ 2 2.4 2.4.6 ^ ^ ^-^3-^ 37-5 -"37577^-^"^^^^^^"^^ , 11.3 1.3.5 ^ , ' 1 + o + ?nt + o ^ /- + ... to 01 terms 2 2.4 2.4.6 when w is infinite, is — ; and the limit of the ratio of the w*^ term of the numerator to the ?i*'' term of the denominator is tt. [Thia may be deduced from the equation 1667. Prove that ^/l+V2 + V3+... + >>^i^<§{(7i+l)^-l}, 1 + I + ^ + . .. + - > log (1 + w) < 1 + log w ; p-i ^ 2''-' + 3'"' + ... + w"-' > - ?i'' < - {(w + l)' - 1}, p p^^ ' 111 1 1 fi _L_\ 1/ 1 ^) pTT + 2P Ti + 3FH -^ • • • + ^' > ^ V - (n + 1 )»• j ^ p F "^ ^ " n'' J ' sin (« + 1) a , sin wa cos a + cos 2a + ... + cos na > — ^^ 1 < ; a o p being a positive quantity, and 2na in the last < v. 286 • DIFFERENTIAL CALCULUS. 16G8. Prove that if a, h, h be eliminated by differentiation from the equation ff,r* + 62/" + 2//z, 1/ + 2zx), I)rovc that and having given tc=/{u{x-t), u(y-t), u{z-()}, prove that du du du ^'^_(x dx dy dz dt 1680. Prove that ©"(^i-^y^=(^4) \,_f^d\ ^ dx"^ ,d"^'y A'-'O^ ,., dr^ [More generally, fd\ / d \ ( d yd'y , \^)K'^''Tx-vy^K'-'''d^)d^'-^ 1681. The co-ordinates of a point referred to axes inclined at an angle w are (x, y), and w is a function of the position of the i:)oint : prove that 1 /d^u d^u _ d^u\ 1 fd'ud^u d^u^^\ sin^coVc^^ dif dxdyj' sin' (ii\dx^ dy^ dxdy\ ) are independent of the particular axes. [Their values in polar co-ordinates are 1 du 1 d-u d^u 1 du d^ii 1 / d^u 1 du\^ - r ^ ' ?d?W^rd^-d?~? \d^6 ~ r do) '^ d^u 1 d^u 1 du 1 dhc d^u 1 du d^u d?'^?de''^ 1682. Having given 1x = r (e^ + c-«), 2y = r (e^ - e"^), prove that d^u d'u _ d^u 1 d^u 1 c?w daf ~ dy^ ~ dr^ ~ r^ dd^ r dr^ dSi ^ / d\ Y _l^f^f^_l^^_l / do^ dy' ~ \dxdy) ~ r" dr' d6^ ~ r Hr dr ~ r' \ d'u 1 dits} drde r dej ' 1683. Having given K + y = X, y=: XY, prove that cZV ahf, du _ y. d'u _ d'u du ^ d^'"'^ dx¥y~ d^^ dX'~ IXdY'dX' DIFFERENTIAL CALCULUS. 28!) 1 684. Having given x + i/ = €^+*, ct* - y E ^7*, prove that (I'll (Fu _ „ /d\t d-u\ cM cVn _ _ag((l^'' cru\ 1685. Having given c'E r<^°''^, t^'Sr^in^, prove that - d"ii - fZ"w „ (I'u d'u du . ^ dif-^'^dxdy^y'd^-d? -^ "^ ^"s"- 1686. Having given 2a;'-" = v'-" 4- lo'-", 2y'-" E w'"" + ?i'-', 2^'"" E tt'"" + v'"', prove that „ c/6 „ d c/c^ d<^ dx, dx„ '" dx_ dX, dX, ' dX/ and, if that «, ,x^^ + a^^.r/ + . . . + 2rtv,a;^.r, + . . . : .4,jA7 + ^ggcc/ + . . . + 2 ^„a;,aj. + . 1 689. Pi-ove that, if w be a function of four independent variables »i> ^,> ^3' »^4' and 03, = r sin 6 sin ^, x^ = r sin ^ cos ^, aJ3 = r cos ^ sin i/^, x^ =r cos ^ cos xj/, d'u d°u d'u d'u d'x, 1 d'n 1 d'xi, '^''^ d^'^ dbe'^'^ 'd^^^ d^ ^ r' dB' ^ 71,in' Q d^' 1 d'u 3 f/j* 2 ^ -.fldu + - -r- + -, cot 2C r' COS' 6 d.{/' r dr »■* dO w. p. 19 290 DIFFERENTIAL CALCULUS. 1690. Prove tliat, if ./•, y, z be three variables coiinected by one dz dz ~d'z d'z d'z , equation only, and ]), q, r, s, t denote ^^^ , j- , -^^^ , ^^- v , ^-, as usual, d.c 1 dx 7 d'x r Tz' 'P' dy P' dz' ' f' d'x ^qr -ps 1^ d'x = - ft- 2jiqS + blr; and that a minimum distance is always (c(j^ — h')-i-aJ2. 1694. Prove that <^{f{x)] is always a maximum or mininnmi when f {x) is so ; but that, if a be the maximum or minimum value of f{x), (f> (a) is not a maximum or minimum value of <^ (.r). 1695. The least area which can be included between two parabolas, whose axes are parallel and at a given distance a, and wliich cut each other /3 at right angles in two points, is or ^ . [The included area may be proved to be a' -.- 3 sin w cos' w, where co is the inclination of the common chord to the axes.] 1690. From a point on the evolute of an ellipse are drawn the two normals OP, OQ (not touching the evolute at 0) : prove that if a" < 26^, PQ will have its minimum value when the excentric angle of the point for which is the centre of curvature is tan~* ( ^y^ „ ] . 1697. Prove that, if vi - 1, n- 1, and m — n be positive, theexpres- "1+1 "j-i sion (cosa; + 7?isin.T)"'~i-f- (cos£C+ « sin .r)»-i will be a maximum when a;= J , and a minimum when x = - + cot"' (vi) \- cot""' (n). Also, if m — n and mn—1 be positive, (cos x + vi sin x)" -^ (cos x + n sin x)"" will be a maximum when as = 0, and a minimum when x — tan"' m + tan"' n - -. DIFFERENTIAL CALCULUS. 2!)1 1698. Prove tliat, if n be an odd integer or a fi'action whose nume- rator and denominator are odd integers, the only maximum and minimum values of sin"a;cos«.r are determined by the equation cos (?i + l)a;= 0. Also, with the same form of n (> 1), the maximum and minimum values of tan ?ix' (cot ;«)" correspond to the values 0, ir, Stt, ... of («- l)a; and 7r, Stt, ^tt, ... of 2 (?i+ I) X, the zero value giving a maximum, and any value of X which occurs in both series being rejected, 1699. Through each point within a parabola ?/"= iax it is obvious that at least one minimum chord can be drawn : jorove that the part from which two minimum chords and one maximum can be drawn is divided from the part through which only one minimum can be drawn by the curve (x - 5a)~^ + {ix - G// + 4rt)~- + (4a; + 6i/ + 4ft)~^ = ; and that, at any pomt on the parabola y- = 4a (x — a), one minimum •chord is that passing through the focus, (a, 0). [The curve has two rectilinear asymptotes 3a; ± 3 J5)/ + 5a = 0, and a pai'abolic asymptote 27>f =32a(3x — a), which crosses the cui've when 3a; = 17a, and is thence almost coincident with the inner branches.] 1700. The maximum value of the common chord of an ellipse and its circle of ciu-vature at any point is Sjsl^.b^^ {(«'' + ^') (^^'^ - ^') («' - ^-^') + 2 (a* - a^h^ + h^f]\ 1701. A chord PQ of an ellipse is normal at P and is its pole : prove that, when PQ is a minimum, its length will be 3 ^3 ccb' - {a' + h')^ and Q will be the centre of curvature at P ; and when OP is a minimum the other common tangent to the ellipse and the circle of curvatui*e at P will pass thi'oiigh 0. [The minimum value of PQ here given will only exist when a' > 26*, the axes being l)oth maximum values of PQ ; when a^ < 2b', one axis is the maximum and the other the minimum value of PQ, and there are no other maximiun or minimum values.] 1702. In any closed oval ciu've, PQ, a chord which is normal at P, will have its maximum or minimum values either when Q is the centre of curvature at P, or when PQ is normal at Q as well as at P, which must always be the case for two positions at least of PQ. 1703. Prove that the expression = — -^-^ has 1 ^^^2 for its maximum and minimum values coiresponding to a; = — 1 ± ^2. 1704. Two fixed points A, B are taken on a given circle, and another given circle has its centre at B and radius greater than BA ; 19—2 202 DIFFERENTIAL' f'ALCULrS. nnv point P boin,£f talcon on tbo socoud circle, PA meets the first circle !iL,Miii in Q : prove that the niaximum lengths P/J^, P,^Q^oi PQ are e(pially inclined to AP and each sul^tends a right angle at 7>, and the mininuun lengths both lie on the straight line through A at right au<,des to AH : also P^, A, B, P^ lie on a cii'cle which is orthogonal to the tu-st circle. 1705. The least acnto angle which the tangent at any point of an elliiitic section of a cone of revolution makes with the generating line through the point is cos"' (cos y8 sec a), where 2a is the angle of the cone and yS the angle which the plane of the section makes with the axis. 1706. Prove that three parabolas of maximum latiis rectum can be drawn circumscribing a given triangle ; and, if a, /3, y be the angles which the axis of any one of them makes with the sides, that cot a + cot j8 + cot y = 0. 1707. Prove that, if x + y + z = ^c, f{x)f{y)f{z) will be a maxi- nnim or mixdmum when x = y =z = c, according as /"(c)>or<{/'(c)p-/"'(c). 1708. The minimum value of {Jx + my + nzf ^ {y% + zx -\- xy) is 2mn + %il+ 11m — f — 11^ —^i",, provided this value be positive; otherwise there is neither maximum nor minimum value. 1709. Prove that the maximum value of sin X J a sin" y + h cos^ y + cos x J a cos^ y + h sin^ y is J a + h ; and the minimiim value of Jet? sin" x + lf cos" X + Ja~ sin* y + ^^ cos* y sin [x — y) is « + 6. 1710. Prove that, when x, y, z vary, subject to the single condition xyz {yz + zx + xy) = x+y + z, the minimum value of (1 + yz) (1 + zx) (1 + xy) {l+x'){l + y'){l+z') is-1. 1711. Find the plane sections of greatest and least area which can be drawn through a given point on a given paraboloid of revolution; proving that, if 9^, 6^ be the angles which the planes of maximum and minimum section make with the axis, 2 tan e^ tan 6^ = 3. 1712. The maximum and minimum values oi f{x, y, z), where X, y, z are the distances of a point from three fixed points (all in one plane), are to be determined from the equations 1 df _ 1 df _ 1 df ^ sin (y, z) dx sin {z, x) dy sin (a-, y) dz ' (y, z) denoting the angle between the distances y, z. DIFFERENTIAL CALCULUS. 293 1713. Pro\-e tliat, if A , B, C, D be corners of a tetrahedron and /-* a point the sum of whose distances from A, li, C, D is a mhiimum, FA.Fa_ P B.Ph _ PC.Pc Pp.Pd,^ '"la 'M Cc Dd ' a, b, c, d being the points in which PA, PB, PC\ PD respectively meet the opposite faces. Also prove that when IPA^ + mPB- + nPC + rPD^ is a minimum, vol . P BCD _ vol . PC DA _ yol.PDAB _ vol. PA BO I '/a ti r 1714. The distances of any variable point from the comers of a given tetrahedron are denoted hyit, x, y, z : jDi'ove that, wheny'(?<, x, y, z) is a maximum or minimum, 1 fdfV_ 1 (df c \dx/ l-a^-b^-c'^ 2abc \duj l-a'-b"- c" + 2ab \ - a'^ — b- - c'- + 2a'bc' \dy) \ - a' - 6'* - c" + 2a'b'c \dz rt, b, c, a\ b\ c' denoting the cosines of the angles between the distances (y, «), (-, x), (x, y), {u, x), {ic, y), {u, z) respectively. 1715. Prove that, if be the point the sum of the squares of whose distances from 7i given straight lines, or jjlanes, is a mininium, will be the centre of mean position of the feet of the perpendiculars from ou the given straight lines or planes. 1716. A convex polygon of a given number of sides circumscribes a given oval, without singular points : prove that, when the i)erimetor of the i^olygon is a minimum, the jxunt of contact of any side is the point of contact of the circle which touches that side and the two adjacent sides produced. 1717. In the cui-ve ^ = 3ax^ — x^, the tangent at P meets the curve again in Q : prove that tan QOx + 2 tan POx=0, being the origin. Also prove that if the tangent at P be a normal at Q, P lies on the curve iy{3a-x)=(2a- x) ( 1 Ga - 5x). 1718. Prove that any tangent to the hypocycloid x^ + t/^ - «3, whicli makes an angle ^ tan"' | with the axis of x, is also a normal to the curve. 1719. The tangent to the evolute of a parabola at a point where it meets the parabola is also a normal to the evolute. 1720. From a point on the evolute of an ellipse a'y' + b'af = (rb' the two other normals to the ellipse are drawn : prove that the straight line joining the feet of these normals will be a normal to the ellipse {rrx' + b'y'){a--by^a'b\ 2t)4 DIFFERENTIAL CALCULUS. 1721. A tangent to a given ellijjse at F meets the axes in two points, througli which are drawn straight lines at riglit angles to the axes meeting in p : prove that the normal at ]) to the loc\;s of p and the .stniight line joining the centre of the ellipse to the centre of curvature at r are equally inclined to the axes. 1722. Trace the curve (-) + [t) =1 when n is an indefinitely lai'ge integer, (1) when 7i is even, (2) when n is odd. [(1) the cui've is undistinguishahle from the sides of the rectangle formed by x' = a^, '}f = i>^; (2) when x' < «", y = h; when if<1f,x = a; and when x" > «^ and >f > h', - + , =-- 0, or the curve coincides with two sides a of the rectangle and with the part of one diagonal which is without the rectangle.] 1723. Trace the curve determined by the equations x — a cos V, y = a — — 7, , and prove that the whole curve can only be obtained by using impossible (pure imaginary) values of 6. [The two curves (1) fl; = acos^, y = a— — -.. (2) a; = acosh^, y= . . „ , give the same differential equation of the first order and starting from the same point {a, a) when ^ = must coincide.] 1 724. Trace the curve 4 {x" + 2if - 2ayY = x' (x' + 2y^), proving that . 47r „ the area of a loop is — r^ (2 — ^3) «", and that the ai-ea included between the loops is (2ir - 3 ^3). 1725. A cui've is given by the equations acos6{a'+{a'-¥)co^''e} _ b sin 6 {¥ + {h'-a ') sin' 6} _ ~ a- cos"* 6 + b'^ sin- 6 ' if ~ ^^2 cog-' Q + i^ gi^^ prove that its arc is given by the eqviation ch (a'Rm'6 + b'coH'e)^ dd d' COS" 6 + ¥ sin" 6 . . . . ds 172G. Prove that the curve whose intrinsic equation is yj=asec2^, if X, y, -y^ , and ^ vanish together, has the two rectilinear asymptotes 2/=^^ = -;;^log(V2 + l). DIFFERENTIAL CALCULUS. 295 1727. Two contiguous points P, P' on a curve being taken, PO, P'O are drawn at right angles to the radius vector of each ])oijit : prove that the limiting value of PO when P' moves up to P is ± -7^ . 1728. Two fixed points *S', *S" being taken, a point P moves so that the rectangle SP, 8'P is constant ; prove that straight lines drawn from S, S' at right angles respectively to ISP, fS'P will meet the tangent at P h\ poiuts equidistant from P. 1729, In a lemniscate of Bemoiilli, the tangent at any point makes acute angles $, 6' with the focal distances r, r' ; prove that ^^^=JJ2t" '"' ^'= 2^727^ ' ^^ «i^— 2 " ^ cos J-2t" -2j2r'^ 2 ~ 2 • 1730. In a family of lemniscates the foci S, S' are given (SS' = 2a) : prove that in any one in which the rectangle under the focal distances (c^) is less than a^, the curvature is a minimum at the points of contact of tangents drawn from the centre, and that these 2)oints all lie on a lemniscate (of Bernoulli) of which *S', aS" are vertices. Also, when c" > a^, the points of inflexion lie on another such lemniscate equal to the former but with its axis at right angles to that of the former. [In any of these curves, if p denote the pei'pendicular from the centre on the tangent, 2c-pr = r* + c* — a*, and the radius of curvature is The points of maximum curvature are the vertices.] 1731. In the curve / 6\ ^ 6 T ( m + n tan -1 = 1+ tan - , the locus of the extremity of the polar subtangent is a cardioid. 1732, Tlie tangent at any point P of a certain curve meets the tangent at a fixed point in T, and the arc OP is always equal to n . TP : prove that the intrinsic equation of the curve is 1 8-c (sin <^)"~' ; and tliat the curve is a catenary when n = i, the evolute of a parabola when n = %i a four-cusped hypocycloid when n — -!}, and a cycloid when, n — 2. 1733, From a fixed point ai-e let fall perpendiculaa-s on the tangent and normal at any point of a curve, and the straight line joining the feet of the perpendiculars passes tluough another fixed point : prove that the curve is one of a system of confocal conies. 29G DIFFERENTIAL CALCULUS. 1734. A circle is drawn to toucli a cardioid and pass tlu'ongli the cusp : ]irovo that the locus of its centre is a circle. If two such circles be di-awn, and throujih their second common i)oint any straight line be drawn, the tangents to the cii'cles at the points where this straight line again meets them will intersect on the cai-dioid. [Of course this and many projierties of the cardioid ai'e most easily proved by inversion from the j)arabola.] 1 735. Two circles touch the curve r" = «" cos mO in the points P, Q, and touch each other in the pole S : prove that the angle F/^^Q is equal to -r , n being a positive or negative integer. 1736. The locus of the centre of a circle toucliing the curve r" = a" cos m6 and passing through the pole is the curve {'2r)" = a" cos n6, where n{l — m) = ?«. 1737. In the curve r = «sec"^, prove that, at a point of inflexion the radius vector makes equal angles with the prime radius and the tangent ; and that the distance of the point of inflexion from the pole increases from a to a Je, as n increases from to oo . li n be negative, there is no real point of inflexion. • 1738. A perpendicular, SY, is drawn from the pole aS' to the tangent to a curve at F : prove that, when there is a cusp at P, the circle of curvature at Y to the locus of Y will pass through S ; also that, w^hen there is a point of inflexion at Y in the locus of Y, the chord of cur- vature at P through *S' will be equal to i/SP. 1739. The equation of the pedal of a curve is r=f[6) : prove that the equation found by eliminating a from the eqiiations rcos a =y*(^ - a), r sin a =/' (^ - a), is that of the curve. 1740. Prove that for any cubic there exists one point such that the points of contact of tangents drawn from it to the curve lie on a cii'cle. If the equation of the cubic be ax^ + Sr/.f^^ + 2>fxif + hi/ + Ax^ + ^Hxy + Bif + . . . = 0, and if — - = '— -— = — j- — > there wull be a straight line such that, if tangents be drawn to the cubic from any point of it, the points of contact will lie on a circle. 1741. The asymptotes to a cuspidal cubic are given : prove that the tangent at the cusp enveloi:ies a curve which is the orthogonal projec- tion of a tlu-ee-cusjied hypocycloid, the circle inscribed in the hypocy- cloid being projected into the locus of the cusp. [The locus of the cusp is the maximum ellipse inscribed in the ti-iangle formed by the asym])totes, and any tangent to the tricusp at a point meets this ellipse in two points P, Q so that OQ is bisected in P ; the point corres])onding to is /'.] DIFFERENTIAL CALCULUS. 297 1742. The equation of a curve of the »t"' oi-der being ^, (2) has two roots fj. and <^., (ju,) = : prove tliat there will be two cor- responding rectilinear asymptotes, whose equations are (2/ - H-^r " (/^) + 2 (y - fix) : (fx) + 2<^3 (/.) = 0. 1743. Two points F, Q describe two curves so that corresponding arcs are equal, and the radius vector of Q is always parallel to the tangent at F : show how to find F's path when Q's is given; and in ^es^jecial prove that when Q describes a straight line F describes a ' catenary, and when Q describes a cardioid, with the cusp as pole, F describes a two-cusped epicycloid. 1744. The rectangular co-ordinates of a point on a given curve being (x, y), the radius of curvature at the i)oint is p, and the angle which the tangent makes with a fixed straight line is <;^ : prove that and, in general, 'd-^'xV /d'-^'y = It' + V where -Mc^-y^G^'-)]^. --{Cv'-)"-(i-')] x-X y-r\ . /x — A y — 1 \ 1745. A curve repi-esented by the equation /I , ) ~ ^ is drawn having contact of the second order with a given curve at a point F : prove that, if be the point (A", Y), FO will be the tangent at to the locus of 0. 1746. A rectangiilar hyperbola whose axes are parallel to the co- ordinate axes has three-point contact with a given curve at the point (x, y) : prove that the co-ordinates (A", Y) of the centre of the liyper- bola are given by the equations —-^-y — 5~' dx djf 298 DIFFERENTIAL CALCULUS. :xnd tliat the central radius to tlie point {x, y) is the tangent at (X, Y) to the locus of the centre. Also, when the given curve is (1) the j>ar:i1>ola //= 4a.r, (2) the ellij)se cr//' + b'x- ==(rb'' ; prove that the locus of the centre of the hyperbola is (1) 4:{x + 2ay = 27a7/, (2) {ax)l + [byf = {a' -^ h'f . 174 7. An cllijtse is described having four-point contact with a given ellipse at P, and with one of its equal conjugate diameters passing through F : prove that the locus of its centre is the curve [The curve consists of four loops, and its whole area is to that of the ellip&e as (a - Vf {(«- + ah + l-f - hiclf] : 2a'6^] 1748. The equation of the conic of closest contact which can be described at any point of a given curve, when referred to the tangent and normal at the point as axes, is ao;" + h]f + Ihxy = 2y, where -1 7__1^ 7.-1 If^^py _I^IjP ''~p' ''~ Bpds' ^~p'^9p\dsj 3ds" and p is the radius of curvatm'e at the point. 1749. The sum of the sqiiares on the semi-axes of the ellipse of five pointic contact at any point of a curve is 3 T-iil the product of the semi-axes is 27p--=- 9 + ( y ) - 3p - '^J , the rectangle under the focal distances is 9p--f- •', 9 4- ( ^ ) - 3p'^~\ , and the excen- ( \ds/ ds'j tricity e is given by the equation 9 (^" - ^) ^ ( \dsj ^ds'i m-'^ ds' 1750. A chord QQ' is drawn to a curve parallel to the tangent at F, a neighbouring point, and the straight line bisecting the external angle between FQ, FQ' meets QQ' in : prove that the limiting value DIFFERENTIAL CALCULUS. 299 1751. In any curve a chord PQ is drawn pai'allel and indefinitely near to the tangent at a point : prove that the straight line joining the middle point of the chord to will make with the normal at an angle whose limiting value is tan"' ( - -y- 1 . Reconcile this result with V3 els) the fact that the segments into which the chord is divided Ijy the normal at are ultimately in a ratio of equality. If the chord meet the normal in R, and P', Q' be respectively the mid point and the foot of the bisector of the angle QOP, the limiting value of the third propor- tional to RP', PQ' will be |p ^ . 1752. A chord PQ of a curve is drawn always parallel to the tangent at a point : prove that the radius of curvature at of the locus of the middle point of this chord is lOpfo 1753. A chord PQ of a curve is drawn parallel to the tangent at and is met in P by the bisector of the angle POQ : prove that the radius of curvature at of the locus of ^ is 3p ^ . '^ dp 1754. The normal chord PQ at any point P of a conic is equal to 18p-r(9 + 2-^| —dp J-, j ; in a parabola, if / be the centre of curvature at P and the pole of PQ, PQ. PI=2P0', and 10 is perpendicular to the focal distance iSP ; and, in all conies, PO = 3p-j- , and the angles I OP, IPC are equal, C being the centre. 1755. Prove that the curves r=a9, r = - 7, have five-point 2 4- cos ^ '■ contact at the pole. 1756. The centre of curvature at a point P of a parabola is 0, OQ is drawn at right angles to OP meeting the focal distance o? P in Q : prove that the I'adius of curvature of the e volute at is equal to 3Q0. 1757. All the ciu-ves represented by the equation a b \a + b) ' for difierent values of n, touch each other at the point {x — y= j , and the radius of curvature is (a' + 6')- -^ n{a + b)-. oOO DIFFERENTIAL CALCULUS, 1 758, At each point P of a curve is drawn the equianguhir spiral of closest contact (four-p^int), and s, a- are cori'esponding arcs of the curve and of the locus of the pole S of the spiral : prove that r [ (1S he the pole corresponding to a point P on the given curve, the tangents at P, iS are equally inclined to PiS, ^^= (-')'> V^k::^:!)' and the locus of S for different values of n is the locus of the foci of the conies which have four-point contact with the given curve at P. DIFFERENTIAL CALCULUS. 801 1762. In the last question, prove tliat, when the given curve is such tliat corresponding arcs traced out by *S' and P are equal, the in- trinsic equation of the given curve is either ds , , J~- ds n-\ , -n =c (cos d>) 01" -T7 = <^ sec — ,- 6 : defy ^ ^' d(fi n+1^ that in the former case FS is constant in direction and the locus of *S' is the image of the given curve with respect to a straight line, and in the n+1 latter that FS is constant in length, f n + 1 c or cj , and the cm-vatures of the two curves at P, *S' will be as 1 +n : 3 — n or n+1 : Sii—l. 1763. At each point of a given curve are drawn the cardioid and lemniscate of foui"-point contact, and the arcs traced out by the cusp and node respectively corresponding to an arc s of the given curve are n{37i+l)\ 3n+lJ ' that the radius of curvature of this curve bears to the common radius of curvature of the osculating cvirves at the corresponding point the ratio (n + If : n (3/t -i- 1). Also the area traced out by the radius vector to the pole is — v— a' (n+l 6 + sm 2$). 1767. At each point of an epicycloid is drawn the equiangular spiral of closest contact: prove that the locus of the pole of this spiral will be the inverse of the epicycloiel with respect to its centre ; and, ti02 DIFFERENTIAL CALCULUS. convpi'soly, the curvos for which this proporty is true arc those whose iutriiisic etiuations are s = a{\ - cos Dxfi), 2s - a (e""^ + e""'* - 2), measuring from a cusp in each case. 17G8. A curve is such that any two corresponding points of its evohite and an involute are at a constant distance : ])rove that the sti'aight line joining the two points is also constant in direction. 17G9. The reciprocal polar of the evoluteof a parahola with respect to the focus is a cissoid, which will he equal to the pedal with respect to the vertex when the I'adius of the auxiliary circle is one-fourth of the latus rectum. 1770. In any epicycloid or hypocycloid the radiiis of curvatui'e is pro])ortional to the perpendicular on the tangent from the centre of the tixed cii'cle. 1771. The co-ordinates of a point of a curve, referred to the tangent and normal at a neighbouring point as axes of co-ordi- nates, are Gp' ' Sp^ ds ds 15 p V (Is '^ ds~J •^ 2p Gp'ds \4p'V Is ^?£W"P) -, the axes of the conic of closest contact at each point are inclined at con- stant angles to the tangent and normal. Also, at any point in the curve ds -j- = asec Z(fi, the rectangle under the focal distances in the conic of closest contact is constant. 1774. Tangents to an ellipse are drawn intei'cepting a given length on a fixed straight line : prove that the locus of their common j)oint is DIFFER"ENTIAL CALCULUS. 303 a qiiai'tic having four-point contact wath the ellipse at the points where the tans^ents are parallel to the fixed straight line ; and trace the curve when the fixed straight line meets the (dlipse, (1) in real points, (2) in impossible points ; the given intercept being greater than the diameter parallel to the fixed straight line. 1775. The curvature at any point of the lemniscate of Bernoulli vai'ies as the difference of the focal distances ; and in the lemniscate in which the rectangle under the focal distances is 2a^, where 2a is the distance between the foci, the curvature varies as 1776. Prove that the three equations 2a cos (^ 2/ = YZcos(i' •'^" + y *^^i *^ = *' 2/ = p cos (/) (1 - cos <^), all belong to the same curve, p being the radius of curvature at the point {x, y) ; (j> the angle which the tangent makes with the axis of x and s the arc. 1777. The curve in which the radiixs of curvature at any point is n times the nonnal cut off by a fixed straight line (the base) is the locus / 6 \"~' of the pole of the curve r = a( cos r 1 rolling along that fixed straight line ; and is also the envelope of the base of the curve, in which the radius of curvature is n — 1 times the normal, when the curve rolls along the same straight line. The two rolling curves may be taken to have alwaj's the same point of contact J*, in which case the pole of the foi-mer, Q, will always lie on the base of the latter at the point where it touches its envelope ; the radius of curvatui-e at Q of the roulette or glissette will be nQF, and the radii of curvature at F will be (n — l) PG in the envelope curve and [1 — J FG in the locus curve, FG being drawn at right angles to the fixed base to meet the moving base in G. [All the curves involved are easily found for the values of ??, — 2, -1,0,1,2.] 1778. The curve in which the radius of curvature is always three times the normal cut off" by the base is an involute of a four-cusped hyjiocycloid which passes through two of the cusps : if 4a be the longest diameter of this curve, 2a will be the shortest, and the curve will lie altogether within an ellipse whose axes are 4a, 2a, the maximum distance cut off on any normal to the ellipse being yV a ; and the mini- mum normal chords in the two curves vnW be of lengths 1-840535 a and 1-859032 a, inclined at angles 51" 33' 39"-4 and Gl"52'28"-2 respectively to the major axis. 1770. A point F being taken on a given curve, F' is the corre- sponding point on an inverse to the given curve : prove that (1) a circle can be drawn touching the two curves at F, F', which will be its own inverse, (2) when the diameter of this circle is always equal to the 304 DIFFEREXTIAL CALCULUS. radius of curvature at P the ^iveu curve is either an ellipse or an epicycloid, and (3) the circle of inversion is the director circle for the ellipse and the circle through the cusps for the epicycloid, 1780. The per{)endicular from a fixed point on the tangent to a certain cui've is ainm — ~ ,A (« + 2cos — -^] , where d> is the angle . which the tangent makes with a fixed straight line : prove that the i-adius of curvature at the point of contact is — — ^ a (sin — - J\ ; ^ n+2 \ n + 2/ ' and identify the ciirvcs when 9i is 1, 0, and — 1 respectively. [If the straight line from which <^ is measured be the axis of x and the fixed point the origin, the curves are (1) the points {x ± a)" + y" = 0, (2) the points sif + (i/^ af = 0, and (3) the parabola y^ = - 4a [x + 3a).] 1781. Prove that the equation of the first negative pedal of the parabola ?/*= 4a (a + x) is 27 ai/' = (a + x) (x - 8aY, and that the equation of the evolute of this curve is (£)' 6a rrrri • x • • . » , . , . 6a 3a , I ine uitrinsic equation of this evolute is s = ^^ —„ - "TT • ^ (1 +cos^)^ 2 -^ 1782. The radius of curvature p of a curve at a point whose areal co-ordinates are {x, y, z) is given by the equation (d-x /di/ dz\ dy /dz dx\ d'z /dx f^y\)" 9 \le Kelt ~dt)^df\di~dt)^ df \di ~ dtj] K^p^ ( 2 dy dz dz dx „ dx dy^ * ~ ' / „dii dz ^„dzdx „dxdii\ (a- -f — + b- ^ -r + c- -j- -^) \ dt dt dt dt dt dtJ where a, b, c are the sides and k is double the area of the triangle of reference. 1783. A circle rolls on a fixed straight line, trace the curve which is enveloped by any tangent to the circle ; proving that the whole arc enA^eloped corresponding to a complete revolution of the circle is 2a f 2 ^3 + o ) > ^^^^^ ^^^ ^i'6^ cut oflT the envelope by the fixed straight line is a^ ( ^ + 4 j , a being the radius. 1784. The cui'ver = -77j r^ can be made to roll outside a parabola {0 -a)- of latus rectum 4a so that its pole always lies on the tangent at the vertex, and the curvatures of the two curves at a point of contact P will be as SP + a : SP — a, where S is the focus. Also the curve r = 2a6 can be made to roll inside the parabola so that its pole always lies on the axis, and its curvature bears to that of the parabola the ratio SP + a : 2a; so that at any point of contact the radius of curvature of the parabola is equal to the sum of the radii of curvature of the two rolling curves. DIFFERENTIAL CALCULUS. S05 ■f a 1785. The curve r = 6sm — rolls within, an ellipse of axes 2a. 2b a ^ ' ' startmg -with its pole at the end of the niiajor axis : prove that the pole will remain always on the major axis and the curvatures of the two curves when touching at P will be as h' : b' + SP.iS'F. Similarly with the curve r = a sin -r- and the minor axis. 1786. The three curves of the last two questions touch at P and 0, 0^, 0^ are the centres of curvature of the ellipse and the two roulettes : prove that {00AP] = '^.- be J)9 a0 00 1787. The curves 2r = b (e* -e «), 2r = a{e'J +e~^) can be made to roll on an hyperbola whose transverse and conjugate axes ai-e 2a, 2b, so that the poles trace out these axes respectively : the curvatures at any point of contact P will be as a'b' : a' (6" + SP . >S'P) : h' (SP . JS'P - cr), and if 0, 0., 0„ be the centres of curvature {00 .0 .P\ = - -p, • 0" 1788. A cardioid and cycloid whose axes ai'e equal roll along the same straight line so as always to touch it at the same point, their vertices being simidtaneously points of contact : prove that the cusp of the cai-dioid will always lie in the base of the cycloid and will be the point where the base touches its envelope. The curvatures of the two curves at their point of contact will be as 3 : 1. 1789. A curve rolls along a fixed straight line : prove that the curvatm'e of a carried point is y- (7), where r is the distance from the carried point to the point of contact and ly the perpendicular from it on the directrix. 1790. The curve a . //, . \ a , cfltana + g-etana — = 1 + sec a sm (tj sin a), or - = 1 H .— ^ , r \ /' J. 2 sec a ' rolls on a straight line : prove that the locus of its pole is a circle. 1791. A loop of a lemnlscate i-olls in contact with the axis of x: prove that the locus of the node is given by the equation and that, if p, p' be corresponding radii of curvature of this locus and of the lemniscate, 2pp' — d'. 1792. The curve r"" = a" cos ?«6> rolls along a straight line: prove that the radius of curvature of the path of the pole is r ( — ) . w. r. 20 306 DIFFERENTIAL CALCULUS. 17'J3. A plane curve rolls nloiig a straight line: prove that the radius of curvature of the path of any point carried by the rolling curve is .— where r is the distance from the carried point to the r - p sin point of contact, <^ the angle which this distance makes with the directrix, and p the radius of curvature at the point of contact. 1794. A curve is generated by a point of a circle which rolls along a fixed curve : prove that the diameter of the circle through the gene- rating point will envelf)pe a curve generated as a roulette by a circle of halt the dimensions on the same directrix. 1795. A parabola rolls along a straight line: prove that the envelope of its directrix is a catenary. 1796. Two circles, of radii 6, a -6, respectively, roll within a circle of radius a, their points of contact with the tixed circle being originally coincident, and the circles rolling in opposite directions in such a manner that the velocities of points on the circles relative to their respective centres are equal : prove that they will always intersect in the point which was originally the point of contact. 1797. In a hypocycloid, the radii of the rolling and fixed circles are as n : 1n-\- 1, where n is a whole number: prove that part of the locus of the common point of two tangents at right angles to each other is a circle. 1798. Prove that a graphical solution of the equation tan a; = a; can be found by drawing tangents to a cycloid from a cusp ; the value of x which satisfies the equation being the whole angle through which the tangent has turned, the point of contact starting at the cusp. 1799. Tangents are drawn to a given cycloid inclined at a given angle 2a (the angle through which the tangent turns in passing from one ])oint of contact to the other) : prove that the straight line bisecting the external angle between them is tangent to an equal cycloid whose vertex is at a distance 2r(atana from the vertex of the given cj^cloid; and that the straight line bisecting the internal angle is normal to another equal cycloid whose vertex is at a distance 2«(1— acota) from the vertex of the given cycloid, a being the radius of the generating circle. 1800. Find the envelopes of (1) £c cos^ ^ + 2/ sin^ ^ = rt, (2) -^^ + -2^ = 1 6 being the parameter in each case. 1801. A perpendicular OY is let fall from a fixed point on any one of a series of straight lines drawn according to some fixed law : prove that, when OF is a maximum or minimum, Y is in general a point on the envelope ; and that, if Y be not on the envelope, the line to which OY is the ]i)erpendicular is an asymptote to the envelope. + 6-- DIFFERENTIAL CALCULUS. 307 1802. Find the envelope of tlie system of circles {x - aXy + (y - 2a\y = a'{l+ Xy, \ being the parameter. 1803. The envelope of the directrix of a pai-abola which has four- point contact with a given rectangular hyperbola is the curve 1804. The envelope of the directrix of a parabola having four-point contact with a given curve is the locus of the point found by measuring along the normal outwai'ds a length equal to half the radius of curvature. 1805. Prove that the envelope of the circle x' + if + a' + ¥— 2ax cos 6 — 2hij sin = f - cos ^ -i-| sin0 j (ft''sin*^+6-cos'^) 01? 1^ , . fy? + '/'^X" V? If is the ellipse —, +"(-,= 1, and its inverse ( —^ — ^ I = - ^ o? h- \a^-¥Vj a 1806. The envelope of the straight line X cos 4* + y sin <^ = a (cos ?^)'• is the curve whose polar equation is r'~" = a'~" cos^; . \—n Q 1807. On any radius vector of the curve r = asec" - is described a n a circle : the envelope is the curve r = c sec""' 7 . Prove this sreometri- ^ n-l *= cally when n = 2, and when n-Z. 1808. A parabola is described touching a given circle and ha\Tng its focus at a given point on the circle : prove that the envelope of its directrix is a cardioid. 1809. A sti-aight line is drawn thi'ough each point of the curve r" = a" cos md at right angles to the radius vector : prove that the envelope of such lines is the curs^e r""' = a"*"' cos -r- 6. 7?i - 1 1810. From the pole S is drawn ^ST" perpendicular upon a tangent to the curve r" = a" cos niO, and with S as pole and Y as vertex is dra\vn a ciu-AC similar to r" = a' cos nd : prove that the envelope of such curves is r""*" = a"*" cos 6. in + n 1811. Tlie negative pedal of the parabola tf^iax with respect to the vertex is the curve 21mf = (x- iaf. 20—2 808 DIFFERENTIAL CALCULUS. 1812. Tlic envelope of the straiglit line px + qy + rz - 0, suliject to the condition q-r r-p p- q IS (f + 2/ + ^) %(f + ^ + a^-) '" + (1 + ^ + 1/) ' = and, when the condition is -^^ + — — + = 0, a, b, c being the q-r r—p p—q sides of the triangle of reference ABC, the envelope is %r^ + v~^ + io~^ - 0, ■where w = # (a; + 2/ + ;5) - a; cos f (5 - (7) - 2/ cos {1 20" + I (C - ^)} -zcosiUO^ + fiB-A)], and similarly for v, w. 1813. The contact of the curve /(a;, v/, a) - with its envelope will be of the second order if, at the point of contact, ^=0 and -i^^=-^^. da^ ' dadx dy dady dx' 1814. Find the envelopes of the rectangular hyperbola x^ — y^ - iax cos' a + iay sin' a + 3a^ cos 2a = 0, and of the parabola {x — a cos' a)" = 2ay sin' a + rt* sin* a (2 + cos" a) ; proving that the conditions for osculation are satisfied in each case. 1815. Given a focus and the length and direction of the major axis of a conic, the envelope of the tangents at the ends of either latus rectum is two parabolas, and that of the normals at the same points two semi-cubical parabolas. 1816. The tangent at a point P of an ellipse meets the axes in T, t, and a parabola is described touching the axes in T, t : prove that the envelope of this parabola is an evolute of an ellipse, and if FJf, PN be let fall perpendicular to the axes, MN will touch the parabola where it has contact with its envelope. The curvatures of the parabola and the envelope at the point of contact ai'e as 2:3. 1817. At each point P {a cos 6, h sin 6) of an ellipse is described the parabola of four-point contact, and S is its focus : prove that the point where PS touches its envelope is (x, y) where X -y _ 2 (g" - ¥) a cos' d " fdix' d~ a' cos^' 6 + b' sin' + {a'- b') (cos' d - sin' 6) ' DIFFERENTIAL CALCULUS. 300 and, if P' be this point, ^p,^ 2CF.CD^- where CP, CT> are conjugate semi-diameters. Also prove that, when a* — 26^, the envelope is the curve 1818. At each point of a given ellipse is described another ellipse osculating the given ellii:)se at P and having one focus at the centre 6' : prove that its second focus will be the point P' found in the last question, and PP' will be the tangent at P' to the locus of P'. 1819. A given finite sti'aight line of length 2c is a focal chord of an ellipse of given eccentricity e : prove that the envelope of the major axis is a four-cusped hypocycloid inscribed in a circle of radius ce\ the envelope of the minor axis is that involute of a four-cusped hypocycloid, inscribed in a circle of radius :j g, which passes through the centre and cuts the given segment at right angles ; the envelope of the nearer latus rectum a similar involute touching the given segment; the envelopes of the farther latus rectum and farther directrix are also involutes of four-cusped hypocycloids j and the envelope of the nearer directrix is a circle of radius ^cq. HIGHEE PLANE CUEVES. 1820. Prove that a cubic which passes through the angular points, the mid points of the sides, and the centroid of a triangle, and also through the centre of a circumscribing conic, will also pass through the point of concourse of the straight lines each joining an angular point to the common point of the tangents to the conic at the ends of the opposite side. [The equation of the cubic will be lyz (l/ — z) + m%x (s — jb) + nxy {x — y) = 0, and, if (X : Y : Z), (X' : Y' : Z') be the centre of the conic and the point of concourse, YZ' + Y'Z= ZX' + Z'X = XY' + X'Y. When the conic is a circle the cubic is the locus of a point such that, if with it as centre be described two conies, one circumscribing the triangle and the other touching its sides, their axes will be in the same directions.] 1821. Two cubics are drawn through four given points A, B, G, D, and through the three vertices of the quadrangle ABCD : prove that, if they touch at A, B, (7, or D, the contact will be three-pointic. [The equation of such a cubic may be taken to be Ix {y^ - z') + my {z^ - x^) + nz (x^ - 3/^) = 0.] 1822. Two cubics are drawn through the four points {A, B, C, D) and the three vertices {E, F, G) of a quadrangle : prove that, if they touch at E, their remaining common point lies on FG and on the common tangent at E, and, if EG be the tangent at E, FG will be a tangent at F. 1823. Two cubics are drawn as in last question, and another common point lies on the axis of homology of the triangles ABC, EFG : i>rove that their remaining common point lies on the conic whose centre is D and which touches the sides of the triangle ABC. HIGHER PLANE CURVES. oil 1824. Prove tliat an infinite number of cubics can be drawn through the ends of the diagonals of a given quadrilateral, and through the three points where the straight lines joining a given point to the intersection of two diagonals meets the third; also that the cubic which passes through will have a node at 0. [The equation of such a cubic will be Xxijz + {Ix + mu + nz) (rc^ + y^ + s^) - 2 (l,x^ + m)/ + nz^) = 0, where a; ± ?/ ± « = are the sides of the quadrilateral, and Ix = my = nz the point ; and the taugents at the node will be real if lie within the convex quadrilateral or in one of the portions of space vertically opposite an angle of -the convex quixdrilateral.] 1825. A conic is drawn through four fixed pornts, and 0, 0' are two other fixed points which are conjugate with respect to every such conic : prove that the locus of the intersections of tangents drawn from 0, 0' to the conic is a sextic having six nodes and two cusps. [The cusps are at 0, 0', three of the nodes are at the vertices A, B, C of the quadrangle, and the other three are points A', B', C" on BC, CA, AB such that the pencils A'{AOO'B}, B'{BOO'C}, C {COO'A}, are harmonic. 1826. The evolute of the parabola if = \ax is its own polar re- ciprocal with respect to any conic whose equation is hf^\(x-1aY='HX'a-\ the cissoid x (y? + y^) = cuf is its own polar reciprocal with respect to any conic whose equation is (a;— a)" = 3A'V- 2A'y. Also the cubic X {tf — x^) — aif is its own reciprocal with respect to each of the latter faii^Jy of conies. 1827. A cubic of the third class is its own reciprocal wdth respect to each of a i'amily of conies, the triangle whose sides are the tangents to the cubic at the cusp and at the j)oint of inflexion and the straight line joini)ig the cusj) and inflexion is self-conjugate to any one of these conies ; and the cubic has double contact with each of the conies, the chord of contact passing through the inflexion : also each of the conies lias double contact with another cubic having the same cusp and inflexion and the same tangents at those points. [The cubics may be taken to be ji? - ± i/'z, and the conies are the family XV-3Ax=+2/-0.] 1828. In any cuspidal cubic, A is the cusp, B the inflexion, C the common point of the tangents at A, B ; any straight line through B meets the cubic again in 1\ Q : pro\e that the pencil A {BPQC\ is harmonic. 312 HIGHER PLANE CURVES. 1829. Tho three asymptotes of the cubic (arcal co-ordinates) a* (y + «) -t- tf (;3 + cc) + 2* (.X + y) = meet in the point (1 : 1 : 1), the cubic touches at each anguhir })oint the minimum ellipse circumscribing the triangle of reference, and its curvature at any point of contact is to that of the ellipse as -2:1. If from any j.oint P on this curve AP, BP, CP be drawia to meet the opposite sides of the triangle of reference in ui', B\ C\ the triangle A'B'C will be equal to the triangle ABC. 1830. The area of the loop of the cubic V5-1 is A'j ' ^ ,±-^{l-x-x')dx, where A' is twice the area of the triangle of reference ; and the radius of curvature of the loop at a point where the tangent is parallel to a side is to that at the point on the side as 5 + ^^5 : 2. 1831. The base BC of a triangle ABC being given, and the re- lation tan" A = tan B tan C between its angles ; prove that the locus of its vertex is a lemuiscate whose axis bisects BC at right angles, and whose foci are the ends of the second diagonal of a square on BC as diagonal. Investigate the nature of the singularity at B and C. [Each is a triple point, two of the tangents being impossible.] 1832. A cix-cle is described on a chord of a given ellipse, passing through a fixed point on the axis, as diameter : prove that the envelope is a bicircular quartic whose polar equation is ( - -f- 2m + «i^ - 1 -, + m^ - 1 = 7?i- - \a' a J Xb" J a ma being the distance from the centre of the fixed iioLnt. [One focus of the envelope is always the fixed point, and the other axial foci are at distances from the fixed pomt given by the equation sm' a' on' {a' - ¥) z' - 1am (1 - m'a' - 1 - 2mV) %' + « (1 - m') (1 - ?«V - 2 - 5mV6'' -f V) - 2am (1 - m') b' (1 - «iV - b') = 0. Hence the origin is a double focus if m = 0, e, or 1. When m = the envelope degenerates into the point circle at the centre and the circular points ; when «i = e the equation for the remaining foci is {ez— 2a (1 — e^)}^= 0, so that there are two pairs of coincident foci, and the envelope breaks up into two circles whose vector equations are r, : r^^e :2±e.] 1833. Given the circumscribed and inscribed cu'cles of a triangle, the envelope of the polar circle is a Cartesian. HIGHER PLANE CURVES. 313 [The given centres being A, B and AC a straiglit line bisected in li, C will be the centre (or triple focus), B one of the single foci, and the distances of the others from C are given by the equation ex' - 2 («'= - 3a& + 6-) a; + c (a - 26)- = 0, where a, h are the radii and c the distance between the centres {=Ja'-2ab).] 1834. The equation of tlie nodal limaqon r = 2 (c cos ^ — a) becomes, when the origin is moved along tbe initial line through^ a space 2 8 c —a r« -. 2r (c + ^%os e) + ("^y = 0, (so that the cui*ve is now its own inverse with respect to the pole); and, if OPQ be any radius vector from this pole, A, B the vertices, arc BQ-2iTcAP= 8a sin | A OP. If S be the node and any circle be drawn touching the axis in aS^ and meeting the curve again in P, Q, OPQ will be a straight line, the tangents to the curve at P, Q will intersect in a point R on the cii'cle such that aS'^ is parallel to the bisector of SOQ, the locus of R will be a cissoid, and that of R' (where RR' is a diameter of the circle) a circle. 1835. In the trisectrix r = a (2 cos ^ ± 1), S \5 the node, RSR' a chord to the outer loop, SpP, SqQ two chords inclined at angles of 60° to the former {RSP = P/SQ = QSR' = GO"}, and A is the inner vertex: prove that P, A, q, R' are in one straight line and R, f, A^ Q in another straight line at right angles to the former. 1836. A circle touches a given parabola at P and passes thi'ough the focus S', and the other two common tangents intersect in T : prove that SP is equally inclined to ST and to the axis of the pai'abola, the diameter of the circle through aS^ bisects the angle PST, and the locus of T has for its equation (/ + 28aa; - 96a')'' = 64a (3a - x) (7a - x)'. 1837. The locus of the common points of circles of curvature of a parabola drawn at the ends of a focal chord is a nodal bicii-cular quartic which osculates the parabola in two points whose distance from the directrix is equal to the latus rectum. [The node is an acnode, and the equation of the cun'e when the pole is at the node is r = 2a (cos 6 + j3 + i cos' 0). Two foci are at infinity, and two are the points (vertex of parabola origin) 2x = 3a, 2y = ± 9a.] 314 HIGHER PLANK CURVES. 1838. The envelope of the radical axis of two circles of curvature of the ellipse a*// + b'x' - «^6* drawn at the ends of conjugate diameters ia the sextic (of the class 6) Laving asymptotes 2(-±^j±3 = 0. [The curve has four cusps, x^=2a', 2/=0; y^ = 2b', x = ; four acnodes, — » = "i^s = f , and two cruuodes at infinity.] 1839. A circle is described with its centre on the arc of a given ellipse and radius Jr'^ — c*, where r is the focal distance and c a constant : prove that its envelope is a bicircular quartic which has a node at the nearer vertex when c = a(l— e), and four real axial foci when c is < « (1 — e) or > a (1 + e). [The polar equation is, focus of ellipse being pole, (''+-:) + 4rte ( r + - j cos ^ - 46^ = 0, and any clioxxl through the pole has two middle points on the auxiliary cii-cle of the ellipse. The distances of the foci from the pole are given by the equation ?• H — = 2 (6' =fc ac), and the points of contact of the double tangent lie on the ellipse CO X , ,„ -, 1840. The straight line joining the points of contact of parallel tangents to the cardioid r = 'Za {I — cos 6) always touches the curve 2rcos ^ = a (1 — 4 COS' ^) ; and an infinite number of triangles can be i ascribed in the cardioid whose sides touch the other curve. [This envelope is a circular cubic having a double focus at the cusp of the cardioid and two single foci oxi the prime radius at the distances — a, 3a respectively from the cusp ; and, if r^ , r^ , r^ be the distances of any point on the curve from these three foci, r^=2r^ + 3>y, ?'j being reckoned positive for the loop and negative for the sinuous n branch. Another form of the equation is 2r cos ^^ = a, the origin being at the centre of the fixed cii'cle when the cardioid is generated as an epicycloid.] 1841. A point P moves so that OP is always a mean propoi-tional between SP, IIP ; 0, S, II being three fixed points in one straight line : prove that, if lie between S and //, another system of three points 0', S', II' can be found on the same straiglit line such that O'P is always HIGHER PLANE CURVES. 315 a mean proportional between S'P and WP ; that 0' will lie without S'Jr, and the ratio 0'>b" : O'W will lie between ^2 - 1 : J'2 + 1 and J2 + l:J-2-l. . [If OS^a, OH^h, 00' = z, OS' = x, Oll'^y; s=-^''^,, and ^ = A - i ; ) • The locus of P is a circular cubic whose a ■¥ CO \a + 0/ real foci are S, H, S', W, and a vector equation is l.SP-m. IIP^ {m-l) S'P = 0, , I in - b{a + b-hja' + b'-6ab) a{a + b- Ja' -h b' - 6ab)'^ 1842. With a point on the directrix of the parabola i/' = iax as centre is described a circle touching the parabola : prove that the locus of the common point of the other two common tangents to the circle and the parabola is the quartic (if - 2axy + ia{x+ laf {\x + 3a) = ; also if on the normal at the point of contact of the circle and parabola be measured outwards a distance equal to one-sixth of the radius of curvature, the envelope of the polar of this point with respect to the circle is 2/^ + 8a(.x--2«)' = 0. 18-43. A circle drawn thi-ough the foci B, C of a rectangular hyper- bola meets tlie curve in P, the tangent sd, P to the cii-cle meets BC in 0, and 0(2 is another tangent to the circle: prove that (1) the locus of Q is the lemniscate (Bernoulli's) whose foci are B, C and that OP is parallel to the bisector of the angle BQC ; (2) if OP'Q' be drawTi at right angles to OP meeting the circle in P', Q', the locus of P' will be a circular cubic of which B, C are two foci, and the two other real foci coincide at J, a point dividing BC in the ratio J '2 — 1 : J 2 + \, {^B being the nearer porat to 0) ; (3) the vector equation is {J2 - 1) CP' - {J2 + \)BF = 2AP', TiC or AB . CF + AC . BP = -f- . AP' ; and (4) the angle Q'QP exceeds the angle PQF by a right angle. 1844. Any curve and its evolute have common foci, and touch each other in the (impossible) points of contact of tangents drawn from the foci. 1845. Trace the cui-ve 4.ry {x + y -a-b) + ab (x + y) = ; and prove that if (x*, , yj, (x^, y^ be the ends of a chord through the origin ^, + ^'2 + y, + y. = a + b. 31 G HIGHER PLANE CUllVES. Also j)rove tliat the area of the loop is 2 (a -by [' Jo n/i-c^:-^")' 1846. A circle is dcscribctl with its centre on the axis, and the points of contact of the common tangents to it and to the fixed circle x'' -^ if - a" lie on two straight lines : prove that the locus of the points of contact on the variable circle is the two cu-rves {x- -if- 2ay +{x^ yy = 2a' {x ± y)', that these curves osculate in the points ± a, 0, and that the area of each common loop is «" ( 7 - log 2 j . 1847. The fixed points S, II are foci of a lemniscate {in8P.IIP^SU% and the points Z7, F, F', V its vertices, a circle throiigh *S', H meets the lemniscate in 7?, R' (on the same side of 811) and UR\ VR', V'R', U'R' meet the circle again in Q, P, P, Q' : prove that the straight lines QQ\ PF intersect 8R in the same point as the tangent to the cii-cle at jB, and each is equally inclined to SR, IIR. Also each of the points Q^ P, P\ Q' is such that its distance from R is & mean proportional between its distances from *S' and //, as also is its distance from the cor- responding vertex. The locus of any one of the points for different circles is therefore an inverse of the lemniscate with respect to one of its vertices, the constant of inversion being the rectangle under the focal distances of the corresponding vertex, sign being regarded, [The curves are axial circular cubics, similar to each other, and any one is the inverse of any other with respect to one of its vertices, the constant of invei'sion being always the rectangle under the distances of the centre of inversion from S, H. The loci of P, P (and of Q^ Q') ai-e images of each other, or theii* centre of inversion is at co . Each of the four curves has one vertex peculiar to itself, and is its own inverse with respect to that vertex, the constant following the same rule as for two different curves. The linear dimensions of the loci of Q, P are as Jn - 1 : Jn + 1. Each curve h&s S, H for two of its foci ; and, for the locus of P, the two other real foci (^3, F^ divide SH in the ratios SF^ : IIF^ = Jn{n-l) + j7i{n + l) + 1 : - Jn {n - 1) + Jn {n + 2) + 1, IIF^ : SF^ = Jn {n + 1) + Jn {n - 1) - 1 : Jn{n+l) - Jn {n - 1) - 1 ; and, if r,, r^, r^, r^ denote SP, EP, F^P, F^P, 2 Jn- lr^ = {Jn-l+Jn+l + 2 Jn) r^ + (Jn -1-Jn + l- 2 Jn) r^, 2 J^iT^r^ = (y?i^T- V?wT+ 2 ^n) r, + {J^i^^^ + Jn + 1 - 2 Jn) r^.] 1848. Tangents inclined at a given angle a are drawn to two given circles, whose radii are a, h and centres at a distance c : prove that the locus of theii- point of intersection is an epitrochoid, the fixed and rolling HIGHER PLANE CURVB^C/j^ ,, ^17 • 11. 1 /. !• /a^ + b' -2abcosa i ^i t . /. circles being each of radius . / -. , and the distance of ° V sin a the generating point from the centre of the moving circle being e. 1849. In a three-cusped hypocycloid whose cusps are A, B, C, a chord APQ is drawn through A : prove that the tangents at P, Q will divide BG harmonically, and their point of intersection will lie on a conic passing through B^ C ; also the tangents to this conic at B, C pass through the centre of the hypocycloid. 1850. A tangent to a cardioid meets the curve again in P, Q : prove that the tangents at F, Q divide the double tangent harmonically, and the locus of their common point is a conic passing through the points of contact of the double tangent and having triple contact with the cardioid (two of the contacts impossible). — J — i — iS [The equation X +Y + Z =0 will represent a cardioid when X=x + iy, Y=x — ii/, Z—a; and a three-cusped hypocycloid when X=x + yJZ, Y=x-i/j3, Z=da-2x.] 1851. Chords of a Cartesian are di-awn through the triple focus: prove that the locus of their middle points is (r* - be) {}•' - ca) (?•- - ab) + a'b'-c- sin* ^ = 0, a, b, c being the distances of the single foci from the triple focus which is the origin. 1852. Two points describe the same circle of radius a with veloci- ties which are to each other as m : n (m, 7i being integers prime to each other and 7i>7n); the envelope of the joining line is an epicycloid whose vertices lie on the given circle and the radius of whose fixed circle is a . When - 7n is put for ?«, the points must describe the circle n + m ^ ^ in opposite senses, and the envelope is a hypocycloid. Hence may be deduced that the class number is 7?i + n. 1853. An epicycloid is generated by circle of radius ?Ha rolling upon one of radius (n — 7h) a, 7n, ?i being integers prime to each other, and in the moving circle is described a reguUir ?n-gon one of whose corners is the describing point ; all the other corners will move in the same epicy- cloid, and the whole epicycloid will be completely generated by these 7/1 points in one revolution about the fixed circle. The same epicycloid may also be generated by the corners of a regular 7i-gon inscribed in a circle of radius na rolling on the same fixed circle with intei-nal contact. 1854. In an epicycloid (or hypocycloid) whose order is 2p and class p + q, tangents are drawn to tlie curve from any point on the circle through the vertices ; their points of contact will be corners of two reguUir polygons of ]) and q sides respectively inscribed in the two moving circles by which the curve can be generated which touch the circle through the vertices in 0. 318 HIGHER PLANE CURVES. IS')."). Tho locus of the common point of two tangents to an epicy- cloid inclined at a constant angle is an epitrochoid, for wliicli the radius of the tixed and moving circles are respectively n +b . a + b ... sm r., a- T / oj\ sin , a a{a + 2b) a+2b b {a + 2ft) a + 2ft a + 6 sin a ' a + ft sin a and the distance of the generating point from the centre of the moving circle is ft sm -ra («+ 2ft) : ; ^ ' sm a where a, ft are the radii of the fixed and moving circles for the epicy- cloid, and a is the angle through which one tangent would turn in passing into the position of the other, always in contact with the curve. 1856. The pedal of a parabola with respect to any point on the axis is a nodal circular cubic which is its own inverse with respect to the vertex A, the constant of inversion being the square on OA. If 06*' be a straight line bisected in A, PQ a chord passing through 0', OY, OZ perpendiculars on the tangents at P, Q, then A^ F, ^will be coUinear and A Y . AZ = A 0^. [If OA - 6 and 4a be the latus rectum, the distances of the two single foci from 0, the double focus, are given by the equation a;^ + \ax — 4a6, and the vector equation is, for the loop, »•„ r, 2 ^a ^ ^a + b- Ja Ja + b + Ja ^ r being the distance from the intex'nal focus. The difference of the arcs s^ , s^ from the node to corresponding points Y, Z on the loop and sinuous branch is determined by the equation ds^ c?Sj sin , ■== INTEGRAL CALCULUS. 1857. The area common to two ellipses wMch have the same centre and equal axes inclined at an angle a is 2ab 2ab tan («* - ¥) sin a * 18.58. Perpendiculars are let fall upon the tangents to an ellipse from a point within it at a distance c from the centre : prove that the area of the curve traced out by the feet of these perpendiculars is ^(a' + b' + c'). 1859. The areas of the curves a'l/^ (x — by = («" - X') {bx — cc^y, x^ + tf = a^, (b > a) A' — A are A, A': prove that the limiting value of -r , as b decreases to a, o — a is Gira. 1860. The sum of the products of each element of an elliptic lamina multiplied by its distance from the focus is ^ Ma (2 + e^), M being the mass of the lamina, 2a the major axis, and e the excentricity ; and the mean distance of all points within a prolate spheroid from one of the foci is ^ a (3 + e'). 1861. Prove that the arc of the curve y = J iij' — b'^ i\ - co& \ between x = 0, x- 'itrb, is equal to the perimeter of an ellipse of axes 2a, 2b : and determine the ratio oi a : b in order that the area included between the curve and the axis of x may be equal to the area of the ellipse. [a:b = 2 : JX] 1862. Find the whole length of the arc enveloped by the directrLx of an ellipse rolling along a straight line during a complete revolution; and prove that the curve will have two cus^^ if the excentricity of the ellipse exceed ^— — . 320 INTEGRAL CALCULUS. [The arc s is determined by the equation ds-aL ecosi/r {I - e^) e cos if/\ #~e I "■yrr77ii?^~(l-e=sin»?r ^ being tlio angle through which the directrix turns.] 18G3. A sphere is described touching a given pLme at a given point, and a segment of given curve sui-face is cut off by a plane parallel to the former : prove that the locus of the circular boundary of this segment is a sphere. 18G4r. Two catenaries touch each other at the vertex, and the linear dimensions of the outer are twice those of the inner ; two common ordinates JfPQ, mpq are dra^vn from the directrix of the outer : prove . that the volume generated by the revolution of the arc Pp about the diiectrix is equal to 27r x area MQqm. 1865. The area of the curve r = a (cos ^ + 3 sin Oy ^ (cos ^ + 2 sin By included between the maximum and minimum radii is to the triangle formed by the radii and chord in the ratio 781 : 720 nearly. 1866. Prove the results stated below, A denoting in each case the whole area, x and y the co-ordinates of the centre of inertia of the area on the positive side of the axis of y ', (1) the curve («^ + x^) if - ia?y + a* = 8a*, ^ = 5W, 5;- = £log(2 + ^/3), y = -^; (2) the curve y' (3a^ + x^) — ia^y + a;* = 0, A 11 2 - 4«4-31og3 _ a ,^^ ,„ _ ., (3) the curve y^ (a' + x') - Wy + (x' - 2ay=0, , - 8a A = Tra-, cc= — {V3-log(2 + V3)}, y = a\ (4) the cui-ve y^ (a? + ^) - Imax^y + a;* = 0, [in > 1), A-^l,,, W^n- - _ 4« («^' - 1)U 3 \^m' - 1 - m log (m + Jm' - 1)} y = {in- 1) a. 1867. Prove that the curve whose equation is y^ (a* + X-) - 2m {m+\) 6?y + 4a;* - 4 {mi? + m - 1) aV + (m + 1) (4ni' - 3m + 1) a* = consists of three loops, the area of one of which is equal to the sum of the areas of the other two. INTEGRAL CALCULUS. 321 18G8. For a loop of the curve x^if - W>/ + {3a' - x')' = 0, A WO -K ,'X^ - iJ^/3- log (2 + ^3) } a 9^3-4^ 1869. The area of a loop of the curve 2^ {ia" - x') - 4a'y + (a' - x'Y = is a" (3 - 2 log 2). [This curve breaks up into two hyperbolas.] 1870. In the curve (ma' + a:') y' - 2ay {a' - x') + '^LZ^ = 0, _ _ a 3 {m + 1)^ ^tn - 3m? -2m + 2 ^"^^ lii 3m + 4: ,„^, _ , a cos 3^ a sin 35 , ., , 1871. Trace the curve x = — 7—^ryr, y= . ^^ , and prove that sm 2d sm 26 the internal area included by the four branches is 2 ^/3 a'. 1872. Trace the curve whose equation is a; = 2a sin-; and prove that each loop has the same area Tra' and is bisected by the straight line joining the origin to the point where the tangent is parallel to the axis of y. 1873. The area of the loop of the curve rt^""''y' = nV" , when n is indefinitely increased, \sj2Tra'] and the area between the curve 0^ ^ a'"'^ if — itaf" and the asymptote, when n is indefinitely inci'eased, is 2 J2^a\ 1874. The areas of (1) the loop of the curve y" {a + x) = of (a - .r), (2) the part between the curve and the asymptote differ by — ^^— . 2n 1875. Prove that the whole arc of the curve Sa'y* = x' (ft* — 2a;') is ira ; and for the part included in the positive quadrant, the centre of gravity of the area is [ y^; — . , ^t^ ) ; the centre of gravity of the arc is (^ — Tjr— , T— ) ; the centre of gravity of the volume generated by revolution about the axis of a; is ( „ — ;:; , ) ; and that of the area of the surface generated is [ ■ . ■ , J . 3a w. p. 21 322 INTEGRAL CALCULUS. 1876. Prove that the curve h^if = s? {2a - x) is rectifiable if 1877. The aix; of the curve ?• (c* + 1) = a (e^— 1) measured from the origin to a point (r, 6) is ad -r ; and the corresponding area is ^a'e-a/r. 1878. Tlic whole arc of the curve .r§ + ?/§ = a! is 5a 1 1 + ^j^ log (2 + ^3)1 . 1879. The arc of the curve ic = « (2 - 3 cos ^ + cos 3^), y = ZaJ-2 {29 - sin 26), from cusp to cusp, is 14a. 1 880. The curve 27 {f/- - 8ax - aj = Sax (9a + 8a;)' is rectifiable. [We may put y = af, 8a; ^ 3a (1 - f)~, and 8s = Zaf (2 + f), measur- ing from the cusp.] 1881. The arc of the curve a; = a (6 sin 6 + sin 3^), y = 3a (2^ + sin 2^) measured from the cusp (at the origin) is a (12 sin 6 + sin 3^). 1882. The curves whose intiinsic equations are ds d4 (1) ^^ = asec^|- ^tan|, ds 1-cos.^ ^-'f d4-'^"'{l+co^4>y' are both quintics : in (1), s" = x^ + ty'^, and in (2), s^ = x~ + \ly^, x, y, s vanishing together and ^ being measured from the axis of x ; also 5I- the area of the loop in (2) is |- — ^ a' and its centre of gi'avity divides 3T the axis in the ratio 63 : 80. 1883. The curve whose intrinsic equation is . = ^tan^f(2 + tan'|) is a quartic. [y" + 8a {x - a) y^ + ■ ^^„ =0.] 1884. In the curve y* - Za'xy + 3a* = 0, the arc, measured from a CI? point of contact of the double tangent through the origin, is x. 1885. The arc of the curve ax= if -2a^\og — a*, measured from 2?/' (0, «) is -^ - a; - 2a. INTEGRAL CALCULUS. 323 1886. The whole area of the curve x = asm6{l5- 5 sin" 9 + 3 sin^ 6), y = 10a cos' 6, is ^"i— 7ra^; the arc in one quadrant is 17«, and its centime of gi'avity lies outside the area. 1887. A hypocycloid is generated by a circle of radius na rolling within a circle of radius {27i + l)a, (n integi-al), and an involute is drawn passing through the cusps : prove that the area of this involute is to that of the fixed circle as 2n(7i+l)(S)i' + 8)i-l) : (2u+l)*; and the arc of one to the arc of the other as hi (/i + 1) : (2n + ly. 1888. If u^ denote I .r"* ^(x - a) {b - x) dx, and m - 1 be positive, 2 (m + 3) z<^^, - (2m + 3){a + h) n^ + 2mdbu^_^ = 0. 1889. Prove that the limiting values of ,-. ( . TT . "tt . .Stt . . ^,7r)S ( 1 ) < sin - sm — sm — ... sm in - 1 ) - > , \^ n n n ^ n) /o\ f • T" ■ 3 Stt . , OTT . „_, , 1 \ "■) "'^ (2) -sm-sni — sm^ — ...sm (u-l)-}- , { n n n 'U <'> {(' * '^" ¥>) (' ^ '■^" a) •• (i * '^" (" - ') £-.)}" • when n is indefinitely increased are each equal to |-. 1890. An arithmetical, a geometrical, and an harmonical progression have each the same number of terms, and the same fii"St and last tei-ms a and I; the sums of their terms are respectively s^, s„, s^, and the continued products p^, p,^, p^: prove that, when the number of terms is indefinitely increased. So l — ct ° a ' s^s, Aal ' ^v [The last of these equations is true whatever be the number of terms.] 1891. Prove that, if 7i + l be positive, the area included by the curve x = a cos"'""^' 6, y-h sin""*" 6 and the positive co-ordinate axes is {n + 1) a6 r (h + 1) r ()i + 2) ^ r {2,1 + 3), and tends to ah J air -=- 2°" *^* as n tends to infinity. Also, by considering the arc of this curve, prove that («+l)/-' , ^b I V«'(l+<" + 6'(l-<'fZc»>2''^'-l<2-*'; and that the limit, when n is infinite, of 112-" \ Ja*{\+xy'' + b\\-xY''dx is 2{a + h). ■ -\ 21—2 S24f INTEGRAL CALCULUS. 1892. The lengtlis of two tangents to a parabola are a, b and the included angle w : prove that the arc between the points of contact is , ,, a*+ J'-oifl -coscd) ^ ' a'' + 6* + 2ab cos w rt'i'sin'ti) , J ^a''+6'' + 2rt6cosa» + 6 + acosa) log («*+6*+2aJcosa))5 a^a*+6'+2a6cosa)— a-6cosa) 1893. The general integi-al of the equation- — 7,= — --may be ^ ° ^ cos 6 coH (f> -^ written sec^ 6 + sec^ ^ + sec^ /x - 2 sec ^ sec ^ sec /a = 1 ; and that of the equation -r - ' — = . may be written s(? ■¥ ]f - 2Xxy + X^ = 1. Jx' - 1 Vy' - 1 [That is, the differential equation of all conies inscribed in the parallelogram whose sides are of = a', y^ = V, is — = —j^ — .] Jx" - a' Jy' - 6' 1894. The complete integral of the equation de dxji may be written in the form cos'' ——^ + ~ — - sm^ —T^ = cos^ a' + \ 2 b' + \ 2 2 ' where e* = 1 — 5 : prove that this is equivalent to the ordinary form cos 6 cos = cos a, where tan k = * / , A , 4 • ^ '^ ^ r-j 2 V b-{a' + X) Also prove that a particular solution is (1 - e' sin' 6) (1 - e' sm' <^) = 1 - e\ 1895. The area of each curvilinear quadrangle formed by the four parabolas r/^ = mux - u', when u has successively the values a, b, c, d, is 2 ,— ,— 3^ iJ^ - v/^O U^ - V<^) (<^ + J^^^i +d-a- Jab - b), {a-5)_(a>-c)|— ' ' h \ a—d b—c ) ,g. /•« {(r^ - a:) (.^' -b){:x- c) (x - d)Y~\lx ^^' I ^ {a-x)(x-d) {x-b)(x-c) Y'"-'' Jb \ a-d b — c J are respectively (1) {r {m)Y - r {2m), (2) {a - c)"""' (a - d)"-' {V (m)}"- -r T {2ni). 1902. Tlie limiting values, when b incx-eases to a, of (1) /•« {a-xy"-'(x-cY-'dx I ({a -x) (x- d) (x -b) (x — c) V Jb \ a-d "*" b-c / n (x-bY~'{. I f {a-x){x-d) ^ Jb \ a-d {x-by'~'{x-dY-'dx {x — b){x- c)] b-c j are respectively (1) -(a-c)""', (2) — («-fZ)"'-'. 1 903. Prove that, i£a>b>c>d, r ± ^ / (..-^)(6-c ) / {a-b){c-d y I (a-a!)(a;-t/) ^ (a;-&)(a;-c) V («-6)(c-rf) Sf {a-d) {b-c)' Jb a-d b-c 1 904. Having given 2a; = r (e^ + e"^), 2?/ = ?• (e^ - e" V P^'O^® that • a> * CO ^ CO - oo I 1 Vdxdy=\ I V'rdrdd, V being a function of a;, y which becomes V when tlieir values are substituted. INTEGRAL CALCULUS. 827 1 905. Having given AV = Yy^ = Z'^ = . . . = xyz . . . , prove that {[{... YdXdYdZ ... = 2"-' («- 2) IJL. V {xyz ...)"-' dx dy dz ..., V being a function of X, F, Z,... which becomes v when their values are substituted, and n being the number of integrations. 190G. Having given x-vy + z = u, y + z = uv, z = tivw, prove that / / I Vdxdydz= I j f V it'v du dv dw; Jo Jo Jo Jo Jo Jo also, having given x^ + x^ + Kg + a;, = w, , x^ + x^+x^ = ti^u^, x.^ + x^ = u^u^u^, x^ = u.u^^^i^^ prove that j j I Vdx^dx^dx^dx^= I / / j V u^Wdu^du^du^du^; and the cori'esponding theorem with n variables. 1907. Having given x^ = r sin ^^ cos 0,^, x^ = r cos 9^ cos 6^, x^ = r sin 6^ sin $^ cos ^^, aj^ = r sin ^^ sin ^^ sin 0^ , x^ = r cos 6^ sin ^3 cos ^g, x^ - r cos ^, sin 9^ sin ^j, prove that I / ... Vdx^ dx„ dx^ dx^ dx^ dx^ Jo Jo jr jr 2 = f 7 7 .'0 Jo Jo ...V'r^ sin- ^, cos- 0, sin ^^ sin 9._^ dr d9^ ...d9^ 1908. Prove that III... dx^ dx^ ... dx^^, taken over all real values of a;,, x,^,... x^^ for which x' + x,,^ + ... + xj + 2m (x^x,^ + x^x^ + x^x^ + . . .) :|- 1, is equal to f "^ \^ I ^ ~ "^ 1 \1 - m) V l+m(«i-l) /?i j\ ' provided that «i lies between and 1. n-\ 1909. Prove that C r dx^dx^+ ...+dv^ _ IT* {n + r- 1 [2r / / ■ " ~ 2n+2r+l ()'„2rVl rx n 1 P" ' •'" •" {a'' + a;.» + cK/+... + Vr'~ lin±_2lzili: n, ?' being positive whole numbers. 328 INTEGRAL CALCULUS. 1910. Prove that jjj--- dx^ dx^ . . . dx^ is equal to _1_ /_JL_f _J_ V^TTi \n{n+l)) Y ("+{)' the limits of the integral being given by the equation X' + Xj' + ... + X ' - X,X^ - X.,X^ - ... - a;„_,X =rr-7 r^ . 1 a a in 2 3 " ' " 2w (W + 1) 1911. Prove that III- "■"'•■•"■ '^ ^'^ and that ///- V 1 -^-^- '■■' -^ ^ ^''^ ^''^ - '^''" (1+1 ■^Ki)V(^0 K^): the integral extending over aU real values for which x^^+xJ'+...+xJ:\' 1. 1912. Prove that / n rn — n ^= o-^(-l) Sin — cos" ' — , Jo (1 +a;)"-(l -a;) 2w ^ '' n ?i ' r having all integral values from 1 to jr - 1 or — jr— according as n ia an even or an odd integer. 1913. Having given the equation and that when £c = - , y = ^—v^ ; prove that, when a; = 2a, 2/ = -^log(2 + V3); and generally that 3/ = a^ cos"' Qj ^ ^^^^I^ when a* < a\ and when x> a. [It would seem that the only form of solution holding generally is 2 /■* dz T Jo i + ^xz + a z INTEGRAL CALCULUS. 329 1914. Haviug defined X, X' by the equations COSX= 1-7H+ ,-7 - ••• +(-1) TTT +(-1) 775 1> [2 ^ ^ ' \2n ^ ' \2n+ 1 prove that I — c?a; = I — fZo; = ^ . Jo aJ Jo ^ 2 1915. Prove that the liiuit, when n tends to co , of Jn I sin" a; I ^^ = J.^°S^'' /o l°g«'log(l4-|;)c/a; = 7r«(log«- 1); . /•! dx _ 2 _ ^ ^ i-i V (l - 2«x + a-) (1 -f 26a. + 6^) " J^b ^"^"' ^^''^^' (1) al, &<1; (3) a>l, 6>1; ,„, f" xdx ira (^ xdx Jo 1+ COS a sin a; sin a' Jq (1 +cosasins;)" (a — sin a cos a) / 7r\ ^— ^' r^2)' = TT ■ sm (7) I a; sin 2ra;€"'ViK = ^ ?•€"'■', 1 a;- cos 2ra,'e~''"f/a; yo 2 Jo :^e--=(l-2/); INTEGRAL CALCULUS. 331 ,^. r x&\nxdx IT. 1+c /■alog(l -c'sin^a;) , (8) -. =- = JT log ij , I — 2_^ . ' (Ix ^ ^ Jo Jl-c'Hin'x 2c ''I -c Jo amx = -(sin-'cn (c^r "^ ,, „. /■"' cos" arc — cos" hx ^ ,, , 3 . 5 . . . (?i - 1) ^ ^ i-« — ^^ '^'''=''^^""^27r::r(^) 3. 5, ..71 ^^"^^-")2T4t::T«^i)' .,^, r"" A coH ax + B cos bx + C cos ex + ... , (1,) j_^ ^. <& = -7r(ila + -S& + Cc+ ...); .. . r" ^ cos ax + B cos 6.r + C cos ex* + (?a; '^^ = log(«-'*t-^c-^..) (tlie two last integi-als are finite only Avlien A + B + C + ... =0); (19) f J -a (20) f A , cos a,.r + A„ cos «,« + . . . + -4„ cos « aj , (when finite) = [J]^ irS (^a'""') i A, sin a, a; + A^ sin ff,.r + ... +A sin « a; , —^ * ^ — ^.:t-, = doc (when finite) = Lil 7r2 (.!,« '") ; 332 (21) / .'0 (20) /; INTEGRAI. CALCULUS. A, cos a.jc + A„cosa,x + ... + A__ con ax ■ ^ — — ^.^ — ^ ^ ^. oT*' ~ dx (-1)"- (when finite) = ^ ' 2 {Aa^" log a) ; A , sin a,x + A^ sin ajK + . . . + A sin a aj , _1 ! S ^^ 5 !i. dx i-iy (when finite) = j^— :fj 2 (^ ,«/""' log«r)^ /■" sin* re /"'^ (23) I — ,- F (sin- a;) c/a; = / F (sin* a;) clx ; J-so ^ Jo /nt\ r log (1 + 7^2. sill* a;) , , ,, ^ ^ ]-«> x' -dx=27r(Jl+m-l), («i^4.^4. I (35) — logT — K— 7172 = 2^*^^ "' ('^^^) — log :; -. — = TT sin ' (m), {m1); ■'0^-1 Lsin^j > 71' n 334! INTEGRAL CALCULUS. (44) I - — tau"' (c sin x) dx = -„- \og(c + Jl+ c") : It (45) J^'log(cota;-l)(/x- = ^log2, (46) f"-log( ^^^^^""'^^-"""^^^^l ^ 'Jo i^" U + 2?i cos (cc + a) + ?i*j \1 + m cos aj ,,„- /""^ €~""'cosa; , /Tr^l+m^+m (4/) I ^ cZa;=Vo^^^-n 3 — , h Jx '^""-^ 2 1+m' ' f 2 Sill. 77/ (48) I c" '^os ^ cos (a; + w sin a;) cZa; = ' : .'0 n (49) / (1 -cc")" fZa;= (1 + a;") «f^^ = | / (l+a;")"f/rB Jo Jo Jo ,r-r\\ /"°° x"'~^dx sm(7i — m)a it ^ ' In X + 2x cos na + 1 sm ?ia . mir n sm — (m < 2n, wa < 7r) ; (51) I (sin a;)""' dx \ (sin a;)" dx= — , Jo Jo n j (sin xY dx I (sin x)~" dx = — tan -^, (n2, 2/ = 1.] imiting value of the iufi 4 1923. The limitine: value of the infinite sei-ies when 9i is 00 , is . e— 1 1924, Prove that r /I , (^ fz , 2esina 2esuiy3 / V A ~^ cosaiaa;— / ^1—e cos x ax = -. — - ■ . •"' ■'0 Vl+^cosa ^l-ecos;8' if a, B be angles < tt such that tan h = a / :; tan - . 1925, Prove that 7 , , 4sm'^ dx ['"ax ^ ra. dx /■" Jn (I- sin B cos a)^ .'0 (1 - Jq (l-sin (icosx)^ .'o (1 -sin /Sees a;)^ cos' /i if /? be any angle between - - and — , and cos a = tan ~ . 2 H 2 1 T"^ 1926, Prove that, if ^ (c) = - I log (1 + c cos x) dx, 2 — 5' is even or odd. 1929. From the identity I {a + sin'* a;)" cos xdx= \ (\ + a— sin* .r)" sin x dx, Jo .'0 or otherwise, prove that «- X ** «"-' ^ ^'^^^~^) „'■-« '^ (w - 1 ) (^ - 2) „_, = («+!)— 3- («+l)" '+ 3^^ ^(a+1)'-' 37577 ^ "^ ^ and prove in a similar manner that ,_i 3?i(n-l) ,_, 3. 5n(n-l) (n-2) ,. = (a + 2)"-w(a+2)- + m)^ ^a + 2)" 3 . 5» (y^ - 1) (71 - 2) ^ W ^ ^ 1930. Prove that, c being < 1, r • -1 / • \ 7 f fx -1 /2c cos x\)' IT \ sm (c sm a?) cZic = I -'.tan ' ( — j- j ^ 0?^. 1931. On a straight line of length a + h + c are measured at random two segments of lengths a + c, b + c respectively : prove that the mean value of the common segment is & + c - 7^- , a beins: > b. * 3a' ° 1932. A point is taken at random on a given finite straight line of length a : prove that the mean value of the sum of the squares on the two parts of the line is | a", and that the chance of the sum being less than this mean value is -^ . \/3 INTEGRAL CALCULUS. 3.'i7 1933. A triangle is inscribed in a given circle whose radius is a : prove that, if all positions of the angular points be equally probable, the mean value of the perimeter is — , and that the mean value of tlip TT '12 radius of the inscribed circle is « f — 5 - 1 1934. The perimeter {2a) of a triangle is given and all values of the sides for which the triangle is real are equally probable : i)rove that the mean value of the radius of the circumscribed circle is five times, and that the mean value of the radius of an escribed circle is seven times the mean value of the radius of the inscribed circle. rmi 1 1 47rtt 47rrt 47r« T The three mean values ai-e :, . _ , -^^ > t^- • ■- lOo 21 lo -^ 1935. The whole perimeter (2o) and one side (c) of a triangle are 77* I given, prove that the mean value of its area is -^c J aici — c); and that the mean value of this mean value, being equally likely to have any value from to a, is ktc n'- 193G. The mean value of the area of all acute-angled triangles inscribed in a given circle of radius a is - - , and the mean value of the area of all the obtuse-angled triangles is - . TT 1937. The mean value of the perimeter of all acute-angled triangles inscribed in a given circle of ladius a is — ^ , and that of the perimeter . , , , , • , . lG(7r-l)a of the obtuse-angled triangles is — 37" 1938. The mean value of the distance from one of the foci of all 3 -I- e° points within a given prolate spheroid is a — — - . 1939. The mean vahie oi J.ryz where x, y, z arc areal co-ordinates 477 of a point within the triangle of reference is , ^_ ; and the mean value of Jwxyz, where 10, r, y, z are tetrahedral co-ordinates of a point within the tetrahedron of reference is ^^ . Also the mean value of (wxyz)"'' is 6 (r (n)y -- r (in). 1940. Prove that the mean valui; of Jx^x^x.^...x_^, ior all positive values of fc, , a;^, . . . such that x^ + Xi+ ... + .r^= 1 is T{h) I T (^j i 4- T f y j ; and, more generally, that oi (x^x^ ...xj~\ r being positive, is T(n){V(r)]'^T{m-). w. P. 22 ,S38 INTEGRAL CALCULUS. 1941. In the equation x^ - qx + r = it is known that q and r both Ho between — 1 and + 1 ; assuming all values between these limits to be e(iually probable, })rove that tlie chance that all the roots of the equation shall be real is 2-^15^3. 1942. A given finite sti-aight line is divided at random in two points : prove that the chance that the three pai'ts can be sides of an acute-angled triangle is 3 log 2 — 2. 1943. A rod is divided at random in two points, and it is an even chance that n times the sum of the squares on the parts is less than the sqiiare on the whole line : prove that n{iTr + 3 J3) = 127r. 1944. On a given finite straight line are taken n points at random : prove that the chance that one of the 7i+l segments will be greater than half the liueis(?i+ 1) 2~". 1945. A straight line is divided at random by two points : prove that the chance that the square on the middle segment shall be less than the rectangle imder the other two is (47r - 3 ^3) ^ 9 ^3 ; and the chance that the sqiiare on the mean segment of the three shall be less than the rectangle contained by the greatest and least is -41841... 1946. A rod is divided at random in three points ; the chance that one of the segments will be greater than half the rod is '5, and the chance that three times the sum of the squares on the segments will be less than the square on the whole is tt -4- 6^3. Also the chance that in times the sum of the squares on the segments will be less than (n+1) times the square on the whole {n > 3) is tt ^ 2n^. 1947. A given finite straight line is divided at random in (1) four points, (2) n points ; the chance that (1) four times, (2) n times the sum of the squares on the segments will be less than the square on the whole line is ^'^ 100^/5' ^-^^ V^TTi W(« + i)| ^/«^ \- 1948. A given finite straight line of length a is divided at random in two points; the chance that the product of the three segments will exceed ^g«Ms f'/I^TT^T^ j^ Z 4 ojil ^'''P^ J I + 2 cos 3xdx. 9 1949. The mean value of the distance between two points taken at random within a circle of radius a is 121aH-457r; the corresponding mean value for a sphere is 36a -^ 35. The mean distance of a random INTEGRAL CALCULUS. 33fi point \vithin a given sphere from a fixed point, (1) without the sphere, (2) within the sphere, is a- 3a ^ c* c being the distance of the fixed point from the centre of the sphere and a the radius of the sphere. 1950. The mean value of the distance of any point within a .sphei'e of ladius a from a point in a concentric shell of radius h is 3 (« + 6)(5a^+ 76-) 20 ce + ah + ¥ ' 1951. A rod is marked at random in three points; the chance that n times the sum of the squares on the segments will be less than tlio ., ... 7r/4-w\l T (36 .. /12 \l| square on the whole is ^ I > or 75— tt, { 10- o r, ^ 2\ n J 6^/3 {n \n J ) according as n lies between 3 and 4 or between 2 and 3. 1952. A point in space is determined by taking at random its distances from three given points A^ B, C : prove that the density of distribution at any point will vary directly as the distance from the plane and inversely as the product of the distances from A, B, C. 1953. Points P, Q, R are taken at random on the sides of a triangle ABC; the chance that tlie area of the triangle PQR will be greater than in + \) of the triangle ABC, {n being positive and < |), is 3 — 4:71 1 2w + 1 '°«4;^i+8»*(i-'"'"'^'") 1954. A rod is marked in four points at random, A bets B £50 even that no segment exceeds -^^ of the whole: prove that A's expectation is 3s. lid. nearly. 1955. A given finite straight line is marked at random in tliree points; the chance that the square on the greatest of the four segments will not exceed the sum of the squares on the other three is 12 log 2 -TT- 5. 1956. From each of n equal straight lines is cut off a piece at random ; the chance that the greatest of the pieces cut oflF exceeds the sum of all the others is 1 : \n—l; and the chance that the square on the greatest exceeds the sum of the squares on all the others is (If-m 1957. A rod AB is marked at random in F, and points Q, R are then taken at random in AP, PB respectively: prove that the chance that the sum of the squares on AQ, RB will exceed the sum of those on QP, PR is "5; but, when Q, R are first taken at random in AB and P then taken at random in QR, the chance of the same event is ^-(3 -2 log 2). 22—2 340 INTEGRAL CALCULUS. 1958. Three points P, Q, It are taken at random on tlie perimeter of a given st'micii-cle (including the diameter): prove tliat the mean value of the area of the triangle VQIi is a being the radius. 1959. A rod being marked at random in two points, the chanco that twice the square on the mean segment will exceed the sum of the squares on the greatest and least segments is "225 nearly. 19G0. The curve jy {la - r) — a^, 2> being the perpendicular from the pole on the tangent, consists of an oval and a sinuoiis branch : the oval being a circle, and the sinuous branch the curve -0-7.)0 l-cos^ = (l-4.Hl---" r r - 1961. Trace the curve r ~ —f- — „, the prime radius passing thi-oiigh a point of the curve where r=2a: discuss the nature of this point and prove that, if pei'pendiculars OY, OZ be let fall from the pole on the tangent and normal at any point, YZ will touch a fixed circle. 1962. Find the difierential equation of a curve sucli that the foot of the perjoendicular from a fixed point on the tangent lies on a fixed circle : and obtain the general integral and singular solution. [Taking the fixed circle to be a;" + y^ = a*, and the fixed point (f, 0), the diflferential equation is which is of Clairaut's form.] 1963. Reduce the equation {x—'py){x—-\^c^ to Clairaut's form, l>y putting ar= X, y" = F, and deduce the general integral and singular solution. cc^ \r 1 [The general integral is -^ — r + y — r = zy^ ^^'^^ '^^ singular solution is ± x ± y = c. ] 1964. Along the normal to a curve at P is measured a constant length PQ; is a fixed point and the curve is such that the circle described about OPQ has a fixed tangent at : find the differential eq\iation of the curve, the general integral, and singular solution. [Taking for origin, and the fixed tangent at for axis of x, the differential equation is x" + 1xyp-y^ = cy J\ + p~ ; the general integral is of + y' - 2ax + 6" = 0, where b^ (6* + c°) = a'c", INTEGRAL CALCULUS. 34 L and the singular solution is x^+rf^^cy. If the singular yolution lie deduced from the general integral, the student should account for the extraneous factors c and a:.] 1965. The ordinate and normal from a point P of a curve to the axis of X are PM and PG : lind a curve (1) in which PM' varies as PG; (2) in which the curvature varies as PM* ^ PG^, and prove that one species of cun-e satisfies both conditions. [The curve (1) is the catenary — =me + m~^(. <; and the curve (2) is — = me + ?ie % which coincides with the fonner when mn = 1 ; or c 2 V = ^ cos - + jB sin - .1 1966. Prove that the equation 2a-?/-^ -a; (-r^ j +i/-^ = is the general equation of a parabola touching the co-ordinate axes ; and deduce (1) that, if in a series of such parabolas, the curvature has a given value when the tangent is in a certain given direction, the locus of the points where the tangent has this direction is an hyperbola with asymptotes parallel to the co-ordinate axes and passing through the origin where its tangent is in the given direction and its curvature is four times the given curvature, (2) that if a straight line from the origin meet one of the parabolas at right angles in the point {x, y) the radius of curvatui'e at (.c, y) will be 2xy J of + y^ + 2xy cos w . „ / ^ w 7 Sin* (o, {x + y cos w) [y + x cos w) where w is the angle between the co-ordinate axes. 1967. Find the general solution of the equation Vdx'^'^\dxjj ^"'d^Tdaf dx\:^dx\)^' and prove that a singular firet integral is [The general solution is a^ + y-=2a (\y + ^x + /u,), and one general fii-st integral is x + y~-= a {X- + 2X ^ ] .] HOC CliC/ 1968. Prove that the equation (x--h/-2.ry;.)^ = 4«y(l-/) can be reduced to Clairaut's form by putting a;" - y' = 2^ ; and obtaia the general and singular solutions. [(,;=• -y'-Xx-ay^.a' K - X=) ; .r' - y' - ^ 2ay.] 342 INTEGRAL CALCULUS. 1969. Find the general and singular solutions of the equation [Reduce by putting x^ + y'=T, dxy = A' : the solutions are a;' + f- 2,ax)j = a\ x" + 6xy + y" = O.J 1970. Find the genei*al differential equation of a circle toucliing tlie parabola ?/ = iax and passing through the focus ; and deduce that the locus of the extremities of a diameter parallel, (1) to the axis of y, (2) to the axis of x, is (1) 8/ (x + 2ay = 27a {x- + yj, (2) (.r -y' + iaxf = 21 ax {x' + yj. 1971. The equation x'^ (-—] +2ab--^ + y-=0 has the singular solution xy = ah and the general solution is formed by eliminating 6 from the equations x' = Xah £-e (1 - sin 6), y' = \~' ah e« {1 + sin 0). dii X — II -^ ir>>Tn CI 1 ,1 ,. X- y^ a-h dx • .i 19^2. Solve the equation — + "f- = j- , : and examine the ^ a h a-\-h dy' x + y -, ^ dx - + ^ -1 nature of the solution — h "y- = 1. a [The general solution is x^+y^ = 1 log \ ri J , so that a + ^ ^ if • — + T- = 1 is the particular integral corresponding to (7= 0.] 1973. Find the general solution of the equation y + cc -y^ = — - ; and dx examine the relation of the curve y" + 4acc = to the family of curves represented by the equation. [The general solution is {if - 1 2axy - \y + (y^ + Aaxf = 0, and each curve of the family has a cusp on the limiting curve 2/^ + 4aaj = O.J 7 Q 1974. The general solution of the equation 2y = x-~ + — is dx {nf - \ax) (x' - 4Xy) + 2a\xy = 27a-y ; and each such curve has a cusp lying on the curve f = Zax. 1975. The general solution of the equation xi-^\- my + a =0 is found by eliminating j^ between the equations 1 _ X -— (2m -\)x = ap"^ + Xp™"^ , (2m - 1) ?/ = 2ap ' + — ^'"-^ ; except when 2m = 1, when p^x = 2a log^) + X, py - ia log 7? + 2a + 2X. INTEGRAL CALCULUS. 343 197G. Find the orthogonal trajectory of the circles «;"+/- 2\y + a' = 0, \ being the parameter. [x- + y- - 2/x^ -a- -^ 0.] 1977. Find the orthogonal trajectory of the rectangular hyperbolas of — y^ — 2Aic + ft* — ; and prove that one solution is a conic. [The general solution is y (.So;" + 'if - 3a*) = 2/a^, which has an oval lying within the conic 3.t;' + y" = Sa", if ft." < a", an acnode when fx^ — d^ and lies altogether without the conic when /x," > «".] 1978. Prove that the oi-thogonal trajectory of the family r" = A." cos nd is r" = /a" sin nd. 1979. The orthogonal trajectory of the system of ellipses 3a;' + ^" + 3a' + 2\x = is y' = fi{x- -y' - a'). 1980. Integrate the equations : ,„, d'x _ - d^y „ (3) j^ J + 2x + 2»/ = -y^ + a; + 3^ = cos )i(, (^) (l-^-^)£?-2(^^+l)x-|^r(H-fl)y, (5) 2xyzfl +2..2/^^i^+2:f^+2y^ + .= =9, ^ ^ -^ dxi/y -^ dxdy dx -^ dy ' „ d'u d^tt d^u d^u d^u d'u ^ ' dx* dif d^ dydz dzdx dxdy ' /^v d'u d'u d'u _ d'u ^ ^ ' dx' dy' dz dydz ._. d^u d^u d^u dhh _ ^ ' dx^ dtf H^ ~ dxdydz ~ d")/ ^ d'lj dif . , 1981. Having given the equation x -j--^ + 2 j^"^^ + x -- = ; and that, when x = 0, y^O, ^=1, 7rT = 0; prove that, when a; is oo , 3-ii INTEGRAL CALCULUS. 1982. lutegrate the equaduu dif . TI (ijcoa y cos x + sin ^ siu x) - 2 cos y sec .'' : It and examine the nature of the solution y = ^ • [The general solution is ?/ + \ = 2 tan a; ^cos ?/.] 1983. Integrate the equation ^y I \ -J- cos X cos {y - x)= cos y ; and examine if the solution y= 2.« + 2?'7r - - is a general solution. [The general solution is X cos o^ = sin {x - ?/).] 1984. Solve the equation (g + 2/)cota.+ 2(1+2/ tan :«)=/(.r), by putting y=zco5 x. 1985. The general solution of the equation u^{2x+\-it.^^;)=x' is = C + 2 © 1986. A complete primitive of the equation is % + 1 = (.-B + C)*; and another is rt^ + 1 = {C (- 1 )' - |-}^ : also deduce one of these as the indirect solution corresponding to the other. 1987. Solve the equations (1) w,,. + 3z6,-4w/ = 0, (2) u,^^ = x{u^ + u,_X (3) K^,-wJ^-w^^^ + w^, (4) 2 («,,, - 2^,)' = («,^^ + 2«„) K + 2«...)- 1988. Prove that the limiting vahxe of ^^-^^-^^"g l. 3.5... (2 ^1)' when n is an indefinitely great positive integer, is log 2. 1089. Prove that, if r lie a positive integer so de} tending upon x ,•00 tliHt ,r - rTT ahvavs lies betwe-^n - ^ and - . I — dx = jr log 2. 2 2 ;o a; 2 - INTEGRAL CALCULUS. 345 1990. Solve the equations : (1) y'|(x-^ + /+cr) + x-(..^ + /-cr)^0, / - \ , 11 — z ilz ^ z- X dz , ic - y (;)) tan ^ ^ + tan — r — j- =^ tan —>, . ^ ' 2 a.« '2 ay 2 [( 1 ) («" + 2/' + a-)^ - 4a V ^ A, (2) ?/ = (.4 cos x + B sin .x-)"', (3) 2/ = «„ + «,x + rt^rs^ + . . . + a _rc" + 6x-" log ;c + a;" (log a)" ; (4) y-z = {x-ij)f{yz + zx + x7j), (5) cos (y + «) + cos (2; + a.-) + cos {x + y) =y{sin {y + z) + sin (« + x) + sin (.c + y)}.] 1991. Prove the following equation for Bernoulli's numbers : x^nB ■..■ ^^(»-l )0^-2) ^ n(H-l)(»-2)(n-3)(^-4) n-\ to --^ terms, Avhere n is an odd integer. The equation will still be 71+1 true when n is an even integer if we multiply the last term by r- , - being then the number of terms. 1992. Two equal cii'cles have radii 2a, and the tlistance between their centres is 4c, a series of circles is drawn, each touching the previous one of the series and touching the two given circles sym- metrically : prove that the radius of the w"' of such a series is c sin^ a 4- sin {na + yS) sin (/i - 1 a + ^), where a = c cos a. Dediice from this the result when c= a, a-f-(/i + A)(n + \- 1); and the result when c 4 {be — ae), and discuss the special case when (b + cy = 4: {be - ae). SOLID GEOMETEY. I. Straight Line and Plane. 1994. The co-ovdinates of four points are a—b,a — c,a — d; b-c, h - d, h - a ; c - d, c — a, c — h; and d-a, d — b, d— c, respectively : prove that the straight lijie, joining the middle points of any two opposite edges of the tetrahedron of which they are the angular points, passes through the origin. 1995. Of the three acute angles which any straight line makes with th]-ee rectangular axes, any two are together greater than the third. 1996. The straight line joining the points (a, b, c), (a', b', c) will pass through the origin if aa + bb' + cc' = pp ; p, p being the distances of the points from the origin, and the axes rectangular. Obtain the corresponding equation when the axes are inclined respectively at angles whose cosines are I, m, n. [aa' + bb' + cc' + {be + b'c) I + {ca + c'a) m + {ah' + a'b) n = pp' yjl - I' - m' — n^ + 2linn.'\ 1997. From any point P are drawn PM, PX perpendicular to the planes of zx, zy\ va the origin, and a, ^, y, Q the angles which OP makes with the co-ordinate planes and with the plane OMN: prove that cosec^ 6 = cosec^ a + cosec^ fi + cosec'^ y. 1998. The equations of a straight line are given in the forms a + Tnz — ny 6 + nx — Iz c + ly — mx (1) (2) I m n a + mz — ny b + nx— Iz c + ly — mx L 21 N obtain each in the standard form X /* /i 348 SOLID GEOMETRY. [(') mc — lib iia — Ic lb — ma r + m'' + li' "^ /' + ///" + u^ ~ Z- + m^ + TO® I m n ' Mc — Nb Ka - Lc Lb — Ma LI -^M III + Sn LI + Mm + Xn Ll + Min+Na^ ^''' l " " 7i " T' 'J 19D9. A straight line moves parallel to a fixed j)lane and intersects' two fixed straight lines (not in one plane) : prove that the locus of a point which divides the intercepted segment in a given ratio is a straight line. 2000. Determine what straight line is represented by the e(|uations a + niz — ny b + nx — Iz c + ly — mx m- 11 n — l I — 111 ' _^. a + mz — ny b + nx — lz c + ly — mx mc — nb' na — lc' lb' — ma [(1) The straight line at infinity in the plane X (m — n) + y {n—l) + z (l — m) = ; unless la + mb + nc = 0, in which exceptional case the line is indeternunate, and the locus of the equations is the plane X [ill - n)+y {n -l) + z{l- m) = a+b + c; (2) the straight line at infinity in the j)lane x (mc — nb') + y {na' — lc) + z (lb' — ma) = ; unless la + mb + wc = 0, when the locus of the equations is the plane X {mc' — nb') + y {tia' — lc') + z {lb' — ma') = aa' + bb' + cc'.] 2001. The two straight lines - yz zx xy ^ b — c c — a a — b * are inclined to each other at an angle ^ . 2002. The cosine of the angle between the two straight lines determined by the equations Ix + my + nz = 0, av? + by' + c:^ = 0, Z^ (6 + c) + m^ (c + a) + w^ {a + b) Jl'{b-cY+ ... + ...+2irMJc(^V)^{a^c)+... + Z' 2003. A straight line moves parallel to the plane y = z and inter- sects the curves (1) y = Q, z' = cx; (2) z = 0, y-=bx: prove that the locus of its trace on the plane of yz is two straight lines. [The locus of the moving straight line is x={y - z) (j^- -) .] SOLID GEOMETRY. 349 2004. The direction cosines of a number of fixed straight lines, referred to any system of rectangiiUir axes, are (l^, m^, ?i,), [l^, m^, n,^), &c. : prove that, if 2 {I') = 2 (m') = 2 {ii"), and 2 (mn) = 2 {nl) = 2 (Im) = 0, when referred to one system of axes, tlie same equations will be true for any other system of rectangular axes. Also prove that, if these conditions be satisfied and a fixed plane be drawn perpendicular to each straight line, the locus of a point which moves so that the sum of the squares of its distances from the planes is constant will be a sphere having a fixed centre which is the centre of inertia of equal particles at the feet of the })erpendiculars drawn from 0, and that the centre of inertia of equal particles at the feet of the perpendiculars drawn from any other point F lies on OP and divides OP in the ratio 2:1. 2005. A straight line always intersects at right angles the straight line X + 1/ = z = 0, and also intersects the curve y = 0, x' = az : prove that the equation of its locus is X' — 1/^ = az. 200G. The equations a.v + hy + gz _ hx + by +fz gx +fy + cz X y z represent in general three straight luies, two and two at right angles to each other; but, if « -^ = 6 - — = c- ^ , they -will represent a plane and a straight Kne normal to that plane. 2007. The two straight lines x^a ± »/ z cos a sin a ' meet the axis of x in 0, 0' \ and points P, P' are taken on the two respectively such that {\)OP = k.O'F; (2) OF.O'P' = c'; (3) OP + O'P' = 2c : prove that the equation of the locus of PP' is ( 1 ) (.'« + a) (y sin a + z cos a) = k (x — a) (y sin a-z cos a) ; X' 1/' z' (2) _--^-.L__+_^ 1; a c" COS" a c sur a xy az c (3) - -.-- = - (x- - a-) ; ^ cos a sin a a ^ the points being taken on the same side of the plane xy. [Denoting OP, O'F by 2A, 2/x, the equations of PP' may be wriltcii X y—(\-fx) cos a z- {X + fi) sin a a (\ + /i.) cos a {^ — H-) ''•'^ °- * so that, when any relation is given between X, fi, the locus may be found innnediately.] ooO SOLID GEOMETRY. 2008. A triangle is projected oi-tliogonally on eacli of three planes mutually at right angles : prove that the algebraical sum of the tetra- hedrons which have these projections for bases and a common vertex in the plane of the triangle is equal to the tetrahedron which has tho triangle for base and the common point of the planes for vertex. [This follows at once from the equation x cos a + y cos ft+z cos y = p on midtiplying both members by the area of the triangle.] 2009. A plane is drawn through the sti-aight line -= — = -: ^ I m n prove that the two other straight lines in which it meets the surface (h - c) y% (?»2 — 'loj) + (c - ci) zx {71X — Iz) + (a - b) xy (ly — mx) = are at right angles to each other. 2010. The direction cosines of three straight lines, which are two and two at right angles to each other, are (l^, m^, n^), [l^, m^, n^, (Z„ m.^, n^, and am^n^ + hnj,^ + cl^m^ = ani;a„ + hnj^ + cl,jni„ = : prove that am.n. + bnj^ + cijn = U ; and ^-j-j- = = . 2011. The equations of the two straight lines bisecting the angles between the two given by the equations Ix + my + nz = 0, ax" + by^ + ez' = 0, may be written Ix + my + nz = 0, I {b — c)yz + m (c — a) zx + n (a - b) xy = 0. 2012. The straight lines bisecting the angles between the two given by the equations Ix 4 my + nz = 0, ax" + hf + cz" + 2fyz + 2gzx + Ihxy = 0, lie on the cone x" (nh — mg) +... + ...+ y^; [mh - ng -\- 1 [b —c)] + ... + . . . = 0. 2013. The lengths of two of the straight lines joining the middle points of opposite edges of a tetrahedron are x, y, w is the angle between them, and a, a' the lengtlis of those edges of the tetrahedron which are not met by either x or y : prove that ixy cos w-a" ~ a". 2014. The lengths of the three pairs of opposite edges of a tetrahe- dron are a, a ; b, b' ; c, c' : prove that, if be the acute angle between the directions of a and a', 2aa' cos 6 ^ {b^ + b'') ~ (c' 4- c"). SOLID GEOMETRY. 351 2015. The locus of a straight line which moves so as always to intersect the three fixed straight lines, y = m{h-a), z = n (c — a); z = n (c-h), x = l {a-h); x=^ l(a-c), y = m {b-c); is It/z (b — c) + mzx (c - a) + nxy (a - b) - miix (b - cY — ... — ... = 2lmn{b-c) (c - a) (a-b): and evei-y such straight line also intersects the fixed line a (x - al) _b {y — bni) c (z- en) I m n ' 2016. The straight line joining the centres of the two spheres, which touch the faces of the tetrahedron ABCD opposite to ^, i^ respectively and the other faces produced, will intersect the edges CD, AB (produced) in points P, Q respectively such that CP : PD = AACB : AAjDB, and AQ :BQ = ACAD : ACBD. 2017. On three straight lines meeting in a point are taken points A, a; B, b ; C, c respectively: prove that the intersections of the planes ABC, abc ; aBC, Abe; AhC, aBc ; and ABc, abC ; all lie on one plane which divides each of the three segments harmonically to 0. 2018. Through any one point are drawn three straight lines each intersecting two opposite edges of a tetrahedron ABCD ; and (t,f; b, g ; c, h are the points where these straight lines meet the edges BC, AD ; CA,BD; AB, CD : prove that Ba . Ch • Dg- -Bg . Ca . Dh, Cb. ■ Af. . Dh = -^Ch. . Ab ■Df, Ac. %• Df- -4/. Be . %, Ab. Be . Ca = -Ac. Ba . Cb. 2019. Any point is joined to the angular points of a tetrahedron ABCD, and the joining lines meet the opposite faces in a, b, c, d : prove that Oa Ob Oc Od Aa^ Bb^C'c'^Dd' ' regard being had to the signs of the segments. Hence [trove that the reciprocals of the radii of the eight spheres which can be drawn to touch the faces of a tetrahedron are the eight positive values of the expression 1111 =fc — ± — ± — ± — • Pi P2 P3 P, Pv Pii Pzi P* '^eirig tli6 perpendiculars from the corners on the opposite faces. 352 SOLID GEOMETRY. 2020. Tho throe diagonals of an octahedron intersect each other in one point, at right angles two and two, and perpendiculars are let fall from this point on the faces : prove that the feet of these perpendiculars lie on a s])here and will be corners of a hexahedron such that the perpendiculars on its faces each from the corresponding corners of the octahedi-on will all meet in a point. [The faces of the octahedron will all touch a prolate conicoid of revolution of which the points of concourse are foci.] 2021. The areas of the faces of a tetrahedi'on ABGD are denoted by A, B, C, D and the cosines of the dihedral angles A A A A A A BC, CA, AB, DA, DB, DC respectively by a, b, c,f, g,h: prove that A' B' C- 1 _/^ _ y^ _ c- - 2fbc l-a'-f-c'- 2agc l-a'-b'- h' - 2abh D- that A^=B'a-C"--^D-- IBCa - 2CDh - IDBg, (fee. ; and that A _^__^ D sin a sin /3 sin y sin 8 ' where a, /?, y, 8 are determined by equations of the type sin' a= 1 -cos»^.4C-cos'(7^Z> - cos'Z».4i? + 2 cos ^^C cos<7^i) eos Z>yl 7?. Also, if I, m, 01, r be any real quantities, prove that l^ + nv + n' + r^ > 2mnf+ 2nlg + 2lmh + 2lr 1 1 2cohBG , /«- y- /3y a- 8^ aS , , be ad II. Linear Tranaformations. General Equation of the Second Degree. [The following simple method of obtaLning the conditions for a surface of revolution is worthy of notice. Wlien the expression ax^ + h}f + cz^ + 2/^« + 2gzx + 2hxy is trans- formed into AX^ + B7- + CZ'\ we obtain the coefficients A, B, C from the equivalence of the conditions that X ix" +y" + '^) - ay? - ... and X {.T- + P + Z') - AX' -BY'-CZ' may break up into (real or impossible) linear factors : which is the case when \ = A, B, or C. But, should two of the three coincide as B= C, then when \ = B the corresponding factors coincide, or either expression must be a complete square. The conditions that the former expression may be a complete square when X = ^ give us [B - a)f=^ - gh, &c., J 9 n. provided/, g^ h be all finite. Should we have /= 0, then gh must = 0; suppose then /and ^r = 0, then B = c, and we must have {c- a)x' + {c -h)if -2hxy a square, whence h" - {c- a) (c - h). In the case of oblique axes, inclined two and two at angles a, P, y, we must have \{x'+'if+z^ + 2^;: cos a + 2zxcosfS + 2xi/ cos y) - ax' - ... - 2/i/z- ... a complete squai-e. SOLID GEOMETRY. 355 It follows that the three equations (\ - a) (\ cos a -/) = (X cos li-g) {X cos y - h), (\ - h) {X cos ^-g) = (X cos y - h) (X cos a -/), (X - c) (X cos y-h) = (X cos a -/) (X cos /3 - g), must be simultaneously true ; and the two necessary conditions may be foimd by eliminating /i.] 2031. Determine the natui-e of the curve traced by the point x = aco%id+ -A, i/ = acos6, z = acos(6-^]. [A cii-cle of radius ^^ a.] 2032. In two systems of rectangular co-ordinate axes, 6^, 6„ 6.^ are the angles made by the axes of x, y', z' with the axis of ~, and cji^, ^,, ^^ the angles which the planes of zx', zy\ zz' make with that of zx: prove that x,,,.fl , C0s(^) cos ( and this will be of the second degi-ee only when /(m) = (.1 + Bm) -^ (^1' + B'm).] 358 SOLID GEOMETRY. 2017. A colic i.s ucsciil)ed liaving a plane section of a given splierc for base and vertex at a point on the sphere ; the subcontrary sections are parallel to the tangent plane at 0. 2018. A cone whose vertex is the origin and base a plane section of the surface ox" + btf + cxr = 1 is a cone of revolution : prove that the plane of the base must touch one of the cylinders (b-a)i/-+(c-a)z- = l, {c-b)z'' + (a-b) x" = 1, (a -c)x' + {b- c) >f - I. 204:9. A cone is described whose base is a given conic and one of whose axes passes through a fixed i)oint in the plane of the conic : prove that the locus of the vertex is a circle. 2050. The locus of the feet of the perpendiculars let fall from a fixed point on the tangent planes to the cone a./f+ bif-v c«^ = is a plane curve : prove that it must be a circle, and that the point must lie on one of the three systems of straight lines cc = 0, b{c — ct)if — c [a — b) z", &c. [One only of the three systems is real.] 2051. Prove also that, when the point lies on one of these straight lines, the plane of the circle is perpendicular to the other ; and that a jjlane section of the cone perpendicular to one of the straight lines will liave a focus where it meets that straight line, and the excentricity -vvill be equal to - ,^(6 — o) (c — a). 2052. A plane cuts the cone ayz + bzx + cxy = in two straight lines at right angles to each other : prove that the normal to the plane at the origin also lies on the cone. 2053. The centre of the surface a {x- + 2yz) + b(y'+ 2zx) + c{z' + 2xy) - 2Ax -2By-2Cz+l=0 is (X, Y, Z) : prove that A^ + B'+C^- 3 ABC = {a' + b' + c' - Zabc) {X^ + Y' + Z' - 3XYZ} ; and that the surface will be a cylinder whose principal sections are rectangular hyperbolas, if a + b + c — 0, A-hB + C — 0. [In this case the axis of the cylinder will be _ Aa _ Bb Gc cf + b^+c^ '^ a- + b' + c"' * a^ + ¥ + c" "-' 2054. The radius r of the central circular sections of the surface ay% + bzx + cxy = 1 is given by the equation abcT''-^{a?-\-¥ + c')r'' = i; and the direction cosines (I : m : n) of the sections by the equations - («r + iv) = -J {n- + r) = - (I- + nr) -= - Imnr: SOLID GEOMETRY. 3o9 2055. The semi-axes of a central section of the surface ayz + hzx + cxy + ahc = 0, made by a plane whose direction cosines are ?, m, n, are given by the equation r* (2hcmn + . . . - aU" -•••)- iahcr' (amn +...) + ia%'c' = 0. 2056. The section of the surface yz + zx + X7/ = a^ by the plane Ix + my + nz=p will be a parabola if I- + m'^ + n- — ; and that of the surface x^ + if + z^— 2ijz — 2zx — Ixy = a? will be a parabola if mn + nl + Im - 0. 2057. Prove that the section of the surface ^t = by the plane Ix + my + M5 = will be a rectangular hyperbola, if I- (6 + c) + vv {c-\-a)->r n^ {a + b) = 2mnf+ 2nlg + llmh ; and a parabola, if r (he -/-) +... + ...+ 2mn {gh - a/) + ... + ... = ; and explain why this last equation becomes identical when gh = af, hf= bg, fg = ch. [The surface when these conditions are satisfied is a pai-abolic cylinder, and every plane section will oljviously be a parabola, reckoning two parallel straight lines as a limiting case.] 2058. Pi'ove that, when bg == A/ and ch =/g, the equation u=ax'' + h7/ + cz^ + 2/yz + 2gzx + 2hxy + 2Ax^- 2By + 2 C^; + Z) = represents in general a paraboloid, the du-ection cosines of whose axis are as (0 : ^f : - h). 2059. Prove that the tangent lines drawn from the origin to the siu'face w = lie on the cone Z)it - {Ax + By+Cz+Dy = Qi; and investigate the condition that the surface u may be a cone from the consideration that this locus will then become two planes. 2060. The generators drawn through the point (X, F, Z) of the surface ayz + bzx + cxy + abc - will be at right angles, if X'+ Y' + Z' = a' + b' + c'. 2061. The generators of the surfixce « = drawn through a point (X, Y, Z) will be at right angles, if fdUV/d'U d'U\ \dx) V/F^'^f^.^V'^"" ^dUdU d'U ^dYdZ dYdZ^ '"^" 360 SOLID GEOMETRY. 20G2. Normals are di-awn to a conicoid at points l}'ing along a gencnitoi' : prove that tliey will generate a hyperbolic paraboloid whose principal sections are equal parabolas. [It is obvious that the surface generated is a right conoid.] 20G3. The axes of the two surfaces Ax- + By' + Cz" - {ax + by + cz)' = e\ /x' ?/ z-\ /a' b^ c- A /ax by cz\' , arc coincident in direction. 20G4r. The two conicoids ax- + by^ + c;r + Ifyz + 2gzx + 'Ihxy = 1, Ax^ + Bif + (7«° = 1, have one, and in general only one, system of conjugate diameters coua- cident in direction ; but, if there will be an infinite number of such systems, the direction of one diameter being the same in all. [If I, 7)1, n be the direction cosines of any one of such a system, we have the equations al + hin + c/71 = XAI, Id + bm -vfn = Xi?/?i, gl +fm + en — kCn, giving for \ the cubic {a - X.1) {b - XB) (c - XC) -/-' {a -XA)- cf (b-XB)- h' (c-XC)+ 2fgh = ; which may be wi-itten in the form 1 . 9^' + '^ + & af- gh - XA/ bg - /if- XB ch -fg - XG •] 20G5. Prove that eight conicoids can in general be drawn contain- ing a given conic and touching foiu' given planes. 206G. The equation of the polar reciprocal of the surface ax- + by^ + cz- + 2fyz + 2gzx + llvxy = 1 with respect to a sphere, centre (X, F, Z) and radius A-, is A{X(a;-jr) + r(y-r) + ^(^-^)P=(k-/-")(^-A7+... + 2(srA-a/)(y-r)(^-^) + ..., where A is the discriminant. SOLID GEOMETRY, 8G1 20G7. Prove that, if l^, l^, ... l^ be constants so determined that the expression where u^, w^, ...Mj ai'e given linear functions, is the product of two factoi-s, the two planes corresponding to these factors will be conjugate to each other ^vith resjiect to any conicoid which touches the seven planes ^l=0 ; and that, when the expression is a complete square, the corresponding plane is the eighth plane which touches every conicoid drawn to touch the other seven. 20G8. Seven points of a conicoid being given, an eighth is thereby determined; eight points A^, A^, ... A^ being given, from every seven is determined an eighth accordingly, gi^dng the points B^, B^ ... B^: prove that the relation between the A points and the B points is reciprocal, and that the straight lines -4,-5,, A^B^, ... all meet in one point. 20G9. The straight line, on which lies the shortest distance between two generators of the same system of a conicoid, meets the two in ^, ^, and any generator of the opposite system meets them in P, Q respectively : prove that the lengths x, y of AP, BQ are connected by a constant rela- tion of the form axy + hx + cy + d = 0. 2070. Two fixed generators of one system of a conicoid are met by two of the opposite system in the points A, B ; F, Q ; respectively, and A, B are fixed : prove that the lengths x, y of AP^ BQ are connected by a constant relation of the form axy + hx +cy = 0. 2071. An hyjierboloid of revolution is dra-\vn containing two given straight lines which do not intersect : prove that the locus of its axis is a hyjierbolic paraboloid, and that its centre lies on one of the generating lines through the vei'tex of the paraboloid. Til. Conicoids re/erred to their axes. 2072. The curve traced out on the surface 'r h — — x by the he *' extremities of tbe latus rectum of any section made by a plane through the axis of x lies on the cone y^ + z^ = ix^. 2073. The locus of the middle points of all straight lines passing through a fixed point and terminated by two fixed planes is a hyperbolic cylinder, unless the fixed planes are parallels. 2074. An ellipsoid and a hyperboloid are concentric and confocal : prove that a tangent plane to the asj-mptotic cone of the hyperboloid will cut the ellipsoid in a section of constant area. 362 SOLID GEOMETRY. 2075. Tlie locus of the centres of all plane sections of a given conicoid drawn through a given point is a similar and similarly situated conicoid, on wliich the given point and the centre of the given surface are ends of a dianicter. 207G. An ellipse and a circle have a common diameter, and on any choi'd of the ellipse parallel to this as diameter is descx'ibed a circle whose plane is i)arallel to that of the given circle : prove that the locus of these circles is an ellipsoid. 2077. Of two equal circles one is fixed and the other moves parallel to a given plane and intersects the former in two points : prove that the locus of the moving circle is an cllii^tic cylinder. If instead of circles we take any two conies of which one is fixed and the other moves parallel to a given plane without rotation in its own plane, and always intersects the fixed one in two i^oints, the locus of the moving conic is a cylinder. [There is no need to use co-ordinates of any kind.] 2078. A given ellipsoid is generated by the motion of a point fixed in a certain straight line, which straight line moves so that three other jioints fixed in it lie always one in each of the principal planes : prove that there are four such systems of points ; and that, if the corresiionding four straight lioies be drawn through any ]:)oint on the ellipsoid, the angle between any two is equal to the angle between the remaining two. 3>2 ^,2 ^ [If X. y, z be the point on the ellipsoid — „ + y^ + ^,= 1, the direction a 6" c" cosmes of the four straight lines will be - , ± f , =^ ^ -1 « 6 c -" 2079. Prove that, when a straight line moves so that three fixed points in it always lie in three rectangular planes, the normals drawn at diffei-ent points of the straight line to the ellipsoids which are traced otit by those i)oints will in any one position of the straight line all lie on an hyperboloid. [When I, m, n are the direction cosines of the straight line, the locus of the normals is ||(5-c) + ...+(6-c)(5 + c-2a)|+...+2(6-c)(«-&)(a-c) = 0, 2a, 2&, 2c being the axes of the ellipsoid. 2080. From a fixed point on an ellipsoid are let fall perpendicu- lars, (1) on any three conjugate diameters, (2) on any three conjugate diametral planes of the ellipsoid : prove that in each case the plane passing thi-ough the feet of the perpendiculars passes through a fixed point, and that this point in (2) lies on the normal to the ellipsoid at 0. SOLID GEOMETRY. 303 [If {X, Y, Z) be the point 0, the fixed poiut in (1) is given by X _ y _ z _ 1 Zr" ¥Y~ 7Z~ a'+h' + c'' 1 and in (2) by X-X y-Y ; X Y z-Z Z a' ¥ c' ill 2081. At each point of a generating line of a conicoid is drawn a .sti'aight line in the tangent plane at right angles to the generator : prove that the loons of such straight lines is a hyperbolic paraboloid whose principal sections are equal parabolas. 2082. The three acute angles made by any system of equal conjugate diameters of an ellipsoid will be always together equal to two right angles, if 2 (6- + c'- 2a') (c' + a' - 2b"-) {a' + b'- 2c') + 27a-b'c' = ; 2a, 26, 2e being the axes. Deduce the condition that an infinite number of systems of three generators can be found on the cone Ax^ + By' + C^ = 0, such that the sum of the acute angles in any such system is equal to two right angles. [The condition is found by eliminating \ from the equations A{A-\)-' + B{B-X)-' + C{C-\)-'^0, \'=2ABC; and, if ^, B, C be roots of the equation the resiUt is ^3'' + 1 2;; j^,^^ + 2\% = 1 6;?/.] 2083. The locus of the axes of sections of the surface ax' + by- + cr - 1 , made by planes containing the straifjht line 7 = — = - , is the cubic ° i rii n cone {b - c) yz (mz - ny) + (c-a) zx {nx - h) + (a-b) xy {ly - inx) = 0. /v" -j/^ 21' 2084. Two generators of the hyperboloid ^ + "^ — i=l di-awn through a point intersect the principal elliptic section in points P, P' at the ends of conjugate diameters : prove that SG-i SOLID GEOMETRY. 2085. Tlic generators of a given conicoid arc orthogonally projected npon a plane perpendicular to one of the generators : prove that their projections all pass through a fixed point, 2086. The orthogonal projections of the generators of the conicoid nx^ + hif + cz° = 1 on the plane l.v + my + nz = in general envelope a conic which degenerates if a^' + 6?>i^ + c)i* = 0; and which is similar to x^ ■?/ «* the section of the reciprocal surface ^ t" "^ — = *'** ^J ^^^ plane. 2087. From different points of the straight line ^ = -- = -, asymp- '■ ° I m n '■ X' y^ ^ totic straight lines are drawn to the hyperboloid — + ^ 2 = 1: prove that they will lie on the two planes \a' b- cy\a- b- c'J \a^ b^ c ) 2088. The asymptotes of sections of the conicoid ax^ + by^ + C5;* = 1 made by planes parallel to Ix + my + nz — ^ lie on the two planes {J.^bc + Wi^ca + rt^ab) {ax' + bif 4- cz^) ^ abc ilx + my + nzy. 2089. The locus of points from which rectilinear asymptotes at right angles to each other can be drawn to the conicoid ax^-\- by^+cz^= 1 is the cone a? (b + c) cc^ + b" (c + a) ?/^ + c^ {a + 5) s" = 0. 2090. The locus of the asymptotes drawn from a point {X, Y, Z) to the system of confocal conicoids X- y^ z^ , a" + A. b~ + \ c" + \ is the cone {x-XW-e) {^j-Y)(^)> the spheres, of •whicli one series of circular sections of the hyperboloid ai-e gi-eat circles, will have a common radical plane. [If the sections be parallel to y Ja^ — b' + z Ja^ + 6^ = 0, the common radical plane will be y Jar -h' - z Jd' + Ir = 0.] 2094. Two generators of the paraboloid a -^* ^^® drawn through the point (X, 0, Z) : prove that the angle between them is _^fa-h + Z^ Ja-b + Z\ \a + b + z)' 2095. The perpendiculars let fall from the vertex of a hypei'bolic paraboloid on the generators lie on two quadric cones whose cii'cular sections are pai-allel to the principal parabolic sections of the paraboloid. oc' tf 2z [The equation of the paraboloid being — 2 — ts = — , those of the two cones are x*^ + y* + 2«* ± .t^ ^- + - j = 0.] 2096. Through A, A' the ends of the real principal axis of an hyperboloid of one sheet are dra^\Ti two generators of the same system, and any generator of the opposite system meets them in P, P re- spectively : prove that the rectangle contained by A P, AP' is constant. u? if ^ [If the equation of the hyperboloid be^+p- — 5=1, and AA' = 2a, the constant rectangle is equal to 6" + c^] 2097. The least distance between two generators of the same system in an hyperboloid of revolution of one sheet cannot exceed the diameter of the jjrincipal cii'cular section. 2098. The equation of the cone generated by straight lines drawn through the origin parallel to normals to the ellipsoid -^+^^+-= = 1 at ^ a c u^ ?y^ ^ points where it is met by the confocal -5 — r- + —- — + -5 — = 1 is a—\b-\c—\ a'x' Vy" c-z- a^-\^b'-\ e-\ 2099. The points on a given conicoid, the normals at which inter- sect the noi-mal at a given point, lie on a quadric cone whose vertex is the given point. [With the usual notation, (.7, F, Z) being the given point, the equation of the cone is x-X y-Y z-Z ■-' 3GG SOLID GEOMETRY. 2100. Normals arc drawn to a central conicoid at the ends of three conjugate diameters: prove that their orthogonal projections on the plane through the three ends will meet in a point. ^ , y ^» 2101. The sLx nonnals drawn to the ellipsoid -r, + ^ + — =1 drawn from the point (.r^,, y^, z^ all lie on the cone 'x-x^ 'y-y^ 'z-z^ and the normals drawn from the same point to any confocal will also lie on the same cone. 2102. The normals at the ends of a chord of a given conicoid inter- sect each other: prove that the chord Mill be normal to some one confocal conicoid. 2103. The six normals drawn to the conicoid ax~ + hy^ + cz^ -~1, from any point on one of the lines a{h — c)x = ^h{G — a)y = ^c{ci — h)z, wDl lie on a cone of revolution. cc* ?/* z^ 2104. The normals to the ellipsoid —^ +^ +-§ = 1 at points on the ijlane ^ - + m f + '* - = 1 all intersect one straight line : prove that normals ^ a 6 c at all points lying on the plane — + ~ + h 1 = also intersect the same straight line; and that the necessary condition is (wiV - 1') {b' - cj + (nT - m') (c' - ay + (IW - n') {a' - bj = 0. Also prove that, when l=^m = n=\, the normals all intersect the ax ib' - c") - by (c^ - a-) = cz {c^ - ¥). •?/* %^ 2105. The normals to the paraboloid ^ h — = 2aj, at points on the plane pa; + gy + ra; = 1, will all intersect one straight line if f {b - cf + 2p {q'b - r'c) (& - c) = 2 {q'b + re). 2106. Prove that a tangent plane to the cone y^— + 4 = will ° ^ b-c b c y' ^ meet the paraboloid ^ H — = 2a3 in points the normals at which all inter- sect the same straight line; and the surface generated by this straight line has for its equation 2 (& - c) {x {hf - cz') - be {y' - z')Y- = (by' - cz') (by- + czy-. straight line SOLID GEOMETRY. 3G7 2107. A section of the conicoid ax' +h]f + c:ii^ = 1 is made by a plane parallel to the axis of z, and the trace of the plane on xy is nonual to the ellipse ax^ by' _ c* prove that the noi-mals to the conicoid at points in this plane all intersect one straight line. 2108. Through a fixed point (.x-^, y^, z^ are drawTi straight lines each of which is an axis of some plane section of the conicoid ax- + hf + c^ = \: prove that the locus of these lines is the cone 2109. In a fixed plane are drawn straight ILaes each of which is an axis of some plane section of a given conicoid : prove that the envelope of these lines is a parabola. 2110. Straight lines are drawn in a given direction, and the tangent planes ch-awn through each straight line to a given conicoid are at right angles to each other: prove that the locvis of such straight lines is a cylinder of revolution or a plane. [With a central conicoid ^ + t^ + — = 1, the locus is ^ a b' c" ar + y^ + ;r — {Ix + my + nzy = a^ +b' + c" —J^'t where I, m, n are the direction cosines of the given direction, and p the central distance of a tangent plane perpendicular to the given direction. ?/ z^ With the paraboloid ^ + — = x, the locus is m {ly - mx) + n {Iz -nx) = b(l^ + of) + c{P + m").] 2111. A cone is described having for base the section of the conicoid ax^ + by^ + cz^=l made by the plane Ix + my + nz = 0, and inter- sects the conicoid in a second plane perpendicular to the former : prove that the vertex must lie on the surface {r + m^ + n") [ax' + by- + cz^ -\) = 2 (Ix + my + nz) [alx + bray + cnz). 2112. The cone described with vertex (X, F, Z) and base the curve determined by the equations orK* + %' + C5^= 1, Ix + my -\- nz — 2h ^^^^ meet the conicoid again in the plane {aX'+bY^+ cZ'-l)(lx+my+nz-p) = 2 {LY+ mY+nZ-p)(axX+hjY+czZ-l). 2113. A chord AB of a conicoid is drawTi normal at A and the central plane conjugate to AJi meets the tangent plane at A in a straight line, through which is di'awn a plane intei'sccting the conicoid in a conic U : prove that the cone whose vertex is A and base the conic U has for its axes the noimals at A to the conicoid and to the two con- focals through A. 868 SOLID GEOMETRY. 2114. Through the vertex of an enveloping cone of a given conicoid ax' + hi/' + c£^ = \ is drawn a similar concentric and similarly situated conicoid: i)rove that this conicoid will meet the cone in a plane curve which will touch the given conicoid if the vertex lie on the conicoid ax' + hy' + cz' = 4. 2115. A tangent plane is dra^^•n to an ellipsoid and another plane dra-wn parallel to it so that the centre of the ellii)Soid divides the distance between them in the ratio 1:4: prove that, if a cone be drawn enveloping the ellipsoid and have its vertex on the latter plane, the c. G. of the volume cut off this cone by the foi-mer plane will be a fixed point. [The equation of the ellipsoid referred to conjugate diameters being — 4-^ + ^ = 1 a; + a = the tangent plane, x = 4a the parallel plane, a^ b' c the c. G. is (|,o,o).] 21 IG. Straight lines are drawn thi'ough the point (x , y^, z^) such ?> z' that their conjugates with respect to the paraboloid ^ + — = 2a' are perpendicular to them respectively : prove that the locus of these straight Lines is the cone -^ • — '^ + = : 2/-2/0 «-~o «^-«^o and that their conjugates envelope the ]:)arabola 2117. A straight line is perpendicular to its conjugate with respect to a cei-tain conicoid : prove that it is also perpendicular to its conjugate with respect to any conicoid confocal with the former. 2118. Any generator of the surface y^ +z^ — x' = m vnll be perpen- dicular to its conjugate with respect to the surface ax' + hf + cs^ + %fyz + 2gzx + 2hxy = 1, if hc-f -ca-g- = ah-h' and af=gh. 2119. An hyperboloid of one sheet and an ellipsoid are concentric and every generator of the hyperboloid is perpendicular to its conjugate with respect to the ellipsoid : prove that theii- epilations, referred to rectangular axes, may be obtained in the foi-ms x'-2yz = m', '^+^+ ^=\; ^ ' 2& 2c 6 + c ' and that the locus of the conjugate straight lines is (& + Cf 2bc 71V ' [If 26 = 2c = in' this locus is the hyperboloid itself, the ellipsoid being a sphere.] SOLID GEOMETRY. 369 2120. In the two conicoids ax^ + bi/' + cz' = 1, Ax' + Bif + Cz' = 1, eight generators of the first are respectively perpendicular to their con- jugates with respect to the second. 2121. A fixed point being taken, P is any point such that the polar planes of 0, P with i-espect to a given conicoid are perpendicidar to each other : prove that the locus of P is the plane bisecting chords which are perpendicular to the polar plane of 0. 2122. A hyperbolic paraboloid whose principal sections are equal is. drawn through two given straight lines not in one plane : prove that the locus of its vertex is a straight line. 21 23. Prove that, when two conicoids have in common two generators of one system, they have also common two generators of the opjDosite system. 2124. Two given straight lines not in one plane are generators of a conicoid : prove that the polar plane of any given point with respect to the conicoid passes tlu'ough a fixed point. 2125. Two conicoids touch each other in three points: prove that they either touch in an infinite number of points or have four common generators. 2126. Generators of the same system of the hyperboloid x' + y^ — m"^ = a' are drawn at the ends of a chord of the principal circle which subtends a given angle 2a at the centre : prove that the locus of the straight line which intersects both at right angles is the hyperboloid of revolution l + »i* \^ x' + y- \1 + m"cos a/ 2127. A cone is described with vertex (X, Y, Z) and base the curve S = ax' + bt/' + cz'- 1, 2^x + qy + rz = l : prove that the equation of the plane in which the cone again meets the conicoid aS^ = 1 is 2 {aXx + bYi/ + cZz - 1) (pX + qY+ rZ - 1) = (aA'-° + bY' + cZ'-l) {px + q>/ + rz - 1). [The cone will intersect the conicoid in two planes at right angles to each other if (p- + q' + r'){aX' + bY'+cZ'-l)=2{apX+bqY+crZ)(j}X + qY+rZ-\); and in two parallel planes if — = — = — , that is if the vertex lie on p q r the diameter conjugate to either plane section.] w. P. 24 370 SOLID GEOMETRY. 2128. Tangent planes are drawn to a series of confocal conicoids parallel to a (/iwn plune: iirove that tlic locus of the points of contact is a rectangular hyjjerbola which intersects both focal curves. [The equations of the locus will be, with the usual notation, h'- c* c'-g' / g- - h- \ -, m n n I I m u? ■?/ z- 2129. Two cii'cular sections of the ellipsoid -, + V5 + -2 = 1 «ii'e such that the sphere on which both lie is of constant radius mh : prove that the locus of the centre of this sphere is the hyperbola 2130. A sphere of radius r has real double contact with the x^ if z- u — H ellipsoid %+%,+ -5=1? ^^^^ l^^s altogether within the ellipsoid : prove £C 1/ T that the locus of its centre is tbe ellipse -^ ^ + r^ g = 1 - -5 , z = 0; Ct ~~' C ^ c c and, if there be real double contact and the sphere lie altogether without the ellipsoid, the locus of the centre is the ellipse 0; (a^>6">c^). rin the first case, r must lie between — , -7- : in the second, r must lie between -j- , —; and, in both cases, only a part of the ellipse can be c traced out by the centre.] 2131. In an hy|->erboloid of revolution in which, the excentricity of the generating hyperbola is ^^, a cube can be placed with one diagonal along the axis of the hyperboloid and six edges lying along generators of the hyperboloid. 2132. A cone whose vertex is meets a conicoid in two plane sections A, B; two other conicoids are described touching the former along A, B respectively and passing through : prove that these two conicoids will touch at 0, and will have a common plane section in the polar plane of with respect to the first conicoid. x^ if z^ 2133. The axes of sections of the conicoid — + ^ + — = 1 made by a c planes parallel to Ix + my + nz = Q lie on the two planes - (6-c) + ... + ... + 2al' ("i- l^ ^ + ... + ... a^ ' \c 0/ mn !„ /I 1\ \ /«2/^ ^^^ ^^2/\ A \ \6 c / / \mn nl Im J SOLID GEOMETRY. 371 2134. Two points are taken in the surface of a polislied liollow ellipsoidal shell and a ray jiroceeding from one after one reflexion passes through the other : ]!)roAe tliat the numljcr of possible points of incidence is in general 8 ; but if the two points be ends of a diameter the number is 4, aiul these four points are the ends of two diameters which lie on a quadric cone containing the axes of the ellipsoid and of the central sec- tion perpendicular to the given diameter. IV, Tetrahedral Co-ordinates. .2135. A plane meets the edges of a tetrahedron in six points and six other points are taken, one on each edge, so that each edge is divided harmonically : prove that the six planes, each passing through one of these six latter points and the edge opposite to it, will meet in a point. 213G. The opposite edges of a tetrahedron ABC D axe, two and two, at right angles : prove that the three shortest distances between opposite edges meet in the point xiAE" + AC + AD' -k)=y{BC'-¥BD- + BA'-h) = ... = ..., k being the sum of the squares on any pair of opposite edges. 2137. Prove that any conicoid which touches seven of the planes dt /a; ± my ± nz + rw = will touch the eighth ; and that its centre will lie on the plane l^x + mSj + n^z + r'w = 0. Prove that this plane bisects the ])art of each edge of the tetrahedron of reference which is intercepted by the given planes. 2138. Determine the condition that the straight line -=- = - p q r may touch the conicoid lyz + mzx + nxy + I'xio + m'yio + n'zio = ; and thence prove that the equation of the tangent plane at the point (0, 0, 0, 1) is I'x + m'y + n'z = 0. 2139. The general equation of a conicoid touching the faces of the tetrahedron of reference may be wi'itten Iqrx^ + tnrpy' + npqz^ + Imnw^ + {Ip — viq — nr) (Ixw + py^) + (mq — nr — Ip) {myw + qzx) + [nr — Ip — mq) (nzw + rxy) = 0. Prove that this will be a ruled surface if 1^2^^ + m* 2mnqr + 2nlrp + 2lmpq ; and that, when lp = mq = nr, the straight lines joining the points of contact each to the opposite comer of the tetrahedron will meet in a point. 24—2 372 SOLID GEOMETRY. 2140. A liyporbolic ]iar:il)oloid is drawn containing the sides -4/?, BC, CD, DA of a quadrangle not in one plane : prove that, if P be any point on this paraboloid, vol. PBCD : xoX.PABC^yoirCDA : xo\. PDAS'. and that, if any tangent i)lane to the paraboloid meet AB, CD in P, Q respectively, AP .BP=DQ -.CQ. 2141. The locus of the centi-es of all conicoids which have in common four genei-ators, two of each system, is a straight line. 2142. Perpendiculars are let fall from the point (a:, y, z, w) on the faces of the tetrahedi-on of reference, and the feet of these j^erpendiculars lie in one plane : prove that J2 ^2 (J2 J)2 — + — + — + — = 0, X y z to A, B, C, D being the areas of the faces of the tetrahedron. 2143. The volume of the ellipsoid which has its centre at the point (X \ Y '. Z : TF), and to which the tetrahedron of reference is self- conjugate, is ^ttY ^ XY ZW -^ {X + Y+Z+ Wy, where V is the volume of the tetrahedron. 2144. A tetrahedron is self-conjugate with respect to a given sphere : prove that each edge is perpendicular to the direction of the opposite edge, and that all the plane angles at one of the solid angles are obtuse. 2145. The opposite edges of a tetrahedron are two and two at right angles to each other, and in each face is described a circle of which the centroid and the centre of perpendiculars of that face are ends of a diameter : ])rove that the four circles so described lie on one sjihere ; and that this sphere (4), the cii-cumscribed sphere (1), the polar sphere or sphere to which the tetrahedron is self-conjugate (2), the sphere bisecting the edges (3), and the s^jhere of which the centroid and the centre of perpendiculars of the tetrahedron are ends of a diameter (5), have all a common radical plane. Taking P, p to represent the radii of the circumscribed and polar spheres and 8 the distance between their centres, 8^ = P^ + 3p^ ; and the distances from the common radical plane of the centres of these fiA'e spheres are 8 ' ^^^ 2S~~' ^^ 38 ' ^^^ 48 ' the radii of the five are (1) 7?, (2) p, (3) IJP^^ (4) |, (5) |; and the centres of the spheres (3), (4), (5) divide the distance between the centres of (1) and (2) in the respective ratios 1 : 1, 2 : 1, 3 : 1. 2146. A tetrahedron is such that a sphere can be drawn touching its six edges : prove that any two of the four tangent cones drawn to this sphere from the corners of the teti-ahedron have a common tangent ])lane and a common plane section ; and that the planes of the common sections will all six meet in a point. SOLID GEOMETRY. .S73 2147. A tetrahedron is such tliat the straight linos joining its angular points to the points of contact of tlie inscribed sphere with the respectively o]iposite faces meet in a point : prove tliat, at any point of contact, the edges of the tetrahedron which bound the corresponding face subtend equal angles. 2148. The tangent planes at A, B, C, J), to the sphere circumscrib- ing the tetrahedron J 567^, form a tetrahedron abed: pi-ove that Aa, £b, Cc, Dd will meet in a point if BC.AD^CA.BD = AB.CD. 2149. Each edge of a tetrahedron is equal to the opposite edge : prove that the diameter of the circumscribed sphere is . / — ^ , where a, &, c are the edges bounding any one face. 2150. A conicoid circumscribes a tetrahedron ABCD and the tangent planes at A, B, C, D form tlie tetrahedron abed : ])roye that, if Aa, Bb intersect, Gc, Dd will also intersect. . 2151. Four points are taken on a conicoid and the straight line joining one of the points to the pole of the plane containing the other three passes through the centre : prove that the tangent plane at that point is parallel to the jjlane of the other three. 2152. The equation of a conicoid being mnyz + nlzx + Imxy + Irxw + mryio + nrzw = ; prove that it cannot be a ruled surface, and that it will be an elliptic paraboloid if 1 1 1 1 - /I 1 1 IV 77. + — 2 + — + - = H 7 + - + - + - ) • I' m w r' "^ \l m n rj 2153. The surface lijz + mzx + nxy + I'xw + m'yw + n'zw - ■will be a cylinder, if W (m-k-7b — l) + mm' (n + l — m) + nn' {l+m — n) = 2bnn, and W {m! + n' -I) + mm {n' + 1- m') + nn {l + m' ~ n') = limn. [The relations IV {m + n' — V) + mm' {n +1' — m) + nn' [V + m- n) = 21' mn, W (m' + w - V) + mm' {n +1' — m') + nn (V + m' - n) = '21'm'n, will of course also be satisfied, the system being equivalent to the two- fold relation {11')^ + (»i»i')- + {nn')^ = 0, \ mn J \ nl J \ im J the first of which is the single condition for the surface to be a cone. See question (141).] 374. SOLID GEOMETRY. 2154. The rectangles under tlio segments of chords of a certain sphere drawn tin ou^li the four points A, B, C, D (not in one plane), are /, m, n, r, and the ratlins of the s]ihcre is p : prove that = 0. 0, 1, 1, 1, 1, 1 1, 0, AB\ AC\ AD% l + p^ 1, BA\ 0, BC\ Biy\ m-¥p' 1, CA\ CB\ 0, CD\ n + p' 1, DA\ DB% DC\ 0, r + p- 1, l + p", 0)1 + p", w + p', r + p*, 2155. The enveloping developable of the two conicoids Ix' + mi/ + nz'^ + rw^ - 0, I'x^ + m'-/ + ?iV + r\o^ = 0, will meet the planes of the faces of the tetrahedron of reference in the comes lo — O, Ir' - I'r mm'if - = 0, &c. 2156. The perpendiculars ^;, q, r, s let fall from the corners of a finite tetrahedron on a moving plane are connected by the equation Ajr + Bq' + Cr- + Ds' + 2Fqr + 2Gr23 + lEpq + 2F';ps + "IG'qs + 2//'rs = ; prove that the envelope is in general a conicoid, which degenerates to a plane curve if A, H, G, F' H, B, F, G' G, F, C, W F', G\ //', D = 0. V. Focal Curves : Reciin'ocal Polars. 2157. The equations of the focal lines of the cone ayz + hzx + cxy = {cy + hzY (az + ex)" (bx + ciyY f + ^ z' + x' X' + y 2158. A parallelogram of minimum area is cii'cumscribed about the focal ellipse of a given ellipsoid, and from its angular points taken in order are let fall perpendiculars p^, 2^->i l^^i Pi o^^ ^^^y tangent plane to the ellipsoid : prove that 2c being the length of that axis of the ellipsoid which is normal to the plane of the focal ellipse. SOLID GEOMETRY. 375 2159. The perpendiculars from the ends of two conjugate diameters of the focal ellipse on any tangent plane to the ellipsoid are zir^, zi:,,, zcr^, cr^, and the perpendiculai" from the centre is 2> '■ prove that cTjCT^ + ra-^,CT-^ =2)^ + c^. 2160. With any two points of the focal ellipse as foci can bo described a prolate spheroid touching an ellii^soid along a plane curve, and the contact will be real wlacn the common point of the tangents to the focal at the two foci lies without the ellipsoid. [The plane of contact is the polar Avith respect to the ellipsoid of this common point.] 21G1. Four straight lines can be drawn in a given direction so as to intersect both focal curves of an ellipsoid, and they will lie on a cylinder of revolution whose radius is Jo? -2^^ '} <^ being the semi major axis and ^j the perpendicular from the centre on a tangent plane normal to the given dii-ectiou. 2162. The cones whose common vertex is (X, T, Z) and whose bases are the real focal curves of the ellipsoid — + f? + - = 1 beinj? ^ a" 6 c" denoted by U^ and ?7, whose discriminants are respectively Z' Y ^ (a* - c') Qf - &) ' (f^ - b^Y^c^^') ' the cone \U'^+ ^3= will be a cone of revolution if l-\ "^ X ^ {a-b)+ ,^_^_xiar-lr) " "' 21G3. With a given point as vertex is described a cone of revolu- tion whose base is a plane section of a given conicoid : prove that the plane of this section will envelope a tixed cone whose vertex lies on one of the axes of the enveloping cone drawn from the given point to the given conicoid. 2164. This straight line joining the points of contact of a common tangent plane to the two conicoids ax- + by- + cs' =1, (a - X) x^ + {b - X) y- + {c - >) z- =\, subtends a right angle at the centre, 2165. Through a given point can in general be drawn two straight lines either of which is a focal line of any cone having its vertex on the straight line and enveloping a given conicoid : and, if two such cones be drawn with their vertices one on each straight line, a prolate conicoid of revolution can be inscribed in them having its focus at the given point. 2166. A point is taken on the umbilical focal conic of a conicoid: prove that there exist two points L such that, if any plane A be drawn through L and a be its pole, Oa will be normal to the plane through containing the intersection of A with the polar of L. S7G SOLID GEOMETRY. 21G7. "With a given point as vertex there can in general be drawn one tetrahedron self-conjugate to a given conicoid and such that the edges meeting in the point are two and two at right angles ; but when the given poixit lies on a focal curve the number of such tetrahedi'ons is intinite. 21 G8. A tetrahedron circumscribes a prolate ellipsoid of revolution whose foci are iS\ S', so that the focal distance (fi'om >S') of each angular point is normal to the opposite face : prove that the diameter of the sphere circumscribing the tetrahedron is three times the major axis of the ellipsoid, and that the centroid of the tetrahedron and the centre of the circumscribed sphere divide SS' in the ratios- 1:3, 3 : - 1 respectively. 21G9. The vertical angles of two principal sections of a quadric cone are a, fi : prove that the ratio of the axes of any section normal to a focal line is cos a : cos (3. 2170. A sphere is described -with centre (X, 0, Z) intersecting the ellipsoid — , + r-o + --3 = 1 iu two circles : prove that the points of contact ■wdth the sphere of common tangents to the sphere and ellipsoid lie on the two planes { a'-b' c'-b' / ' 2171. The circumscribing developable of two conicoids, which have not common plane sections, will in general contain four plane conies, which are double lines on the developable. 2172. In a given tetrahedron are inscribed a series of closed sui-faces each similar to a given closed surface without singular points : prove that the one of maximum volume will be such that the normals at the points of contact will be generators of the same system of an hyperboloid. 2173. Two conicoids having for their equations Z7=0, U' = 0, the discriminant of kU + V is X''A + X^0 + X^ $ + A0' + A': prove that the condition that hexahedra can be described whose six faces touch U and whose eight corners lie ujDon V is 0* _ 4©2 $ A + 800'A' - 1 6 A' A' = 0, and the condition that hexahedra can be described whose twelve edges are tangent lines to U and whose eight corners lie upon U ' is 20* - 90'«I>A + 2700'A' - 81 A' A' = 0. SOLID GEOMETRY. 377 YI. General Functional and Differential Equations. 2174. A surface is generated by a straight line wliich always inter- sects the two fixed straight lines x = a, y — mz ; x = — a, y = — mz : prove that the equation of the sui'ftice generated is of the form maz — xy . fmxz — ay^ - /mxz — ay\ \ a^ — X' ) ■ 2175. The general functional equation of surfaces generated by a straight line which intersects the axis of z and the circle ;: = 0, a;' + y' = a', is and the general differential equation is (x- + /) (px + qy-z)=a" (px + qy)-. 2176. The genei-al functional equation of surfaces generated by a straight line which always intersects the axis of z is and the difierential equation is rx' + Isxy + ty" = 0. 2177. The differential equation of a family of surfaces, siich that the perpendicular from the origin on the normal always lies in the plane oi xy, is z (jr + q-) +px + qy = 0. 2178. The diflferential equation of a family of surfaces, generated by a straight line which is always parallel to the plane of xy and whose intercept between the planes of yz, zx is always equal to a, is (px + VjY {v" + r) = (^'P"(f' 2179. The general differential equation of surfaces, generated by a straight line, (1) always parallel to the plane Ix + my + nz = 0, (2) always X 1/ z . intersecting the straight line -^ = — = - , is ° ^ I m n (1) {m + nqY r - 2 (»i + nq) {I + up) s + {l + npY t = 0, (2) {ly - rnxf (q-r - 2pqs +p^t) + 2{ly- inv) (na; - Iz) (qr -ps) + 2 (ly- mx) (ny - mz) (qs —pt) + (iix - h)- r 4- 2 {nx. - h) (ny - m-) s + (ny - mzf t = 0. 378 SOLID GEOMETRY. VII. Envelopes. 2 ISO. Tlie envelope of tlic i)I:ine lx + my + nz = a] I, m, n being piiranietc'i'S connected by the equations 1+ m + n = 0, I- + m' + n^ = 1, is the cyluidei' (i/-zy+{z-xy + {x-yy=3a\ 2181. Find the envelope of the planes (1) - cos (^ -(/)) + 1 cos (^ -) + J sin {9 + cji) = sin {0 - ), (2) - cos (6-6) + T (cos ^ 4- cos d)) + - (sin 6 + shi(i>) = 1; 'a c both "when 6, are parameters, and when 6 only is a i:)aranieter. [The envelope of (1) when both 0, are parameters is the hyperboloid 2 1,2 """ 2 — > a c and when 6 only is a parameter, the plane (1) always passes through a fixed generator of this hyperboloid ; the envelope of (2) when 6, 4> ^i'© parameters is the ellipsoid a a be when d alone is a parameter the envelope is a cone whose vertex is the point (—a, bcos(f}, csin^).] 2182. The envelope of the plane X y z sin 6 cos (f> sin 6 sin cos is the sm-face 2 2 2 2 2183, The envelope of all paraboloids to which a given tetrahedron is self-conjugate is the planes each of which bisects thi-ee edges of the tetrahedron. [More generally, if a conicoid be drawn touching a given plane and such that a given tetrahedron is self conjugate to it, there will be seven other fixed planes which it always toviches, the equations of the eight planes referred to the given tetrahedron being ±^j>a; ^qy^rz + w — 0.] 2184. A prolate ellipsoid of revolution can be described having two opposite umbilics of a given ellipsoid as foci and touching the given ellipsoid along a plane curve : and this will be the envelope of one system of spheres, each of which has a circular section of the ellipsoid for a great cii'cle. SOLID GEOMETRY. 371) 2185. Spheres are described on a series of parallel chords of a given ellipsoid as diameter : prove that they will have double contact with another ellipsoid, and that the focal ellipse of this envelope will be the diametral section of the given ellipsoid which is conjugate to the chords. Also, if a, 5, c be the axes of the given ellipsoid, and a, /?, y of the envelope, y being that axis which is perpendicular to the focal ellipse. 2186. A series of pai-allel plane sections of a given ellipsoid being taken, on each as a principal section is described another ellipsoid of given form ; the enveIo})e is an ellipsoid touching the given one along a central section at any point of which the tangent plane is pei-pendicular to the planes of the parallel sections. 2187. The envelope of a sphere, intersecting a given conicoid in two planes and passing through the centre, is a quartic which touches the given conicoid along a sphero-conic. VIII. Curvature. 2188. From any point of a curve equal small lengths s are measiu'ed in the same sense along the curve, and along the circle of absolute curvatui-e at the point, respectively: prove that the distance between the ends of these lengths is ultimately 1^ /1+V-^Y Cp V 0-- p- \dsj ' p, o- being the radii of curvature and torsion respectively at the point. 2189. Find the radius of absolute cui-vature and of torsion at any point of the curves (1) x = a{Zt-f), y=Zat\ z^a{Zt^e); (2) a; = 2a<'(l+<), y = ae(t + 2), z = at' {^ +2t+ 2). 2190. The radius of absolute curvature (p) at any point of a rhumb line is a cos 6-^ Jl — sin* 6 cos* a, where 6 is the latitude, and a the angle at which the line crosses the meridians ; and the ratlins of torsion is « ,, . , /, o V a^tana -; (1 - sur 6 COS' a) or -; r, r~ • sin a cos a ' a' — p' cos a 2191. Two surflices have complete contact of the n^^ order at a point : prove that there are « + 1 directions of normal section for which the ciu-ves of section have contact of the n + V^ order ; and hence prove that two conicoids which have double contact with each other intersect in plane curves. 3S0 SOLID GEOMETRY. 2192. Prove tliat it is in gcnoral jiossible to deteiinine a paraboloid, whoso i>rincii);il sections are ecjual parabolas, and Avliich has a complete contact of the second order with a given surface at a given point. 2193. A paraboloid can in general be dra^vn having a complete contact of the second order with a given surface at a given point, and such that all normal sections through the point have four-point contact. 2194. A skew surface is capable of generation in two ways by the motion of a straight line, and at any point of it the absolute magnitudes of the principal radii of curvature are a, b : prove that the angle between the generators which intersect in the point is cos"' ( — j 2195. The points on the surfaces (1) xiiz= a {yz + z.v + xy), ■ , (2) xyz = a^ {x-vy + z), (3) SI? + y^ + ^- Sxyz = ft^ at which the indicatrix is a rectangular hyperbola lie on the cones (1) x'(y + z) + y'{z + x)+z'{x + y) = 0, (2) x'^ + y^ + z^ + xyz = 0, (3) yz +zx + xy = 0, respectively ; and in (3) these points lie on the circle X + y + z = a, x" + y- +z^ = a'. 2196. A surface is generated by a straight line moving so as always to intersect the two straight lines a , a a , a ^=2' 2/ = ^tan-; o:^--^, y = -ztan-; and X, fj. are the distances of the points where the generator meets these straight lines from the points where the axis of x meets them ; prove that the principal radii of curvature at any point on the first straight line are given by the equation a*p' sin^ a - 2ap sin a -^ (\ - /x cos a) J a" + /x^ sin* a ^j(«+^-sm^a)-. 2197. A surface is generated by the motion of a variable circle, which always intersects the axis of x, and is parallel to the plane of yz. At a point on the axis of a;, r is the radius of the circle, and 6 the angle which the diameter through the point makes with the axis of z : prove tliat the principal radii of curvature at this point are given by the equation SOLID GEOMETRY. SiSl 2198. A surface is generated by a straight line which always in- tersects a given circle and the normal to the plane of the circle drawn through its centre ; $ is the angle which the generator makes with this noi-mal, and ^ the angle which the projection of the generator on the plane of the circle makes with a fixed radius : prove that the principal radii of curvature at the point where the generator meets the normal are a ^- 4- sin 6 (cos ^ ± 1 ) ; and that at the point where it meets the cii'cle, the pi'incipal radii are given by the equation 2199. A surface is generated by a straight line, which is always parallel to the plane of xy, and touches the cylinder of -^y^ = a? : prove that, if p be a princiijal radius of curvature at the point whose co-ordi- nates are (a cos Q -^ r sin ^, a sin Q — r cos Q, z) 2200. A straight line moves so as always to intersect the circle X' + y^ = of, 3 = 0, and be parallel to the plane of zx ; prove that the m.easure of specific curvature at the point {a cos ^, a sin ^, 0) is 1 COS' ^ ('^^\' 6 being the angle which the generator through the point makes with the axis of z. 2201. A circle of constant radius a moves so as to intersect the axis of X, its plane being parallel to the plane of yz : prove that, at the point (x, a sin ^ + a sin — ^, « cos — 6), the measure of specific curvature of the surface generated is dx (dx d-x . .\ ifdx\- , . )' -^ I ^- cos - -— -„ sin 6 ] ^ -A ,-] + cr sm- (9 / 2202. In a right conoid whose axis is the axis of z, prove that the radius of curvature of any normal section at a point (r cos 6, r sin 6, z) is d'z _ d)' dz ^dd'~ dO dd and deduce the equation for the principal radii of curvature at the point. 382 SOLID GEOMETRY. 2203. A stvaiglit lino moves so as always to intersect the axis of z and make a constant angle a with it : prove that, if p be a ])rincipal i-adius of cnrvature of the surface generated at the point whose co- ordinates are (?• sin a cos <^, r sin a sin <^, z + r cos a), P" '^'^ " ($^ -^ '^ ^^" '^ J"-' ■" dcOX"^ '°'^ a + 2cosa (J) 4- r ^,) 2204. Investigate the natui-e of the contact of the surfaces X1/Z = a^ (x + y + z), x{y- zy + 4fr {x + y-\-z) = 0, at any point on the line x = 0, y + z=Q; proving that the j)rincipal ,.. . r ■ ^ . (a' + y-y + ^a" radii of cui'vature of either surface are -^^ „-„ . ±2a^2/ 2205. Prove that, in the surface {y- + z') (2« - 2/ + z') = Aa'z", (1) the points where the indicatrix is parabolic lie on the cylinder a^ + !^ — a^; (2) the lines y = 0, z = 0; y = 2x, z=0, are nodal lines, the tangent planes at any point being respectively z' (a' - X') = Xy, z' (a' - X') =X'{y- 2xf . 2206. An ellipsoid is described with, its axes along the co-ordinate axes and touching the fixed plane px + qy -^rz—\ : prove that the locus of the centres of principal curvature at the point of contact is the surface whose equation is (poc + qy -^-rz—l) {l^^yz + q^zx + r^xy) - xyz (p^ + q' + r'y. 2207, The direction cosines of the normal to the conicoid x^ y^ z^ , a c at a certain point are I, m, n, and the angle between the geodesies joining the point to the umbilics is rove that {Pa (c -b) + m'b {c + a-2h) + n'c (a -jb)Y cos <^ {- I'a {b-c) + m^b (c - a) + n^c{a—h)Y+ ^nvn^ be {a - 5) (a - c) * STATICS. I. Composition and Resolution of Forces. 2208. A point is taken in the plane of a triangle ^l^Cand a, h, c are the mid points of the sides : prove that the system of forces Oa, Ob, Oc is equivalent to the system OA, OB, OC. [The result is true when is not in the plane ABC] 2209. Forces P, Q, R act along the sides of a triangle ABC and their resultant passes through the centres of the inscribed and circum- scribed circles: prove that p _ Q _ ^ cos B — cos G cos C — cos A cos A — cos B ' 2210. Four points A, B, C, D lie on a circle and forces act along the chords AB, BO, CD, DA in the senses indicated by the order of the letters, each force being inversely proportional to the chord along which it acts: prove that the resultant passes through the common points of (1) AD, BC; (2) AB, DC; (3) the tangents at B, D; (4) the tangents at A, C. [Of course this proves that these four points are collineai-.] 2211. In a triangular lamina ABC, AD, BE, CF are the perpen- diculars, and forces BD, CD, CE, AE, AF, BF are applied to the lamina : prove that theii' resultant passes through the centre of the circumscribed cii'cle and through the jioint of concourse of the straight lines each joining an angular point to the intersection of tangents to the circle ABC at the ends of the opposite side. [The equation of the line of action of the resultant in trilinear co-ordinates is CK sin (7? - C) + 2/ sin (C - ^) + 2 sin (^ - ^) = 0, which passes through the points (cos A : cos B : cos C), (sin A : sin 5 : sin C).] 384 STATICS. 2212. 'J luce equal forci-s act at tlio coniors of a triangle ABC, oacli ])orpoiuliciilai- to the o]>po.site side : i)rove that, if the magnitude of each force be re))re.sented by the radius of the circle ABC, the magnitiule of the resultant will be reiiresented by the distance between the centres of the inscribed and circumscribed circles. 2213. Tlie resultant R of any number of forces P,, /"„ P^,... is determined in magnitude by the equation A E' = % (P") + 2%1\P, cos (P^, P,), A where P , P, denotes the angle between the directions of P, P,. 2214. The centre of the circumscribed circle of a triangle ABC is 0, and the centre of perpendiculars is L: prove that the resultant of forces LA, LB, LC will act along LO and be equal to 1L0. 2215. Three parallel forces act at the points A, B, C and are to each other as h + c : c -\- a : a + h, where a, h, c are the lengths of the sides of the triangle ABC : j^rove that their resultant passes through the centre of the circle inscribed iii the triangle wiiose corners bisect the sides of the triangle ABC. 2216. The position of a j)oint P such that forces acting along PA, PB, PC, and equal to I . PA, m . PB, n . PC may be in equilibrium is determined by the areal co-ordinates (I : m : n). 2217. Forces act along the sides of a triangle ABC and are pro- poi-tional to the sides; A A', BB', CC bisect the angles of the triangle : prove that, if the forces be turned in the same sense about the points A', B', C I'espectively, each through the angle , -i/ ,B-C C-A ,A-B\ tan ( — cot — ^— cot — -^ — cot — ^ — I , there will be equilibrium. 2218. Forces in equilibrium act along the sides AL), BB, CD, BC, CA, AB of a frame ABCD, prove the following construction for a force diagram : take any one of the points {D) as focus and inscribe a conic in the triangle ABC ; let d be the second focus and let fall da, db, dc per- pendiculars on the sides of the triangle ABC, then abed will form the force diagram ; that is, be will be perpendicular to AD and proportional to the force along AD, and so for the other sides. 2219. Four points A, B, C, D are taken in a plane, perpendiculars are drawn from D on BC, CA, AB and a circle drawn through the feet of these perpendiculars, and another circle is drawn with centre D and radius equal to the diameter of the former circle ; other circles are similarly determined with their centres at A, B, C. Prove that these four circles Avill intersect by threes in four points a, b, c, d, and that the diagrams abed, ABCD will be reciprocal force diagrams. STATICS. 385 2220. A triangular frame ABC is kept in equilibrium by three forces at right angles to the sides, and *S^ is the point of concourse of their lines of action, the centre of the circle ABC; /SS' is a straight line bisected in : prove that the stresses at A, B, C are perpendicular and proportional to /S'A, S'B, S'G. 2221. A number of light rigid rods are freely jointed at their extremities so as to form a polygon, and are in equilibrium under a system of forces perpendicular and proportional to the respective sides of the polygon and all meeting in one point : prove that the polygon is inscribable in a circle, and, if be the centre of the circle, S the point of concourse of the lines of action, SS' a straight line bisected in 0, that the stress at any angular point P of the polygon is perjicndicular and l^roportional to S'P. [The points *S', aS" will be foci of a conic which can be drawn to touch the lines of action of all the stresses at the angular points, and the circle circumscribing the polygon is the auxiliary circle of this conic. If A B C D ...ha the comers of the polygon, and A' B' CD'... those of the polygon formed by the lines of action of the stresses at A, B, C,..., the diagrams SA'B'C'B' ..., S'ABCD... will be reciprocal force diagrams.] 2222. Two systems of three forces (P, Q, R), (P', Q', P') act along the sides of a triangle ABC : prove that the two resultants will be parallel if P, Q, P =0. F, Q', E' BC, CA, AB 2223. A lamina rests in a vertical plane with one corner A against a smooth inclined plane and another point B is attached to a fixed point C in the plane by a fine string, G is the c. G., and the distances of A, G, C from B are all equal: prove that, when the inclination of the inclined plane to the horizon is half the angle ABG, every position is one of equilibrium. 2224. Perpendiculars SK, SK' are drawn from a focus on the asymptotes of an hyperbola, and P is a point such that the rectangle KP, K'P is constant: jirove, from Statical considerations, that the tan- gent to the locus of P at a point where it meets the auxiliary circle of the hyperbola will touch the hyperbola, and that the normal will pass through S. 2225. Forces proportional to the sides a,, a„,... of a closed jwlygon act at points dividing the sides taken in order in the ratios 7?^^ : ?*,, 7??2 : **2)"- and each force makes the same angle 6 in the same sense with the cori'espondmg side : prove that there will be equilibrium if 2 I a" W 4 cot ^ X area of the polygon. \n + m J 2226. The lines of action of a system of forces are generatoi-s of the same system of a hyperboloid : j)rove that the least distance of any generator of the opposite system fi-om the central axis of the forces is proportional to the cotangent of the angle between the dix-ections of the two straight lines. W. P. 25 386 STATICS. 2227. A system of co-planar forces whose components are (JT,, }",), (Xj, r^), ... act at the points (.k,, y,), (x^, y^), ... and are equivalent to a sim'le couple : prove that there will be equilibrium if each force be turned about its point of application in the same sense through the angle 0, where 2228. The sums of the moments of a given system of forces about three rectangular axes are respectively Z, M, iV; and the sums of the compouents in the directions of these axes are A", Y, Z : prove that LX+MY^NZ is independent of the particular system of axes. [It is equal to EG, where R is the resultant force and G the mini- mum resultant couple.] 2229. Forces P, Q, R, F, Q', R' act along the edges BC, CA, AB, DA DB, DC of a tetrahedron respectively : prove that there will be a single resultant if BG AD CABD^ ABCD ' and that the forces will be equivalent to a single couple if F' _jR ^ Q^_P__J^ K.-A ^ 'AD~~AB~ VA' BD~ BC~ AB' CD~ CA W 2230. The necessary and sufficient conditions for the equilibrium of four equal forces acting at a point (not necessarily in one plane) are that the angle between the lines of action of any two is equal to that between the lines of action of the remaining two. 2231. Necessary and sufficient conditions of equilibrium for a system of forces acting on a rigid body are that the sum of the moments of all the foi'ces about each edge of any one finite tetrahedron shall severally be equal to zei'o. 2232. Forces acting on a rigid body are represented by the edges of a given tetrahedron, three acting from one angular point towards the opposite face and the other three along the sides taken in order of the opposite face : prove that the product of the resultant force and of the minimum resultant couple will be the same whichever angular point be taken. [Tlie product will be represented on the same scale by 18F, V being the volume of the tetrahedron.] 2233. A portion of a curve sui'face of continuous curvature is cut off by a plane, and at a point in each element of the portion a force propoi-tional to the element is applied in direction of the normal : prove that, if all the forces act inwards or all outwards, they will in the limit have a single resultant. STATICS. nS? 2234. A system of forces acting on a rigid body is reducible to a single couple : prove that it is possible, by rotation about any proposed point, to bring the body into such a position that the forces, acting at the same points of the body in the same directions in space, shall be in equilibrium. 2235. A given system of forces is to be reduced to a force acting through a proposed point and a couple : prove that if the proposed point lie on a fixed straight line and through it be draAvn always the axis of the couple, the extremity of this axis will lie on another fixed straight line. 2236. A given system of forces is to be reduced to two, both parallel to a fixed plane; straight lines representing these forces aro drawn from the points whei-e their lines of action are met by a fixfd straight line which intersects both at right angles : prove that the locus of the other extremities of these straight lines is a hyperbolic paraboloid. 2237. Prove that the central axis of two forces P, Q intersects the shortest distance c between their lines of action, and divides it in the ratio Q(Q + F cos 6) :F{P+Qcos6), 6 being the angle between their directions. Also prove that the moment of the principal couple is cPQsmO JF' +Q' + 2PQ cos 6 ' 2238. A given system of forces is reduced to two, one of which F acts along a given straight line : prove that 1 _ cos 6 c sin 9 F^'~R^~G~' 6 being the angle which the given straight line makes with the central axis, c the shortest distance between them, R the resultant force, and G the principal couple. 2239. A given system of forces is to be reduced to two at right angles to each other : prove that the shortest distance between their lines of action cannot be less than '2G^R. More generally, when the two are inclined at an angle 2a, the shortest distance cannot be less than 26-' -i? tan a. 2240. A given system of forces is reduced to two P, Q, and the shortest distances of their lines of action fi-om the central axis are x, y respectively : prove that r (R'x' + C-) = Q' (Ry + G'). 2241. Two forces act along the straight lines x = a, y = z tan a; x = - a, y=-z tan a : prove that their central axis lies on the surfiice X (y' + z") sin 2a = 2ayz, the co-ordinates being rectangular. 25—2 388 STATICS. 2212. Two forces given in magnitude act along two straight lines not in one piano, a third force given in magnitude acts through a given point, and the three have a single resultant : prove that the line of action of the third force must lie on a certain cone of revolution. [If R be the resultant force and G the principal couple which are together equivalent to the two given forces, F the third force, and a the distance of its jjoint of application from the central axis of the two, the semi- vertical angle of the cone is -. / GR \ . cos { r^ " ) ; \FjG' + R'ay from which the conditions necessary for the possibility are obvious.] 2243. Forces X, T, Z act along the three straight lines y — 'b,z--c; z= c, x = -a', x = a, y = ~b ', respectively : prove that they will have a single resultant if aYZ+bZX+cXY=0', and that the equations of the line of action will be any two of the three Y'Z'^X^^' Z~X^Y~ ' X'Y^Z"^- II. Centre of Gravity {or Inertia). 2244. A rectangular board of weight W is supported in a horizontal position by vertical strings at three of its angular points ; a weight 5 W being placed on the board, the tensions of the strings become W, 2 IF, 3 IF: pi'ove that the weight must be at one of the angular points of a hexagon whose opposite sides ai'e equal and parallel, and whose area is to that of the board as 3 : 25. 2245. Particles are placed at the corners of a tetrahedron respect- ively proportional to the ojiposite faces : prove that their centre of gi-avity is at the centre of the sphere inscribed in the tetrahedron. 2246. A uniform wire is bent into the form of three sides of a polygon AJ5, BO, CD, and the centre of gravity of the whole wire is at the intersection of AC, BD : prove that, if U be the common point of AB, DC produced, UB : BC : 01 = AB' : BC : CI)\ 2247. A thin uniform wire is bent into the form of a triangle .45(7, and particles of weights P, Q, R are placed at the angular points : prove that, if the centre of gravity of the particles coincide with that of the wire, P \ Q : R=b + c : c + a : a + b. 2248. The straight lines, each joining an angular point of the triangle ABC to the common point of the tangents to the circle ABC at the ends of the opposite side, all meet in : prove that, if perpendiculars STATICS. 389 be let fall from on the sides, will be the centx'oid of the triangle formed by joining the feet of these perpendiculars. 2249. Prove that a point can always be foimd within a tetra- hedron ABCD such that, if Oa, Ob, Oc, Od be perpendiculars from on the respective faces, will be the centroid of the tetrahedron abed ; and that the distances of from the faces will be respectively proportional to the faces. [The point 0, for either the triangle or the tetrahedron, is the point for which the sum of the squares of the distances from the sides or faces is the least possible.] 2250. Two uniform similar rods AB, BC, rigidly united at B and suspended freely from Ay rest inclined at angles a, /3 to the vertical : prove that AB /\ sin^ , BC V sm a 2251. Two unifoi-m rods AB, BC are freely jointed at B and moveable about A, which is fixed; find at what point in BC a smooth prop should be applied so as to enable the rods to rest in one straight line inclined at a given angle to the horizon. [If the weights of the rods be W, W, the point requii-ed must divide BC in the ratio W : W ■¥ W'.] 2252. Four weights are placed at four fixed points in space, the sum of two of the weights being given and also the sum of the other two : prove that their centre of gi-avity lies on a fixed plane, and within a certain parallelogram in that plane. 2253. A polygon is such that the angles a„ a^, %, ... which its sides make with any fixed straight line satisfy the equations 2 (cos 2a) - 0, 2 (sin 2a) = : prove that if be the point which is the centre of gravity of equal particles placed at the feet of the perpendiculars from on the sides, then the centre of gravity of equal particles, placed at the feet of the perpendiculars from any other point P, will bisect OP. [Such a polygon has the property that the locus of a point, which moves so that the sum of the squares on its distances from the sides is constant, is a circle.] 2254. The limiting position of the centre of gravity of the ai*ea included between the area of a quadrant of an ellipse bounded by the axes and the corresponding quadrant of the auxiliary circle, when the ellipse approaches the circle as its limit, will bo a point whose distance from the major axis is twice its distance from the minor. 2255. A curve is divided symmetrically by the axis of x aiul is such that the centre of gravity of the area included between the ordinates 2n - 1 jg = 0, x-h, is at a distance ;z 1 h from the origin : prove that the ' 3?i - 1 equation of the curve is 2/" = ex""'. 390 STATICS. 2256. The circle is the only curve in which the centre of gravity of the area included between any two radii drawn from a fixed point and the curve lies on the straight line bisecting the angle between the radii. 2257. Obtain the differential equation of a curve such that the centre of gravity of any arc measured from a fixed point lies on the straight line bisecting the angle between the radii drawn to the ends of the arc ; and prove that the cui-\'e is a lemniscate of Bernoidli, with its radii drawn from the node, or a circle. [The equation is r . / r* + ( -77, ) = a*, the general solution of which is r' = a* sin 2 (^ + a), and a singular solution is r = a.] III. Equilibrium of Smooth Bodies. 2258. A rectangular board is supported with its plane vertical by two smooth pegs and rests with one diagonal parallel to the straight line joining the pegs : prove that the other diagonal will be vertical. 2259. A I'ectangular board whose sides are a, b, is supported with its plane vertical on two smooth pegs in the same horizontal line at a distance c : prove that the angle 6 made by the side a with the vertical when in equilibrium is given by the equation 2c cos 26 = b cos 9 — a sin 6. 2260. A uniform rod, of length c, I'ests with one end on a smooth elliptic arc whose major axis is horizontal and with the other on a smooth vertical plane at a distance h from the centre of the ellipse : prove that, if 6 be the angle which the rod makes with the horizon and 2a, 2b the axes of the ellipse, 26 tan 6 — a tan <^, where a cos a J2, the only jiositions of equilibrium are when one axis is vertical ; and that, when oh ^2 and < a J2, the positions in which an axis is vertical are both stable and there are jjositions of unstable equilibrium in which the pegs are ends of conjugate diametei'S. 2265. A rectangular lamina rests in a vertical plane with one comer against a smooth vertical wall and an opposite side against a smooth peg : the position of equilibrium is given by the equation c sja' ■{■¥ =^2h{b%va.0-a cos 6) + sin (6 cos ^ + a sin 6)'] where 2a, 26 are the sides (the latter in contact "with the peg), 6 the angle which the diagonal through the point of contact makes with the vertical, and c the distance of the jieg from the wall. 2266. Two similar uniform straight rods of lengths 2a, 2b, rigidly imited at their ends at an angle a, rest over two smooth pegs in the same horizontal plane: prove that the angle which the rod 2a makes with the vertical is given by the equation c {a + b) sin (2^ -"■) = f*" sin a sin $ — b" sin a sin (a — 6), c being the distance between the pegs. 2267. A uniform lamina in the form of a parallelogram rests with two adjacent sides on two smooth pegs in the same horizontal plane at a distance c, 2h is the length of the diagonal through the intersection of the two sides, a, /?, 6 the angles which this diagonal makes with the sides and with the vertical : prove that h sin 6 sin (a + /3) = c sin (/? - a + 2^). 2268. A uniform triangular lamina ABC, rough enough to prevent sliding, is attached to a fixed point by three fine strings OA, OB, OC, and on the lamina is placed a weight w : prove that the tensions of the strings are as OA{W+^xtv) : OB {W + 3>/tc) : OC{W+'dzic), where W is the weight of the lamina and x, y, z the areal co-ordinates, measured on the triangle ABC, of the point where w is placed. Also prove that the least possible value of w ior which the tensions can be equal is '^"'•^^(o^"^ 06'~(il)' where OA is the longest string. 2269. A lamina in the form of an isosceles triangle rests with its plane vertical and its two equal sides each in contact with a smooth peg, the pegs being in the same horizontal plane : prove that the axis of the triangle makes with the vertical the angle or cos"' ( ~^ — ) ; h being the length of the axis, a the vertical angle, and c the distance between the pegs. 302 STATICS. 2270. A uniform rod AB of length 2a is freely moveable about A ; a smooth ring of weight 7^ slides on the rod and has attached to it a fino string which ])asses over a jmlley at a height a vertically above A and supports a Aveight Q hanging freely : find the position of equilibrium of the system ; and prove tliat, if in this position the rod and string are equally inclined to the vertical, 2Q{Qb-Way = FWab. 2271. A portion of a parabolic lamina, cut off by a focal chord inclined at an angle a to the axis, rests with its chord horizontal on two smooth pegs in the same horizontal line at a distance c : prove that the latus rectum of the pai'abola is c ^5 sin^ a, that the distance between the pegs is -pr x length of the bounding chord, and that the centre of gravity of the lamina bisects the distance between the mid points of the bounding chord and of the straight line joining the pegs. 2272. A portion of a parabolic lamina cut off by a focal chord inclined at an angle a to the axis rests on two smooth jiegs at a distance b, Avith its chord c parallel to the distance between the j)egs and inclined at an angle (3 to the vertical : prove that 56''_ 3cos(2a+/3) + 17cos y3 c^ ~ 3 cos (2a + /3) + cos /3 2273. A small smooth heavy ring is capable of sliding on a fine elliptic wire whose major axis is vertical; two strings attached to the ring pass through small smooth rings at the foci and sustain given weights : pi'ove that, if there be equilibrium in any position in which the whole string is not vertical, there will be equilibriiun in eveiy position. Prove also that, Avhen this is the case, the pressure on the wii-e will be a maximum when the slidina: ring is in the hiL'hest or lowest positions, and a minimum when its distances from the foci are i-espectively as the weights sustained. [The maximum pressvires are w^ (1 + e) + w^ (1 - e), ty, (1 - e) + w^ (1 + e), and the minimum is 2jl-e~Jiv^w^; where w^, w^ are the weights sustained at the upper and lower foci, and e the excentricity of the ellipse. When tv^(l -e)> w^{l + e), the pressure will be a maximum in the highest position, and a minimum in the lowest, and there will be no other maximum or minimum pressures.] 2274. A uniform i-egular tetrahedron has three corners in contact with the interior of a fixed smooth hemispherical bowl of such magnitude that the completed sphere would circumscribe the tetra- hedron : prove that every position is one of equilibrium ; and that, if P, Qy R be the pressures at the corners and W the weight of the tetrahedron, 2{QR + RP + rQ) = 2,{P"- + Q'A.R'-W'). STATICS. 303 2275. A heavy uniform tetrahedron rests witli three of its faces against three fixed smooth pegs and the fourth face horizontal : prove that the pressures on the pegs are as the areas of tlie faces respectively in contact. 2276. A heavy uniform ellipsoid is placed on three smooth pegs in the same horizontal plane so that the pegs are at the extremities of a system of conjugate diameters : prove that there will be equilibrium, and that the pi'essures on the pegs will be one to another as the areas of the corresponding conjugate central sections. 2277. Seven equal and similar uniform rods AB, BC^ CD, DE, EF, FG, GA are freely jointed at their extremities and rest in a vertical plane supported by rings at A and C, which are capable of sliding on a smooth horizontal rod: prove that, $, , t// being the angles which BA, AG, 6'i'^ make with the vertical, tan 6 — b tan ^ = 3 tan i//^. 2278. Two spheres of densities p, o- and radii «, &, rest in a para- boloid whose axis is vertical and touch each other at the focus : prove that p''a"'=o-'6"'; also that, if IF, W be their weights and li, W the pressures at the points of contact with the paraboloid, W W'~^\W'~w)' 2279. Four uniform similar rods freely jointed at their extremities form a parallelogram, and at the middle points of the rods are small smooth rings joined by light rigid bars. The parallelogram is suspended freely from an angular point ; find the stresses along the bars and the pressures of the rings on the rods, and prove that (1) if the parallelogi'am be a rectangle the stresses "will be equal, (2) if a I'hombus the pressures will be equal. IV. Friction. 2280, Find the least coefficient of friction between a given elliptic cylinder and a pai-ticle, in order that for all positions of the cylinder in which the axis is horizontal, the particle may be capable of resting vertically above the axis. [If the axes of the transverse section be 2a, 26, the least coefficient of friction is tan"' ( , , ) .1 \ 2ao J -" 2281. Two given weights of different material are laid on a given inclined plane and connected by a string in a state of tension inclined at a given angle to the intersection of the plane with the horizon, and the lower weight is on the point of motion : determine the coefficient of friction of the lower weight and the magnitude and direction of the force of friction on the upper Aveight. o04r STATICS. 2-82. A weight w rests on a rough iucliued j)]aiie (/x < 1) supiwrted l)y a string which, passing over a smooth j)ulley at the highest point of the phme, sustiiins a weight > fxw and X') : px'ove that the inclination of the string to the plane in limiting equilibrium, when P is a maximum or minimum, is \ - X'. 2293. A weight W is supported on a rough inclined plane of inclination a by a foi*ce F, whose line of action makes an angle i with the plane and whose component in the plane makes an angle y8 with the line of greatest inclination in the plane : prove that equilibrium will be impossible if fi" (1 + cos 2a cos 2t - sin 2a sin 2t cos (3)>2 sin'a sin* /? cos' i. 2294. A heavy particle is attached to a point in a rough inclined plane by a fine weightless rigid wire and rests on the plane with the wire inclined at an angle 6 to the line of gi-eatest inclination in the plane ; determine the limits of 6, the angle of inclination of the plane being tan"' (/x cosec (3). [The Ihuiting values of 6 are (3, Tr-fS, and 6 must not lie between these limits.] 2295. Two weights A, B connected by a fine string are lying on a rough horizontal plane ; a given force F {> fi. J A' + B^ and < ixA + fiB) is continually applied to -dt so as just to move A and B very slowly in the plane ; prove that A and B will describe concentric circles whose radii F^ — a' A' — LL^B^ ai-e a cosec (3, a cot {3, where cos /8 = ^ ' i ji — * 39G STATICS. V. Mastic Strings. 229G. A string whose extensibility varies as the distance from one end is stretched by any force : prove that its extension is equal to that of a string of equal length, of uniform extensiljility equal to that at the centre of the former, when stretched by an equal force. 2297. An elastic string rests on a rough inclined plane with the upper end fixed to the plane : prove that its extension will lie between the limits -rr. ^ '■, « being the inclination of the plane, c the angle 2A. cose of friction, and I, X the lengths of the whole string and of a portion of it whose weight is eqiial to the modulus. 2298. Two weights P, Q are connected by an elastic string without weight which passes over two small rough pegs A, B in. the same hori- zontal line at a distance a, Q is just sustained by P, and AP — h, BQ = c : P and Q are then interchanged, and AQ = b', BP = c': obtain equations for detei'mining the natural length of the string, its modulus, and the coefficients of friction at A and B. 2299. A weight P just supports another weight Q by means of a fine elastic sti-ing passing over a rough cylinder of revolution whose axis is horizontal ; IF is the modulus and a the radius of the cylinder : prove that the extension of the part of the string in contact with the cylinder is ^ log (-p^). 2300. A heavy extensible string, uniform when unextended, hangs symmetrically over a cylinder of revolution whose axis is horizontal, a portion whose length in the position of rest is a — h hanging vertically on each side : prove that the natural length of the part of the string in contact with the cylinder is 2 J '2ah log (J 2 + I); a being the radius of the cylinder, and 2h the length of a portion of the string whose weight (when unextended) is equal to the modulus : also prove that the extension of either of the vertical portions of the string is (Ja— ijh)'. 2301. An extensible string is laid on a cycloidal arc whose plane is vertical and vertex upwards, and when stretched by its own weight is just in contact with the whole of the cycloid, the natural length of the string being equal to the perimeter of the generating circle : prove that the modulus is the weight of a portion of the stiing whose natural length is twice the diameter of the generating circle. 2302. A heavy elastic string whose natural length is 21 is placed symmetrically on the arc of a smooth cycloid whose axis is vei-tical and vertex upwards, and a portion of string whose natiu'al length is x hangs vertically at each cusp : prove that 2j^={x + X)tan-^; 2a being the length of the axis of the cycloid, and X the natural length of a portion of the string whose weight is equal to the modulus. STATICS. 397 2303. A smootli right cylinder whose base is a cardioid is placed with the axis of the cardioid vertical and vertex upwards, and a heavy extensible string rests symmetrically upon the upper part in contact with a portion of the cylinder whose length is twice the axis of the cardioid, and the length of string whose weight is eqiial to" the modulus is equal to the length of the axis : prove that the natural length of the string is to its length when resting on the cylinder as log (2 + ^3) : ^3. 2304. An extensible string of natural length 21 just surrounds a smooth lamina in the foi'm of a cardioid, its free extremities being at the cusp, and remains in equilibrium under the action of an attractive force varying as the distance and tending to the centre of the fixed circle (when the cardioid is described as an epicycloid) : prove that y2i=^-°*(iyi)^ a being the radius of the fixed circle, 2Jcl the mass of the string, \ the modulus, and fir the acceleration of the force on unit mass at a distance r. VI. Catenaries, Attractioiis, dr. 2305. An endless heavy chain of length 21 is passed over a smooth cylinder of revolution whose axis is horizontal ; c is the length of a portion of the chain whose weight is equal to the tension at the lowest point, and 2(f> the angle between the radii drawn to the points where the chain leaves the cylinder : prove that tan d> + . — f log tan ( t + ? ) = - • 2306. In a common catenary A is the vertex, P, Q two points at which the tangents make angles , 2(^ respectively with the horizon, and the tangents &t A, Q meet in : prove that the arc AP is equal to the horizontal distance between and Q. 2307. Four pegs A, B, C, D are placed at the corners of a square, EC being vertically downwards, and an endless uniform inextensible sti-ing passes round the four hanging in two festoons : prove that _i i— = 2, sin a log cot \ sin ^ log cot ^ 1 1 I tan -^ log cot - tan ^ log cot ^ A A Z J a, ^ being the angles which the tangents at B, C make with the vertical, I the length of the string, and a the length of a side of the square. 308 STATICS. 2308. A lieavy unifonn cliaiii rests in limiting equilibrium on a rough circular arc whose i)]ane is vertical, in contact with a ([uadrant of the circle one end of which is the highest point of the circle : prove that 2309. A heaAT" uniform chain rests in limiting equilibrium on a rough cycloidal arc whose axis is vertical and vertex upwards, one extremity being at the vertex and the other at a cusp : prove that IT {l + ix')e''-^ =3. 2310. A uniform inextensible string hangs in the fonn of a common catenary, the forces at any point being X, Y perpendicular and i^arallel to the axis : prove that sm fb J, + cos qt-jj + 2A sec = ; d

being the angle which the normal at the point makes with the axis. 2317. A heavy uniform chain fastened at two points rests in the form of a parabola under the action of two forces, one (A) parallel to the axis and constant, and the other (F) tending from the focus : prove that 3F = A + B cos^ (f), <^ being the angle through which tfce tangent has turned since leaving the vertex and £ a constant. 2318. Find the law of repulsive force tending from a focus vmder which an endless uniform chain can be kept in equilibrium in the form of an ellipse ; and, if there be two such forces, one in each focus and equal at equal distances, prove that the tension at any point varies inversely as the conjugate diameter. 2319. A uniform chain rests in the form of a cycloid whose axis is vertical under the action of graA^ity and of a certain normal force, the tension at the vertex vanishing : prove that the tension at any point is proportional to the vertical height above the vertex, and that the normal force at any point bears to the force of gravity the ratio (3cos»^-l) : 2cos^; where 6 is the angle which the normal makes with the vertical. 2320. A heavy chain of variable density suspended from two points hangs in the form of a curve whose intrinsic equation is s =/{<(>), the lowest point being origin : prove that the density at any point wall vary inversely as cos' <{>/'{(f)). 2321. A stn'ng is kept in equilibrium in the form of a closed curve by the action of a repulsive force tending from a fixed point, and the density at each point is proportional to the tension : prove that the force at any point is inversely proportional to the chord of curvature through the centre of force. 2322. A uniform chain is in equilibrium under the action of certain forces ; from a fixed point is drawn a straight line Op parallel to the tangent at any point F of the chain and proportional to the tension at F : prove that, ( 1 ) the tangent at p to the locus of p is parallel to the resultant force at F, (2) the ultimate ratio of small corresponding arcs at p, F is proportional to the resultant force at P. 400 STATICS. 2323. A uniform heavy chain rests in contact with a smooth arc in a vertical i)hine of such a form that the pressure at any point per unit of len<,'th is equal to m times the weight of a unit of length : prove that the intrinsic equation of the curve will be (U d(}> {m + cos <}>y ' that, -svhcn 7?i>l, the horizontal distance between two consecutive 27ra ,, , ^ ^, . , 2)mra vertices is r , the arc between the same points » , K-l)5 _ (m^-1/ and the vertical distance between the lines of highest and lowest points 2a^{m'-l). When m = l, the curve is the first negative pedal of a parabola from the focus. 2324. A uniform heavy string is attached to two points in the surface of a smooth cone of revolution whose axis is vertical and rests with every point of its length in contact with the cone : prove that the curve of equilibrium is such that its differential equation, when the cone is developed into a plane, is p {r + c) = a^, the vertex of the cone being pole. 2325. A uniform chain is laid upon the arc of a smooth curve which is the evolute of a common catenary so that a portion hangs vertically below the cusp of a length equal to the diameter of the catenary at the vertex : prove that the resolved vertical tension at any point of the arc is constant, and that the resolved vertical pressure l)er unit of length is equal to the weight of a unit of length of the chain. Also, in the curve whose intrinsic equation is s = a sin ^ -^ y i + cos"^ <^, where <^ is measiu-ed from the horizontal tangent, if a uniform chain be bound tightly on any portion of it so that the tension at a vertex is equal to the weight of a length a J'2 of the chain, the resolved vertical pressure per unit will be equal to the weight of a unit of length and the resolved vertical tension at any point will be twice the weight of the chain intercepted between that point and the vertex. [The height above the directrix of the c. G. of the portion of chain included between two cusps is ^2 log (\/2 + !) + », and the area included between the directrix, the curve, and the tangents at two consecutive cusps is tvo' -^ J^J] 2326. A heavy uniform chain just rests upon a rough curve in the form of the arc of a four-cusped hypocycloid, occupying the space between two consecutive cusps at wliich the tangents are horizontal and vertical respectively : prove that rr STATICS. 401 2327. Find a curve such that the c. o. of any ai'c lies in a straight line dra^vn in a given direction through the intersection of the tangents at the ends of the arc. [It is obvious that the common catenaiy satisfies the condition, and it will be found that, when the arc is uniform, no other curve does so. ds When the density is variable and the curve such that — cos* ^ varies inversely as the density, the condition will be satisfied if the direction from which ^ is measured be at right angles to the given direction.] 2328. A uniform chain rests in a vertical plane on a rough cui'^'e in the form of an ei^uiaugular spiral whose constant angle between the normal and radius vector is equal to the angle of friction, one end being at a point where the tangent is horizontal : prove that, for limiting equilibrium, the chain will subtend at the pole an angle equal to twice the angle of friction. (The chain makes an obtuse angle with the radius vector to the highest point.) 2329. A uniform wire in the form of a lemniscate of Bernoulli attracts a particle at the node, the force varying as the distance : prove that the attraction of any arc is the same as that of a circular arc of the same material touching the lemniscate at its vertices and inter- cepted between the same radii from the node. The same property will hold for an equiangular spiral when the force varies inversely as the distance, and for a rectangular hyperbola when the force varies inversely as the cube of the distance, and generally for any curve r"==a"sinw^, if the force vary as r"~'. W. P. 26 DYNAMICS, ELEMENTAEY. I. Rectilinear Motion : Imindses. 2330. A ball A impinges on another ball B, and after impact the directions of motion of A and B make equal angles 6 witli tlie previous direction of A : determine 6, and prove that, when A = B, tan 6 = ,Je, where e is the coefficient of restitution. [In general Bi\ -e)co^ 26 -A + eBJ] 2331. A smooth inelastic ball, mass m, is lying on a horizontal table in contact with a vertical wall and is struck by another ball, mass m , moving in a direction normal to the wall and inclined at an angle a to the common normal at the point of impact : prove that the angle 6, through which the direction of motion of the striking ball is turned, is given by the equation 7>i cot 6 cot a = m + m. 2332. Two equal balls A, B are lying very nearly in contact on a smooth horizontal table; a third equal ball impinges directly on A, the three centi-es being in one straight line : prove that if c> 3 — 2 J2, the final velocity of B Avill bear to the initial velocity of the striking ball the ratio (1 + e)- : 4. 2333. Equal particles A^, A^, ... J__ are fastened at equal intervals a on a fine string of length (?i— 1) a and are then laid on a horizontal table at n consecutive angular points of a regular polygon of p sides (p > n), each equal to a; a blow F is applied to A^ in direction A^A^ : prove that the impulsive tension of the string A^A^_^^ is ^ , (1 + sin a)""'' — (1 — sin a)"""" ^^os a'-^^j r-~ — jz -. —-; (1 + sin a) — (1 — sm a) where pa is equal to 2;r. 2334. A circle has a vertical diameter AB, and two particles fall down two chords AF, PB respectively, starting simultaneously from A, P : prove that the least distance between them during the motion is equal to the distance of P from AB. DYNAMICS, ELEMENTARY. 403 2o^o. A niirabcr of heavy particles start at once from tlie vertex of an oblique circular cone whose base is horizontal and fall down generat- ing lines of the cone : prove that at any subsequent instant they -will all lie in a subcontrary section. 2.33G. The locus of a point F such that the times of falling down PA, PB to two fixed points A, B may be equal is a rectangular hyperbola in which AB is a diameter and the normals at A, B are vertical. 23.37. The locus of a point F such that the time of falling down FA to a fixed point A is equal to the time of falling vertically from A to a fixed straight line is one branch of an hyperbola in which one asymptote is vertical and the other perpendicular to the fixed straight line. [The other branch of the hyperbola is the locus of a point F such that the time down AF is equal to the time fi'oni the sti-aight line verti- cally to F.] 2338. A parabola is placed with its axis vertical and vertex down- wards : prove that the time of falling down any chord to the vertex is equal to the time of falling vertically through a space equal to the parallel focal chord. 2339. An ellipse is placed with its major axis vertical : prove that the time of descent down any chord to the lower vertex, or from the higher vertex, is proportional to the length of the parallel diameter. 2340. The I'adii of two circles in one vertical plane, whose centres are at the same height, ai'e a, h and the distance between their centres is c (which is greater than a + b) : prove that the shortest time of descent from one circle to the other doA\Ti a straight line is . / ^— i . ° \ g a + b 2341. The radii of two circles in one vertical plane are a, h, the distance between their centres c, and the inclination of this distance to the vertical is a : prove that, when c> {a ■>>- b), the time of shortest descent down a straight line from one circle to the other is equal to the time of fiilling vertically through a space j-^ -^ ; and, when a>(b + c), /■J, (a-bY-c^~ the shortest time from the outer to the inner is . / — ^ , ' - -, V g a — b + c cos a and from the inner to the outer . / — —^^, — , a being the aiiirle /2 («-6) V y a-b- c cos a which the lino of centres makes with the vertical me:isured upwards from the centre of the outer circle. Also prove that, when r cos a > (a + 6), there will be a maximum time of descent from one circle to the other down a straight lino, and this time will be . / ^^, — -~. . V ^r ccosa— (a+6) 26—2 404 DYNA!^IICS, ELEMENTARY. [lu the first case, the length of the line of shortest descent is Jc* + (a + by + 2 (a + 6) c cos a and the angle which it makes with the vertical is a + 6 + c cos a cos' Jc^ + (« + 6)* + 2 (a + 6) c cos a and similarly in the other cases.] 2342. A parabola is placed with its axis horizontal : prove that the length of the straight line of shortest descent from the curve to the focus is one third of the latus rectum. 2343. A parabola is placed with its plane vei-tical and its axis inclined at an angle 3a to the vertical: prove that the straight line of shortest descent from the curve to the focus is inclined at an angle a to the vertical. 2344. An ellipse is placed with its major axis vei-tical : prove that the straight line of quickest descent from the cun^e to the lower focus (or from the higher focus to the curve) is equal in length to the latus rectum, presided the excentricity exceed ^-. 2345. Two straight lines OP, OQ from a given point to a given circle in the same vertical plane are such that the times of falling down them are equal : prove that PQ passes through a fixed point. 2346. Two cii'cles A, B in the same vei-tical plane are such that the centre of A is the lowest point of B ; through each point P on B are drawn two straight lines to A such that the times down them are equal to the time from P to the centre of A : prove that the chord of A joining the ends of these lines will touch a fixed circle concentric with A. 2347. There are two given circles in one vertical plane and from each point of one are drawn the two straight lines of given time of descent (t) to the other : prove that the chord joining the ends of these lines envelopes a conic, whose focus is vertically below the centre of the former circle at a depth Igi". 2348. Two weights W, W move on two inclined planes, and are connected by a fine string passing over the common vertex, the whole motion being in one plane : prove that the centre of gi-avity of the .weights describes a straight line with uniform acceleration eqi;al to W HinB-W sin a , ^ — (IrTTrT — -J '^' +w"+2 ww cos (a + yS) ; where a, /? are the inclinations of the planes. DYNAMICS, ELEMENTARY. 405 2349. When there is equilibrium in the single moveable pulley, the weight is suddenly doubled and the power is halved : prove that, in the ensuing motion, the tensions of the strings are the same as in equilibrium. 2350. In the system of pullies in which each hangs by a separate string, P just supports W: prove that, if P be removed and another weight Q be substituted, the centre of gravity of Q and W will descend with uniform acceleration wjQ-py 2351. In any machine without friction and inertia a weight P supports a weight W, both hanging by vertical strings ; these weights are removed and weights F', W, respectively substituted : prove that, if in the subsequent motion /*' and W always move vertically, their centre of gravity will descend with acceleration (WF'-W'Fy ^ {F'+ W'){W'F' + F']V')' 2352. Two weights each of lib. support each other by means of a fine string passing over a moveable pulley to which is attached another string passing over another pulley and supporting a weight of 2 lbs.; to this pulley is similarly attached another string sujiporting a weight of 4 lbs., and so on, the last string passing over a fixed pulley and suppoi't- ing a weight of 2" lbs. : prove that, if the o-^^ weight, reckoning fi-om the top, be gently raised through a space of 2' - 1 inches, all the other weights will each fall one inch ; and, if the r*** weight be in any way gi-adually brought to rest, all the weights will come to rest at the same instant. (The pulleys are of insensible mass.) 2353. A fine uniform string of length 2a is in equilibrium, passing over a small smooth pulley, and is just displaced : prove that the velocity of the string when just leaving the pulley is Jag. 2354. A large number of equal particles are fastened at unequal intervals to a fine string and then collected into a heap at the edge of a smooth horizontal table with the extreme one just hanging over the edge ; the intervals are such that the times between successive particles being carried over the edge are equal : prove that, if c,, be the length of string between the w"^ and n + 1"' i)article, and v^ the velocity just after the n + V^ particle has been carried over, c = nc,, V = nv,. Deduce the law of density of a string collected into a he^ip at the edge of the table with the end just over the edge, in order that equal masses may always pass over in equal times. [The density must vary inversely as the square root of the distance from the end.] 406 DYNAMICS, ELEMENTARY. 2355. A large number of equal jiarticles are attached at equal intervals to a string; and the whole is lieai)ed up close to the edge of a smooth horizontal table with the extreme particle just over the edge : prove that, if v^ denote the velocity just before the n+ V^ particle is set in motion, ag (n + l){2n+l) " 3 n •where a denotes the length between two consecutive particles. Calculate the dissipated energy, and prove that, when « is indefinitely diminished, the end of the string, in the limit, descends with uniform acceleration ^g. [The whole energy dissipated, just before the « + !"' jiarticle is set in motion, is Aaw(^n^ — 1), -where lo is the weight of each particle.] 2356. A large number of equal particles are attached at equal interA'als « to a fine string which passes through a very short fine tube in the form of a semicircle, and initially there are 2r particles on one side of the tube, the highest being at the tube, and r particles on the other side, the lowest being in contact with a horizontal table wdiere the remaining particles are gathered together in a heap : prove that, if v^ denote the velocity just before the n^^ additional particle is set in motion, 3 \ {n^-Zr-iy}' and deduce the corresponding result for a uniform chain hanging over a small pulley. II. Parabolic Motion. 2357. A heavy particle is projected from a given point J. in a given dii-ection : determine its velocity in order that it may pass through another given point B. [If the polar co-ordinates of B referred to A be (or, a), and fi be the angle which the given direction makes with the horizontal initial line, the space due to the velocity of projection will be a cos^ a ^ 4 cos ;8 sin {/3 - a).] 2358. A particle moving under gravity passes through two given points : prove that the locus of the focus of its path is an hyperbola whose foci are the two given points. 2359. The distances of three points in the path of a projectile from the point of projection are r,, r^, r^, and the angular elevations of the three points above the point of projection are a^, a^, a^: prove that r, COS^ a^ sin (a^ - aj + r^ cos'' a, sin (a^ - a,) + r^ cos^ a^ sin (a^ - a^) = 0. 2360. A number of heavy particles are pi'ojected from the same point at the same instant: prove that their lines of instantaneous motion at any subsequent instant will meet in a point, a]id that this point will ascend with unifoim acceleration g. DYNAMICS, ELEMENTARY. 407 2361. A number of heavy particles are pi'ojected in a vortical piano from one point at the same instant with e(}nal velocities : prove that at any subsequent instant tliey will all lie on a circle whose centre descends with acceleration y and whose radius increases uniformly with the time. Also if, instead of having equal velocities, the velocity of any particle whose angle of projection is 6 be that due to a height a sin^^, the particles will at any subsequent instant all lie on a circle. 2362. Two points A, B in the path of a projectile arc such that the direction of motion at B is parallel to the bisector of the angle between the direction of motion at A and the direction of gravity : prove that the time from A to B is equal to that in which the velocity at A would be generated in a particle falling from rest under gravity. 2363. A number of particles ai'e projected from the same point with velocities such that their components in a given direction are all equal : prove that the locus of the foci of their parabolic paths is another parabola whose focus is the point of projection, semi latus rectum the space due to the given component velocity, and the direction of whose axis makes with the vertical an angle which is bisected by the given direction. 2364. A particle is projected from a given point so as just to pass over a vertical wall whose height is h and distance from the point of projection a : prove that, when the area of the parabolic path described before reaching the horizontal plane through the point of projection is a maximum, the range is ~a and the height of the vertex of the path ^ b. 2365. A particle is projected from a point at the foot of one of two parallel vertical smooth walls so as after three reflexions at the walls to return to the point of projection, and the last impact is direct: prove that e^ + tr + e— 1, and that the vertical heights of the three points of impact above the j^oint of projection ai'e as e^ : 1 — e* : 1. 2366. A heavy particle, mass m, is projected from a point A so as after a time t to be at a point B : prove that the action in passing from A to B is m I '' + Vr ?■ i and is a minimum when the focus of the path lies in AB. 2367. In the parabolic path of a projectile, AB is a focal chord : prove that the time from i4 to ^ is always equal to the time of falling vertically from rest through a space equal to AB ; and that the action in passing from A to B is also equal to the action in falling vertically from rest thix>ugli a space equal to AB. 2368. A heavy particle is projected from a point in a horizontal plane in such a manner that at its liighest point it impinges dii-ectly on a vertical plane from which it rebounds, and after another rebound from the horizontal plane returns to the point of projection : pi'ove that the coetEcient of restitution is i. [The equation for e is 26* + e - 1 = ; the student should account for the root - 1.] 408 DYNAMICS, ELEMENTARY. 23G9, A heavy particle, for wliich e = \, falls down a chord from the hi;:,'liest jxtiut of a vertical circle, and after reflexion at the arc describes a parabolic path passing through the lowest point : prove that the inclination of the chord to the vertical is | cos"' ( T ) ' ^^ ^^^® pai-ticle fall from the centre down a radius and after reflexion pass through the lowest i>oint, the inclination to the vertical will be cos"' ^. 2370. A particle is projected from a given point with given velo- city up an inclined plane of given inclination so as after leaving the plane to describe a parabola : prove that the loci of the focus and vertex of the jiarabola for different lengths of the plane are both straight lines. 2371. A particle, for which e = l, is projected from the middle j)oiut of the base of a vertical square towards one of the angles, and after being reflected at the sides containing that angle falls to the opposite angle : prove that the space due to the velocity of projection beai-s to the length of a side of the square the ratio 45 : 32. [More generally, when the particle is projected from the same point at an angle a to the horizon, the space due to the velocity of projection must be to the length of a side as 9 : 16 cos a (3 sin a — 4 cos a); and 3 tan a must lie between 4 and 9.] 2372. A particle (6=1) is projected with a given velocity from a given point in one of two planes equally inclined to the horizon and intersecting in a horizontal line, and after reflexion at the other plane returns to its starting point and is again reflected on the original path ; determine the direction of projection and prove that the inclination of each plane must be 45°. Also, if the planes be not equally inclined to the horizon, prove that they must be at right angles and that the incli- nation of projection to the horizon (6) is given by the equation cos (6 + 2a) cos + J sin a cos^ a = 0, where h is the space due to the velocity of projection, a the distance from the line of intersection, and a the inclination of the plane from which the particle starts. [This equation has two roots 6^, 0^, and the times of flight in the two paths will be as cos (6^ + 2a) : cos {0^ + 2a).] 2373. A pai'ticle being let fall on a fixed inclined ]ilane bounds on to another fixed inclined plane, the line of intersection being horizontal, and the time between the jilanes is given: prove that the locus of the l)oint from which the particle is let fall is in general a parabolic cylinder, but will be a i)lane if tan a tan (a + (S) ~ e, where a, ^ are the angles of inclination of the planes. 2374. A heavy particle projected at an angle a to an inclined plane whose inclination to the vertical is i, rebounds from the plane : prove that, if 2 tana = (1 — e)tan t, the successive pai^abolic paths will be similar arcs of parabolas, and will all touch two fixed straight lines, one of wliich is normal to the plane and the other inclined to it at an angle tun" C^r^^"")- DYNAMICS, ELEMENTARY. 409 2375. A pai-ticle pi'ojocted from a point iii au inclined plane at the ,.th iixipact strikes the plane normally and at the w*** impact is at the point of projection ; prove that e" — 26' + 1 = 0. 237G. A particle is projected from a given point in a horizontal plane at an angle a to the horizon, and after one rebound at a vertical plane i-eturns to the point of projection : prove that the point of impact must lie on the straight line 2/(1 +e) = x tan a, X, y being measured hoi'izontally and vertically from the point of pro- jection. When the velocity of projection and not the direction is given, the locus of the jioint of impact is the ellipse x^ ■{■ y- {\ + ef = iehy, where h is the space due to the velocity of projection. 2377. A particle is projected from a given point with given velocity so as, after one reflexion at an inclined plane passing through the point, to return to the point of projection : prove that the locus of the point of impact is also the ellipse X' + {\ + e)- y^= iehy, with the notation of the last question. 2378. A heavy particle is projected from a point in a plane whose inclination to the horizon is 30" in a vertical plane perpendicular to the inclined plane : prove that, if all directions of projection in that vertical plane are equally probable, the chance of the range on the inclined plane being at least one-third of the greatest possible range is '5. 2379. A particle is projected from a point midway between two smooth parallel vertical walls, and after one impact at each wall retui-ns to the ])oint of projection : prove that the heights of the points of impact above the point of projection will be as e(2e+l) : 2 + e, their depths below the highest point reached by the particle as (l+2e-e=)- : (l-Se-eY; and that this highest point lies in a fixed vertical straight line whose distance from the point of projection is the less of the two lengths a being the distance between the walls. Also, if the three parabolic })aths be completed, each will meet the horizontal plane thi'ough the jioint of projection in fixed points. III. Motion on a smooth Curve tmder the action of Gravity. 2380. A heavy particle is projected up a smooth parabolic arc whose axis is vertical and vertex upwards with a velocity due to the depth below the tangent at the vertex : prove that, whatever be the length of the arc, the parabola described by the particle after leaving the arc, will pass through a fixed point. 410 DYNAMICS, ELEMENTARY. 2381. A lieavy particle falls down a smootli cui've in a vertical plane of such a form that the resultant force on the pai'ticle in every ])0.sition is equal to its weight : jn-ove that the radius of curvature at any point is twice the intercei:)t of the normal cut ofi' by the horizontal line of zero velocity. 2382. A heavy i)article is projected so as to move on a smooth j>arabolic arc whose axis is vertical and vertex upwards : prove that the pressure on the curve is always proportional to the curvature. 2383. A heavy particle is projected from the vertex of a smooth ]iarabolic arc whose axis is vertical and vertex downwards with a velocity due to a height h, and after passing the extremity of the arc proceeds to describe an equal parabola freely : prove that, if c be the vertical height of the extremity of the arc, the latus rectum is 4 {h - '2c). 2384. A pai-abola is placed with its axis horizontal and plane vertical and a heavy smooth particle is projected from the vertex so as to move on the concave side of the arc : prove that the vertical height attained before leaving the arc is two-thirds of the greatest height attained ; and that, if 26 be the angle described about the focus before leaving the curve h=-a (tan' ^ + 3 tan 0), and the latus rectum of the free path will be 4a tan" 6 ; h being the space due to the initial velocity and 4rt the latus rectum of the parabolic arc. 2385. Two heavy particles, connected by a fine string passing through a small fixed ring, describe horizontal circles in equal times : prove that the circles must lie in the same horizontal jilane. 2386. A heavy particle P is attached by two strings to fixed points ^, ^ in the same horizontal plane and is projected so as just to describe a vertical cii'cle ; the string PB is cut when P is in its lowest position, and P then proceeds to describe a horizontal circle : prove that ?> co^2PAB =2 ; and that, in order that the tension of the string PA may be unaltered, the angle APB must be a right angle. 2387. Two given weights are attaehed at given points of a fine string which is attached to a fixed point, and the system revolves with uniform angular velocity about the vertical through the fixed point in a state of relative equilibrium : prove the equations tan = — [a sin b + a sm 6 ) = tan 6 H r — ■ sm d ; (J ' m-\- m g whei-e a, a are the lengths of the upper and lower strings, m, m' the masses of the particles, 0, 0' the angles which the strings make with the vei-tical, and O the common angular velocity. 2388. A heavy particle is projected so as to move on a smooth circular arc whose plane is vei-tical and afterwards to describe a parabola freely : prove that the locus of the focus of the parabolic path is an DYNAMICS, ELEMENTARY. 411 epicycloid formed by a circle of radius a rolling on a circle of radius 2a ; 4a being the radius of the given circle. 2389. A cycloidal arc is placed with its axis vertical and vertex upwards and a heavy 2)article is projected from the cusp up the concave side of the curve with the velocity due to a height h: prove that the latus rectum of the parabola described after leaving the arc is h^ -h 4o, where a is the radius of the generating circle ; also that the locus of the focus of the parabola is the cycloid which is enveloped by that diameter of the generating circle which passes tkrough the generating point. 2390. In a certain curve the vertical ordinate of any point bears to the vertical choi"d of curvature at that point the constant ratio 1 : 7n, and a particle is projected from the point where the tangent is vertical along the curve with any velocity : prove that the vertical height attained before leaving the curve bears to the space due to the velocity of projection the constant ratio 4:4 + /«. 2391. A smooth heavy particle is pi'ojected from the lowest point of a vertical circular arc with a velocity due to a space equal in length to the diameter 2«, and the length of the arc is such that the range of the particle on the horizontal plane through the jjoint of projection is the greatest possible : prove that this range is equal to a J'Q + Gj'i. NEWTON. 2332. Two triangles CAB, cAb have a cononion angle A and the sum of the sides containing tliat angle is the same in each ; BC, be intei-sect in D : prove that in the limit when b moves up to B, CD : DB = AB : AC. 2393. Two equal parabolas have the same axis and the focus of the outer is the vertex of the inner one, MPp, NQq are common ordinates : prove that the area of the surface generated by the revo- lution of the arc PQ about the axis bears to the area MpqN a con- stant ratio. 2394. Common ordinates from the major axis are drawn to two ellipses which have a common minor axis and the outer of which touches the directrices of the inner ; prove that the area of the surface generated by the intercepted arc of the inner ellipse revolving about the major axis will bear a constant ratio to the coiTesponding intercepted area of the outer. [In general if PM be the ordinate and PG the normal to any given curve at P both terminated by the same fixed straight line, and MP be produced to 2> so that Mp = PG in length, the area of the surface generated by an elementary arc PP' will bear the constant ratio Stt : 1 to the corresjionding area Mpp'M'J] 2395. A diameter AB of a circle being taken, P is a point on the cii'cle near to A and the tangent at P meets BA jDroduced in T : prove that ultimately the difference of BA, BP bears io AT the ratio 1 : 2. 2396. The tangent to a curve at a point B meets the normal at a point A in T ; C is the centre of curvature at A and a point on ^C: l)rove that, in the limit when B moves up to A, the difierence of OA and OB bears to ^IT the ratio OC : OA. 2397. In an arc PQ of continued curvature i? is a point at which the tangent is parallel to PQ : prove that the ultimate ratio PH : PQ when PQ is diminished indefinitely is one of equality. 2398. The tangents at the ends of an arc PQ of continued cur- vature meet in : ])rove that the ultimate ratio of OP + OQ-avcPQ : arc P(2 - chord P^, as PQ is indefinitely diminished, is 2 : 1. NEWTON. 413 2399. Three contiguous points being taken on a curve, tlie tangents form a triangle and the normals a similar triangle : prove that the ultimate ratios of these triangles when the points tend to coincidence (J \ 2 j-j;p being the radius of curvature at P and s the arc to F from some fixed point of the curve. 2400. A point is taken in the plane of a given closed oval, -P is any point on the curve, and QFQ' a straight line drawn in a given direction so that QP = PQ' and that each bears a constant ratio ?i : 1 to OP : prove that, as P moves round the curve, Q, Q' w:ill trace out two closed loops the sum of whose areas is double the area of the given oval. [Wlien is within the oval, the loops will intersect if n>l, and touch if 7i=l; when is without the curve, the loops will intersect if n be less than a certain value (always < 1) which depends on the position of 0.] 2401. Two contiguous points 0, 0' are taken on the outer of two confocal ellipses and tangents OP, OQ, O'P', O'Q' drawn to the inner, P' coinciding with P when 0' moves up to : prove that in the limit PP' : QQ' = 0P"- : OQ'. 2402. At a point -P of a curve is drawn the circle of curvature, and small arcs PQ, Pq are taken such that the tangents at Q, q are parallel : prove that Qq generally varies as PQ^, but, if P be a point of maximum or minimum curvature, Qq will vary as PQ^; also that the angle which Qq makes with the tangent at P is, in the fox'mer case two-thirds and in the latter thi'ee-fourths of the angle which the tangent at Q or q makes with that at P. 2403. Three equal particles A, B, C move on the arc of a given circle in such a way that their centre of gi-avity remains fixed : prove that, in any position, their velocities ai-e as sin '2A : sin 2B : sin 20. 2404. The velocities at three points of a central orbit are in- versely as the sides of the triangle formed by the tangents at these points : prove that the centre of force is the point of concourse of the straight lines joining each an angular point of this triangle to the common point of the tangents to its circumscribed circle at the ends of its opposite side. 2405. A parabola is described under a force in the focus S, and along the focal distance SP is measured a given length SQ; Qli drawn parallel to the normal at P meets the axis in It : prove that the velocity at P bears to the velocity at the vertex the ratio QR : 2SQ. 2406. Pi-ove that the equation 2 r' = i^.Pr is true when a body is moving in a resisting medium, F being the extraneous force and F V the chord of curvature in the direction of F. 414 NEWTOX. 2407. Two points P, Q move :is f(»llo\vs; P describes an ellipse Tinder aceelenitiou to the centre, and Q describes relatively to P an ellipse of which P is the centre under acceleration to P, and the periodic times in these ellijwes are equal: prove that the absolute path of Q is an ellipse concentric with the path of P, 2408. Two bodies are describing concentric ellipses under a centre of force in the common centre : jirove that the relative orl>it of either with respect to the other is an ellipse, and examine under what circum- stances it can be a circle. [The bodies must be at apses simultaneously, and either the sums of the axes of their two paths equal, or the differences.] 2409. In a central orbit the velocity of the foot of the perpendicular from the centre of force on the tangent varies inversely as the length of the chord of curvature through the centre of force. 2410. Different points describe different circles uniformly, the accele- ration in each vai-ying as the radius of the cii'cle : prove that the periodic times will be equal. [Kinematic similarity.] 2411. A particle describes an hyperbola under a force tending to a focus : prove that the rate at which areas are described by the central radius vector is inversely proportional to the length of that racUus. 2412. A rectangular hyperbola is described by a point under acceleration parallel to one of the as\anptotes : prove that at a point P the acceleration is 217". MP-7-C'JlP, MP being drawn, in dii-ection of tlie acceleration, from the other asymptote, C the centre, and U the constant component velocity parallel to the other asymptote. 2413. A point describes a cycloid under acceleration tending always to the centre of tlie generating circle : prove that the acceleration is constant and that the velocity varies as the radius of ciu'vature at the point. 2414. A particle constiaincd to move on an equiangular spiral is attracted to the pole by a force propoi-tional to the distance : prove that, in whatever position the particle be placed at stai'ting (at rest), the time of describing a given angle about the centre of force will be the same. [This follows at once from pi-operties of similar figures.] . 241.5. An endless string, on which runs a small smooth bead, encloses a fixed elliptic lamina whose perimeter is less than the length of the string ; the bead is projected so as to keep the string in a state of tension : prove that it will move with constant velocity, and that the tension of the string will vary inversely as the rectangle under the focal distances, 2416. A small smooth bead runs on an endless thread enclosing a lamina in the foi'm of an oval curve, and the bead is projected so as to NEWTON. 415 describe a curve of contiinious curvature in the plane of the lamina under no forces but the tensions of the thread : prove that the tension will vary inversely as the harmonic mean between the lenf^hs of the two parts of the string not in contact with the lamina ; and apply this result to prove that the chord of curvature of an ellipse at a point P in a {>iven direction is twice the harmonic mean between the tan_irents from P to the confocal which touches a straight line drawn through P in the given direction ; any tangent which is drawn from P outwaixls Ijeing reckoned negative. 2417. A ])arabola is described with constant velocity under the action of two equal forces one of which tends to the focus : prove that either force varies inversely as the focal distance. 2418. A particle is describing an ellipse about a centre of force \x.r~^; at a certain point /x receives a small increment A/x and the excentricity is unaltered : pro^■e that the jioint is an extremity of the minor axis and that the major axis 2a is diminished by - A/i, 2419. A particle is describing an ellipse about a centre of force lkr~' and at a cei-tain point p. receives a small increment A/x : prove the following equations for determining the corresponding alterations in the major axis 2rt, the excentricity e, and the longitude of the apse or, rAa ^Ae r «At 6- {2a -r)~ ah J a'e" - {r - af~ M«« (2a - r) ' 2422. In an elliptic orbit about the centi-e the resolved part of the velocity at any point perpendicular to one of the focal distances is constant ; and if the whole velocity be resolved into two, one pei-- pendicular to each focal distance, each will vary as the rectangle under the focal distances. 2423. A j)aii;icle moves along AP a roxigh chord of a circle under the action of a force to B vai-ying jis the distance and AJi is a diameter; the particle starts from rest at A and conies to rest again at P : prove that the co-efficient of friction is | tau PA B. 416 NEWTON. 2421. A niinibor of particles start from the same point witli tlie Bame velocity ami are acted on by a central force varying as the distance : j)rove that the ellipses described are enveloped by an ellipse liaving its centre at the centre of force and a focus at the point of projection. 2425. An ellipse is described by a particle nnder the action of two forces tending to the foci and each varying inversely as the square of the distance : prove that 2a' _ {fiw' + fi' io") (cj + w'Y a, h being the axes of the ellipse, and w, w' the angular velocities at any point about the foci. 2426. Two fixed points of a lamina slide along two straight lines fixed in space (in the plane of the lamina) so that the angular velocity of the lamina is constant : prove that (1) every fixed point of the lamina describes an ellipse under acceleration tending to the common point of the two fixed straight lines and proportional to the distance; (2) every straight line fixed in the lamina envelopes during its motion an involute of a four-ciisped hypocycloid ; (3) the motion of the lamina is completely represented by supposing a circle fixed in the lamina to roll uniformly with internal contact on a circle of double the radius fixed in space ; (4) for a series of points in the lamina lying in one straight line the foci of the ellipses described lie on a rectangular hyperbola. 2427. A lamina moves in its own jjlane so that two fixed points of it describe straight lines with accelerations f, f : pi'ove that the accele- ration of the centre of instantaneous rotation is Jf^ +f" - W cos a -f sin a, where a is the angle between the straight lines. [The accelerations f, f must satisfy the equations cos^ e ~ (/ sec e) = cos^ 6' ~ (/' sec $'), f cos 6 + /' cos 6' = CO)', where B, 6' are the angles which the straight line joining the two points makes with the fixed straight lines, and w is the angular velocity of the lamina.] 2428. Two points ^, ^ of a lamina describe the two straight lines Ox, Oy fixed in space (in the plane of the lamina), F is any other point of the lamina, and QQ' any diameter of the circle AOB ; PQ, PQ' meet the cii-cle again in R, R : prove that OR, OR' will be the directions of two conjugate diameters of the locus of P. 2429. Two points fixed in a lamina move iipon two straight lines fixed in space and the velocity of one of the points is uniform : prove that every other point in the lamina moves so that its acceleration is constant in direction and varies inversely as the cube of the distance from a fixed straight line. NEWTON. 417 [If A describe Ox with uniform velocity U and B describe Oy at right angles to Ox, then if F be any other point fixed in the lamina and PA, FJJ meet the circle on AB in a, b, the acceleration of P will bo always parallel to Oa and vary invei-sely as the cube of the distance from Ob; and, if i-'J/ bo drawn parallel to Oa to meet Ob, tlie acceleration of F will be U '. AF* ^ AB\ FM\'\ 2430. A lamina moves in its own plane so that two points fixed in the lamina describe straight lines with equal accelerations: prove that the acceleration of the centre of instantaneous rotation is constant in direction, and that the acceleration of any point fixed in the lamina is constant in direction. 24:31. Two ellipses are described about a common attractive force in their centre; the axes of the two are coincident in direction and the sum of the axes of one is equal to the difference of the axes of the other: prove that, if the describing particles be at corresponding extremities of the major axes at the same instant and be mo\'ing in opposite senses, the straight line joining them will be of constant length and of iiniforni angular velocity during the motion. 2432. A lamina moves in such a mamicr that two sti'aight lines fixed in tlie lamina pass through two points fixed in space: prove that the motion of the lamina is completely representetl by supposing a ciix-le fixed in the lamina to roll with internal contact on a circle of half the radius fixed in space. 2433. A lamina moves in its own plane Avith uniform angular velocity so that two sti-aight lines fixed in the lamina pass each thi'ough one of two points fixed in space : pi'ove that the acceleration of any j)oint fixed in the lamina is compounded of two constant accelerations, one tending to a fixed point, and the other in a direction which revolves with double the angular velocity of the lamina. 2434. A triangular lamina ABC moves so that the point A lies on a straight line be fixed in space, and the side BC passes through a point a fixed in space, and the triangles ABC, abc are equal and similar: prove that tlie n\otion of the lamina is completely represented by su})posuig .a j)arabola fixed in the lamina to roll upon an equal parabola fixed iu space, similar points being in contact. 2435. A particle describes a parabola under a reinilsive force from the focus, varying as the distance, and another force parallel to the axis which at the vertex is three times the former ; find the law of this latter force ; and prove that, if two particles describe the same parabola under the action of these forces, their lines of instantaneous motion will intersect in a point which lies on a fixed con focal parabola. [The second force is always three times the first.] 2436. Two particles describe curves under the action of central attractive foi'ces, and the radius vector of either is always parallel and proportional to the velocity of the other : prove that the curves will be similar ellipses described about their centres. W. P. 27 DYNAMICS OF A PARTICLE. I. Rectilinear Motion, Kinematics. 2437. A heavy particle is attaclied by an extensible string to a fixed point, from wliicli the j^article is allowed to fall freely; when the particle is in its lowest position the string is of twice its natural length: prove that the modulus is four times the weight of the particle, and find the time during which the string is extended beyond its natural length. [The time is 2 /- tan"' ^2.] 2438. A particle at B is attached by an elastic string at its natural length to a point A and attracted by a force varying as the distance to a point C in BA produced, A dividing BC in the ratio 1 : 3, and the particle just reaches the centre of force : prove that the velocity will be greatest at a point which divides CA in the ratio 8 : 7. 2439. A particle is attracted to a fixed point by a force ju,(dist.)~^, and repelled from the same point by a constant forced; the pai-ticle is placed at a distance a from the centre, at which point the attractive force is four times the magnitude of the repulsive, and projected directly from the centre with velocity V : prove that (1) the particle will move to infinity or not according as V^ > or < 2af; (2) that, if .-c, a? + c be the distances fi-om the centre of force of two positions of the particle, the time of describing the given distance c between them will be greatest when x{x + c) = ^a^. Also, when V=j2af or Zj2af, determine the time of describing any distance. [When V—J'2af, the time of reaching a distance x from the centre of force is and, when F= Z J'laf, the time is J^{V2^-V2-^.2tan-i-^-2tan-'y|J.] 1. DYNAMICS OF A PARTICLE. 419 2440. The accelerations of a point describing a curve are resolved, into two, along the radius vector and parallel to the prime radius: prove that these accelei-ations are respectively cote d / ,dd\ d'r /dO\' - 1 d / ^dO\ — dtV dtJ'-d^-'U) ^'''^-r.inedtV'-dV- 2441. The motion of a point is referred to two axes Ox, Oi/, of which Ox is fixed and Oi/ revolves about the origin : prove that the accelerations in these directions at any time t are (Px 1 d/,de\^ d-y cot 6 d / „ d6\ f'_^Y . where denotes the angle between the axes. 2442. A point F is taken on the tangent to a given curve at a point (?, and is a fixed point on the curve, the arc OQ = s, QP = r, and ^ is the angle through which the tangent revolves as the point of contact passes from to Q: prove that the accelerations of P in dii-ection QP and in the direction at right angles to this, in the sense in which (f> increases, are respectively di'^dt-' *' Vdi) ' r dt \ dt ) "*" dt dt 2443. A point describes a curve of double curvature, and its polar co-oi-dinates at the time t are (r, 6, ^) : prove that its accelerations (1) along the radius vector, (2) perpendicular to the radius vector in the plane of 6 and in the sense in which 6 increases, and (3) perpendicular to the plane of ^ in the sense in which <^ increases, are respectively 2444. A point describes a parabola in such a manner that its /•2f velocity, at a distance r from the focus, is ./ -^ (r^ - c^), where f, c are constant: prove that its acceleration is compounded of /parallel to the axis and/— 2 along the radius vector from the focus. 2445. A point describes a semi-ellipse bounded by the minor axis, and its velocity at a disttince r fx'om the focus is a . / ,\ .- , > J V ?• (2a - r) where 2a is the length of the major axis and / a constant acceleration: prove that the acceleration of the point is compounded of two, each varying inversely as the square of the distance, one tending to the nearer focus and the other from the farther focus. 420 DYNAMICS OF A PARTICLE. 244G. A i>oint is describing a circle, and its velocity at an angular distance 6 from a fixed point on the circle varies as ^1 + cos^^ -^ sin'' 6: prove that its acceleration is coinj)Ouud€d of two tending to fixed points ut the extremities of a diameter, each varying inversely as the hfth power of the distance and ec^ual at equal distances. 2447. A point describes a circle under acceleration, constant, but not tending to the centre : prove that the point oscillates through a quadrant and that the line of action of the acceleration always touches a certain epicycloid 2a [The radius of the fixed circle of the epicycloid is ~ - and of the a moving circle ^ , a being the radius of the circle described by the point.] 2448. A parabola is described with accelerations F, A, tending to the focus and parallel to the axis respectively : prove that r' dr^ ' dr r being the focal distance. 2449. A point describes an ellipse under accelerations F^, F^ tending to the foci, and r^, r^ are the focal distances of the point : prove that P dr ^^'■^' 'V^dr ^ "'** '' '\ "'i '2 "'2 2450. The parabola y^ = '^ax is described under accelerations X, Y j)arallel to the axes: prove that dx dx 2451. A point describes a parabola under acceleration which makes a constant angle d with the normal, and 6 is the angle described from the vertex about the focus in a time t : prove that etnn ,fde\ (f)=c(l.cos(>r; and find the law of acceleration. [The acceleration varies as cos" ^€~^ta,no^ which is easily expressed as a function of the focal distance.] 2452. A point P describes a circle of radius 4a with uniform angular velocity w about the centre, and another point Q describes a circle of radius a wdth angular velocity 2w about P: prove that the acceleration of Q varies as the distance of P from a certain fixed point. DYxVAMICS OF A PARTICLE. 421 2453. The only curve wliich can be described under constant acceleration in a direction making a constant angle with the normal is an equiangular si)iral. 24.54. An equiangular spiral is described by a point with constant acceleration in a direction making an angle cji with the irormal; prove that sin d> - = 2 sin (jt + cot a cos J cos 6) = J cos 6. 2460. A point describes an epicycloid under acceleration tending to the centre of the fixe«l circle : inove that the ix>dal of the ei»icycloid with respect to the centre will also be described under acceleration tend- ing to the same point. 422 DYNAMICS OF A PARTICLE. 24G1. Tlio intrinsic equation of a curve is s—/{(f)) and the curve is described under accelerations X, Y parallel to the tangent and normal at the origin (where ^ = 0) : prove tliat cos *(s^'-'-^')--'K<7^'"''')"foW^'''°°^*-'''^"*) = '- 2462. The curve s=/{(f)) is described by a point with constant acceleration which is at the origin in direction of the normal : prove that its inclination 6 to this direction at any other point is given by the equation (3-^)tan(<^-^)/'(cA)=/"(<^). 2463. A catenary is described by a point under acceleration whose vertical component is constant (/) : prove that the horizontal comjionent when the tangent makes an angle (ft with the horizon is ycos (fi cosec^ (f)(l + m cos «^ + cos^ ^). 2464. A curve is described under constant acceleration parallel to a straight line which revolves uniformly : prove that the curve is a prolate, common, or curtate cycloid ; or a cii'cle. 2465. A point describes a certain curve and initially the accelera- tion is normal ; when the direction of motion has turned through an angle ^ the dii'ection of acceleration has turned through an angle 2(f) in the same sense : prove that the acceleration varies as cos are the angles which the direction of the acceleration at any point and the tangent at that jDoint make respectively with the dii-ectrix : prove that 3 — , (tan (b cos^ 6) = cos^ 6. 2467. A point moves under constant acceleration which is initially normal, and when the direction of motion has turned through an angle the direction of acceleration has turned throiigh an angle m^ (m constant) in the same sense : prove that the intrinsic equation of the ciu've described is ds ^^ = c(cosm-l«^)'«-i; and determine the curve when m= 1, 2, or 3. 2468. A cycloid is described under constant acceleration and 0, <^ are the angles which the directions of motion and of acceleration at any DYNAMICS OF A PARTICLE; 423 point make with tlie tangent and normal at the vortex respectively : prove that sin ^ = cos esm{- 26) log In tan (^ - ^)| J or that <^ = 2$. 24G9. A point describes an ellipse under accelerations to the foci which are, one to another at any point, inversely as the focal distances ; find the law of either acceleration, i)rove that the velocity of the point varies inversely as the conjugate diameter, and that the periodic time is f - ) , where o) is the angular velocity about the centre at the end of either axis. [This path so described is also a brachystochrone between any two points for a certain force in the centre.] 2470. The cusp of a cardioid is /S and the centre of the fijxed circle (by which it can be generated as an epicycloid) is C, and the cardioid is described under accelerations F, F' tending to H, C respectively : prove that r" dr^ ^ r' dr' \ r' ] "' •where r, r are the distances from S, C, and a = SC. Also prove that, if the angular velocity about the cusp be constant, F will be constant, F' will vary as r\ and at the apse 2F + F' = 0. 2471. A point P starts from A and moves along a straight line with uniform velocity F ; a i)oint Q starts from B and moves always towards P with uniform velocity v : prove that, if V > v, the least distance c between P and Q is l+m \—m a (sin a)'""* (1 - cos a)" ^ (1 + m) " (1 - m)~^ , and, if < be the time after which they are at this distance, _^ 2 //ic — a (m — cos a) \ —m V where «i = — , a = AB, and a is the angle which AB makes with the path of P. 2472. A point P is describing a parabola whose focus is *S' under acceleration always at right angles to SP, the plane in which the motion takes place having a constant velocity i)arallel to the axis, and equal to the velocity of P parallel to the axis in the parabola at the end of the latus rectum: prove that the path of i^ in space is a "curve of pursuit" to S described with a constant velocity equal to that of S. 2473. A point describes a cui-\'e which lies on a cone of revolution and crosses all the generating lines at a constant angle, under accelei-a- tion whose direction always intersects the axis : prove that the accelera- tion makes a constant angle with the axis and varies invei-sely as the cube of the distance from the vertex. 42+ DYNAMICS OF A PARTICLE. 2474. The straight lines AT, 7? /'joining a moving point P to two fixed points A, B liave constant angular velocities 2o>, 3o) : prove that the acceleration of F is compounded of a constant acceleration along AP and an acceleration varying as BP along PB. [These accelerations are 12o)°yl/?, and KSta^PB respectively.] 2475. A point describes a rliunib line on a sphere so that the longi- tude increases uniformly : jirove that the whole acceleration varies as the cosine of the latitiule and at any point makes with the normal an angle equal to the latitude. [If a be the constant angle at which the curve crosses the meridians, and to be the rate at which the longitude increases, the three accelera- tions resolved as in (2443) will be -/sill' 9, /cos 2a sin 6 cos 0, /sin 2a sin 6 cos 6, where /sin^ a = w* x radius of the sphere.] 2476. A point P describes a circle under acceleration tending to a point aS' and varying as SP, S being a point which moves on a fixed diameter initially passing through P : prove that, if 6 be the angle described about the centre in a time t, /JmsmO = Jm -i-l sm(t ij/x), and the distance of S from the centre = — sec' 6 ; where a is the radius m of the circle and 7n constant. 2477. A point describes an arc of a ciixle so that its acceleration is always proportional to the n^^ power of its velocity : prove that the direction of tlie acceleration of the point always touches a certain epicycloid generated by a cii'cle of radius a -f- 2 (3 — ?i) rolling on a circle of radius a {2 — 7i) -i- (3 — n) ; where a is the radius of the described circle. II. Central Forces. 2478. Prove that the parabola y" = inx can be described under a constant force parallel to the axis of y and a force proportional to y j)aral]el to the axis of x ; also, under two forces 4/a (c + x), fxy parallel to the axes of x and y respectively. 2479. A particle is acted on by a force parallel to the axis of y whose acceleration is fxy, and is initially projected with a velocity a Jfx. ]iarallel to the axis of a; at a point where y = a: prove that it will describe a catenary. 2480. A particle is acted on by a force parallel to the axis of y whose acceleration (always towards the axis of x) is /x?/"', and, when y - a, is projected parallel to the axis of x with velocity \/ — '• prove that it will describe a cycloid. DYNAMICS OF A PARTICLE. 425 2481. Two equal particles attract each otlier with a force varjiiig inversely as the square of the distance and are projected simultaneously with equal velocities at right angles to the joining line : jirove that, if each velocity be equal to that in a circle at the same distance, each particle will describe a semi-cycloid. 2482. A cai'dioid is described with constant angular velocity about the cusp under a constant force to the cusp and another constant force : ]>rove that the magnitude of the latter is double that of the former and that its line of action always touches an ejdcycloid generated by a circle of radius a rolling upon one of radius 2a; 8a being the length of the axis of the cardioid. 2483. The force to the origin under which the hyj^erbola r cos 2^ = 2 ^2 a cos 9 can be described will vary as {Jar + r'^ + cCf -r r\ 2484. The perpendicular SY is let fall from the origin upon the tangent at any point P of the cui've r' = cc sin 20, and the locus of Y is described under a force to IS: prove that this force will vaiy as 2485. In a central orbit the resolved velocity at any point perpen- dicular to the radius vector is equal to the velocity in a circle at that distance : prove tliat the orbit is a reciprocal spiral. 2486. A particle moves under a constant repulsive force from a fixed point, and is projected with a velocity which is to that in a circle at the same distance under an equal attractive force as ^2 : 1 : prove that the orbit is the curve whose equation is of the form r^ = a^ sec 1 9. 2487. The force to the pole under which the pedal of a given curve r=f[p) can be described will vary as rp'^ i2r — j) -j-) > ^^^y if the given curve be r^sin f ^ = a--, this- force will be constant. 2488. An oi'bit described under a constant force tending to a fixed point will be the pedal of one of the curves represented Uy the equation aV = 2^^ + ^i''^ where a and h are constants. 2489. A parabola is described about a centre of force in C, the centre of curvature at the vertex A : prove that the force at any point F of the parabola varies as CP {AS + SP)~^, where *S' is the focus. 2490. The force tending to the pole under which the evolute of the curve r =f{T>) can be described will vary inversely as 426 DYNAMICS OF A PARTICLE, 2491. A jmrticle P is projected from a point A at riglit angles to a straight line SA and attracted to the fixed point a? by a force varying as cosec PSA : prove that the rate of describing areas about A will be viniformly accelerated. 2492. A particle is projected at a distance a -with velocity eqnal to that in a circle at the same distance and at an angle of 45° with the distance, and attracted to a fixed point by a force which at a distance r is equal to fj.r~^ (a^ + 3r^) : prove that the equation of the path is r = a tun ( j — .-> ) , and that the time to the centre of force is 2493. A particle is atti-acted to a fixed point by a force which at a distance r is equal to /tr-"(3«V3aV*-r''), and is projected from a point at a distance a from the centre with a velocity equal to that in a circle at the same distance and in a direction making an angle cot"* 2 with the distance : prove that the equation of the orbit is r^ = a^ tan {l-^-e), and that the time to the centre of force is 1 ^sjy. l02 2. 2494. A particle is describing a circle under the action of a constant force in the centre and the force is suddenly increased to ten times its former magnitude : prove that the next apsidal distance will be equal to one fourth the radius of the circle. 2495. A particle is describing a central orbit in such a manner that the velocity at any point is to the velocity in a circle at that distance as 1 : Jn : prove that p co r", p being the perpendicular from the centre of force on the tangent at a point whose distance is r, and that the force will vary inversely as j-^""^*. If the force be repulsive and the velocity at any point be to that in a circle at that distance under an equal attractive force as 1 : Jn, the particle will describe a path having two asymptotes inclined at an angle n + \ 2496. A particle acted on by a central force fxr~^{ir-3a) is pi'ojected at a distance a, an angle 45", and with a velocity which is to the velocity from infinity as J'2 : Jb : prove that the equation of the path is a = r (1 + sin 6 cos 6), and that the time from projection to an apse is — j^r- (47r - 3 ^3). DYNAMICS OF A PARTICLE. 427 2497. A portion of an epicycloid is described under a force tending to tlie centre of tlie fixed circle : prove that, if a straight line be drawn from any fixed point always parallel and proportional to the radius of curvature in the epicycloid, the extremity of this line will describe a central orbit. 2498. The curve whose intrinsic equation is « = «(€'"* — €"'"''') is described under a central attractive force, the describing point being initially at an apse at a distance c= 2ma-^(l 4-m^) from the centre of force : prove that the force varies as r (r* + c*)~*, and that the end of a straight line drawn from a fixed point always parallel and pro- portional to the radius of curvature in the path will also describe a central orbit. 2499. A particle describing a parabola about a force in the focus comes to the apse at which point the law of force changes, and the force varies inversely as the distance until the particle next comes to an apse when the former law is restored ; there are no instantaneous changes in magnitude : prove that the major axis of the new elliptic orbit will be i/f a -r {m' — 1), where 4« is the latus rectum of the parabola and ni is that root of the equation x' (I —\ogx) = 1 which lies between 2 Je and e, and that the excentricity will be 1 ; . 2500. In an orbit described under a central force a straight line is drawn from a fixed point perpendicular to the tangent and pro- ])ortional to the force, and this straight line describes equal areas in equal times : prove that the difierential equation of the orbit is of the form and that the rectangular hyperbola described about the centre is a l)articular case. 2501. A uniform chain rests under normal and tangential forces which at any point of the chain are —n, t per unit of length of the chain : prove that a particle whose mass is equal to that of a unit of length of the chain can describe the same curve under the action of normal and tangential forces %i, t at the same point. 2502. A centre of force vai-ying inversely as the ^t*** power of the distance moves in the circumference of a circle and a particle describes an arc of the same circle under the action of the force : prove that the velocity of the centre of force must bear to the velocity of the particle the constant ratio ^ —n : 1 — ??, and that, when the ac- celeration of the force at a distance r is fir"^, the time of describing a semicircle is 4 "^ V /^" 2503. A particle P is repelled from a fixed point aS' by a force varying as (distance)"* and attracts another particle Q with a force vai-ying as (distance)"'; initially /* and Q are equidistant from aS' in 428 DYNAMICS OF A PARTICLE. opposite directions, P is at rest, and tlie accelerations of the two forces are equal : ])rove that Q, if ])rojecte(l at right angles to SQ with proper velocity, will descrilje a parabola with aS' for focus. 2504. A particle P is re])elled from two fixed points .S', S' by forces, varying each as (distance)"'' and equal at equal distances, and attracts another particle Q with a force varying as (distance)"^ ; initially P, Q divide SS' internally and externally in the same ratio, P is at rest, and the accelerations of the forces on the two ]»articles are equal: 2)rove that, if Q be projected at right angles to >S' Jaf) along the interior of the cii'cle : prove that the normal pressiu-e on the curve will be diminished one half after the time ^V / ''V V+Jaf 'J' where a denotes the radius of the circle and /a the coefficient of friction. 2521. A heavy pai-ticle is projected horizontally so as to move on the interior of a smooth hollow sphere of radius a and the velocity of projection is Jiga : prove that, when the j^article again moves hori- zontally, its vertical depth below the highest point of the sphere is equal to its initial distance from the lowest point. 2522. A heavy particle is attached to a fixed point by a fine inextensible string of length «, and, when the string is horizontal and at its full length, the particle is projected horizontally at right angles to the string with the velocity due to a height 2a cot 2a : prove that the greatest depth to which it will fall is a tan a. 2523. A particle slides in a vertical plane down a rough cycloidal ■ arc whose axis is vertical, starting from the cusp and coming to rest at the vertex : prove that the coefficient of friction is given by the equation [More generally, if the particle come to rest at the lowest point and 6 be the angle which the tangent at the starting point makes with the horizon, fxf}^^ = sin - iJ. cos 0.] DYNAMICS OF A PARTICLE. 431 2524. A rough wire in the form of an arc of an eqniangnlar spiral whose constant angle is cot"' (2/u,) is placed with its plane vertical and a heavy particle falls down it, coming to rest at the first point where the tangent is horizontal : prove that at the stai-ting point the tangent makes with the horizon an angle double the angle of friction, and that during the motion the velocity will be gi-eatest when the angle which the tangent makes with the horizon is given by the equation (2/A* - 1) sin <^ + Six cos °4V^')- where a is the radius of the generating ch-cle. 2529. An elliptic wire is placed with its minor axis vertical and on it slides a smooth ring to which are attached strings which pass through smooth fixed rings at the foci and sustiiin each a particle of weight equal to the weight of the ring : determine the velocity which the jiarticle must have at the highest point in order that the velocity at the lowest point may be equal to that at the end of the major axis. [The required velocity is that due to a height 432 DYNAMICS OF A PARTICLE. '2:>^0. Two jiai-ticlcs of masses p, q Jire connected by a fine inex- tensible string which jiasses through a small fixed ring ; f hangs vertically and f/ is held so that the adjacent string is horizontal : prove that when q is let go the initial tension of the string is J)VJ -^kP ^/l)-' and the initial radius of curvature of the path of q bears to the initial distance of q from the ring the ratio 3 {/+ (/3 + !7)'}'^ -.p {p+q) {^P + ^)' 2531. A particle in motion on the surface z^^{x, y) under the action of gravity describes a curve in a horizontal plane with velocity w: prove that, at every point of the path, ij\\di/] dx^ "dxdydxdy \dx) df] \\dxj \di/J j the axis of ;; being vertical. . 1 -n • • / /t-TCOSa , 7 • ,1 time of a vertical oscillation is tt . / —yz — - — ^ s-t , where k is tlie 2532. In a smooth surface of revolution whose axis is A-ei-tical a heavy i)article is projected so as to move on the surface and describe a path which difiers very little from a horizontal circle : prove that the Icr cos a • + 3r sin a cos' a) ' distance from the axis, r the radius of curvature of the meridian curve, and a the inclination of the normal to the vertical in the mean position of the particle. 2533. A heavy particle is projected inside a smooth paraboloid of revolution whose vertex is its lowest point and the greatest and least vertical heights of the particle above the vertex are A,, h^, the velocities at these points being V^, V„: prove that V^' = 2gh,, V^' = 2gh^, and that throughout the motion the pressure of the particle on the paraboloid will vary as the curvature of the generating parabola. [The pressure = 2TF (h^ + a) (h^ + a) -=- ap, Avhere W is the weight of the particle, 4a the latus rectum, and p the radius of curvature of the generating parabola. Also p', the radius of absolute curvature of the path of the particle, is given by the equation C^) - where z denotes the height of the particle.] 2534. A heavy particle is moving upon a given smooth surface of revolution under the action of a force F ))arallel to the axis : prove that the equation of the projection of the path on a plane perpendicular to the axis is where ( - , 6j are polar co-ordinates measured from the trace of the axis dz and the equation of the surface is u" j- =/{'u), the axis of the surface being the axis of z. DYNAMICS OF A PARTICLE. 48:1 2535. Two particles of masses m, m' lying on a smooth liorizontal table are connected by an inextensible string at its full length and passing through a small fixed ring in the table ; the particles are at distances «, a from the ring and are projected with velocities V, V at right angles to the string so that the parts of the string revolve in the same sense : prove that eitlier particle will describe a circle uniformly if m Y^a = m' V'^a ; and that the second apsidal distances will be a, a respectively if m Y'd' = m F'"«'^ 2536. Two particles m, m, connected by a string which passes through a small fixed ring, are held so that the string is horizontal and the distances from the ring are a, a; the particles ai-e simultaneously set free and proceed to describe paths whose initial radii of curvature are p, p' : prove that m m' I I 1 1 — = — , - + -,= - + -. p p p p a a 2537. Two particles vi, m are connected by a string, m! lies on a smooth horizontal table and in is held so that the part of the string (of length a) which is not in contact with the table makes an angle a with the horizon : prove that, when m is set free, the initial radius of curvature of its path is 3a {m* + m' (2/71 + m') cos^a}^ m {m + m') [m cos a + l^2m + 'irn') cos'a} ' 2538. Two particles A and B are connected by a fine string; A rests on a rough horizontal table and B hangs vertically at a distance a below the edge of the table, A being in limiting equilibrium ; B is now projected horizontally with a velocity V in the plane normal to the edge of the table : prove that A will begin to move with acceleration p-Y' -=r{p. + 1 ) rt, and that the initial radius of curvature of the path of B will be « (/u, + 1), where p. is the coefficient of friction. 2539. A smooth surface of revolution is generated by the curve x^y — a? revolving about the axis of y, which is vertically do^saiwards, and a heavy particle is projected with a velocity due to its depth below the horizontal plane through the origin so as to move on the surface : prove that it will cross all the meridians at a constant angle. 2540. A heavy pai-ticle is projected so as to move on a smooth curve in a vertical plane starting from a point where the tangent is vei'tical ; the form of the curve is such that for any velocity of projection the particle will abandon the curve when it is at a vertical height abovo the point of projection which bears a constant ratio 2 : 77i + 1 to the greatest height subsequently attained : prove that the equation of the curve is y^ ~ ex"""', where c is constant. 2541. A smooth wire in the form of a circle is made to revolve uniformly in a horizontal plane aboiit a point A in its circumference Avith angular velocity ^^ any time from its mean position is A^ sin {n^t + B^) + A^ sin {nj, + B^ ; where n^, n^ are the positive roots of the equation in x \x^ - — (1 - cos a)\ Ix" — (1 + 3 COS" a)|- = "', cos a (1 - cos a) ; \ C J \ ^ COS Ot J c where c is the mean distance from the ring and q cos a e J)- 2545. A heavy particle is projected so as to move on a rough inclined plane, the coefiicient of friction being n tan a and the inclination of the plane a : prove that the intrinsic equation of the path will be d8_^J^ 1 ** the radii of curvature, at these points and at the highest point, F,F^cos>=F^ p,p,cos''«^ = r^ DYNAMICS OF A PARTICLE. 435 2540. In tlie last question, in tlie particular case when n - 1, prove that the time of moving from one of the two points to the other is — -. — ^Jog tan - + ^ + — 5^, > ; ^-sma i *= V4 2/ cos' 11 y^y^ 4-tti^ I I'll '1 ' - — ^log tan , + I^ + — :r iina\ * V-i 2/ cos". 2g sin a \ * V-i 2 / cos^ <^ cos'* (/> the horizontal space described is 2F" sin <^ (3 - sin' <^) ^ 3^ sin a cos"^<^ ' and that 114 sin ^ cos cf> Pi P2~ »* 2547. A heavy particle moves on a smooth curve in a vertical plane of such a form that the pressure on the curve is constant and equal to m times the weight of the particle : prove that the intrinsic equation of the path is ds a d ~ (to + cos ^)' ' <^ being measured downwards from a fixed horizontal line ; that the difference of the greatest and least vertical depths of the particle is 2ma -r {vi^ — 1)- ; the time from one vertex to the next in the same horizontal plane is 2Trm - -^ (m' - 1)^; and the arc between these points Tra (1 + 2m^) h- («i^ — !)'• Also the greatest breadth of a loop is « |(1 + 2m') JiiT^ - Znf cos- (-)}- "i K - 1)^. 2548. A particle is placed at rest in a rough tube (4/m = 3) which revolves uniformly in one plane about one extremity and is acted on by no force but the pressures of the tube : prove that the equation of the path of the particle is 5r=«(4€»« + c-2<'). 2549. A rectilinear tube inclined at an angle a to the vertical revolves with uniform angular velocity w about a vertical axis whicli intersects the tube, and a heavy particle is projected from the stationary point of the tube with a velocity ^ cos a -f to J&m a ; find the position of the particle at any given time before it attains relative equilibrium ; and pi'ove that the eqiiilibrium is unstable. [The particle will describe a space s along the tube in the time u ^sm a V' - 8/ 28 2 43 G DYNAMICS OF A PARTICLE, where aw' siu a = g cos a ; and the equation of motion is -T^i- = w* (s - a) sin a.l 2550. A smooth parabolic tube of latus rectum I is made to revolve about its axis, which is vertical, with angtJar velocity /| , and a heavy particle is projected up the tube : prove that the velocity of the particle is constant and that the greatest height to which the particle rises in the tube is double that due to the velocity of projection. 2551. A smooth parabolic tube revolves with uniform augular velocity about its axis, which is vertical, and a heavy particle is placed within the tube very near the lowest point j find the least angular velocity which the tube can have in order that the particle may rise ; and prove that, if it rise, its velocity will be proportional to its distance from the axis; also that, if one position be one of relative equilibrium, every position will be such. 2552. A curved tube is revolving imiformly about a vertical axis in its plane and is symmetrical about that axis ; the angular velocity is /- , where a is the radius of curvature at the vertex : prove that the equilibrium of a particle placed at the vertex will be stable or unstable according as the conic of closest contact is an ellipse or hyperbola. 2553. A cii'cular tube of radius a revolves uniformly about a vertical diameter with angular velocity ^ — , and a particle is projected from its lowest point with such velocity that it can just reach the highest point : prove that the time of describing the first quadrant is y {n + \)g ]og (Jn + 2 + JnTl). 2554. A circvilar tube containing a smooth particle revolves about a vertical diameter with uniform angular velocity w, find the position of relative equilibrium ; and prove that the particle will oscillate about this position in a time 27r -^ w sin a, a being the angle which the normal at the point makes with the vertical. 2555. A heavy particle is placed in a tube in the form of a plane curve which revolves with uniform angular velocity w about a vertical axis in its plane, and the particle oscillates about a position of relative equilibrium : pi-ove that the time of oscillation is k — r sin a cos^ a ' 7c being the distance from the axis, r the radius of curvature, and a the inclination of the normal to the vertical, at the point of equilibrium. DYNAMICS OF A PARTICLE. 437 2556. A straight tube inclined to the vertical at an angle a revolves with uniform angular velocity w about a vertical axis whose shortest distance from the tube is a and contains a smooth heavy particle which is initially placed at its shortest distance from the axis : prove that the space s which the particle describes along the tube in a time t is given by the equation Q cos Ct to" sin" a ^ ^ 2557. A heavy particle is attached to two points in the same horizontal plane at a distance a by two extensible strings each of natural length a, and is set free when each string is at its natural length : prove that the radius of curvature of the initial path of the pai'ticle is 2 ;^/3a -^ (fti ~ n), the moduli of the strings being respectively m and n times the weight of the particle. 2558. Three equal particles P, Q, Q', for any two of which e = 1, move in a smooth fine circular tube of which AB is a vertical diameter; P starts from A, and Q, Q' at the same instant in opposite senses from B, the velocities being such that at the first impact all three have equal velocities : prove that thi-oughout the whole motion the straight line joining any two particles is either horizontal or passes through one of two fixed points (images of each other with respect to the circle) ; and that the intervals of time between successive impacts are all equal 2559. A point P describes the curve y = a log sec - with a velocity which varies as the cube of the radius of curvature and has attached to it a particle Q by means of a string of length a ; when P is at the origin, Q is at the corresponding centre of curvatui'e and its velocity is equal and opposite to that of P : prove that throughout the motion the velocity of (J will be equal in magnitude to that of P, and that Q is always the pole of the equiangular sjjiral of closest contact with the given curve at P. lY. 2Iotion of Strings on Curves or Surfaces. 25G0. A uniform heavy chain is placed on the arc of a smooth vertical circle, its length being equal to that of a quadrant and one extremity being at the highest point of the circle : prove that in the beginning of the motion the resultant vertical pressure on the circle beara to the resultant horizontal pressure the ratio tt^ — 4 : 4. 2561. A string of variable density is laid on a smooth horizontal table in the form of a curve such that the curvature is everywhere proportional to the density and tangential impulses are applied at the ends : prove that the equation for determining the impulsive tension T at any point is T = J€* + i/e"*^, where <^ is the angle which the tangent makes with a fixed direction ; and that, if the curve be an equiangular spiral, the initial direction of motion of any point will be at right angles to the radius vector. 438 DYNAMICS OF A rAllTICLE. 2562. A number of material particles 1\, P^, ... of masses wi,, m^, ... connected by inexteusible strings are i)laced on a liorizontal plane so tliat the strings are sides of an unclosed polygon each of whose angles is TT - a, and an impulse is applied to F^ in the direction J\Pj '• prove that m^ {r ^^ cos a - rj - m^^ , (T^ - T^_^ cos a), ■where T^ is the impulsive tension of the r^^ string; and deduce the equation (Id /x ds ds p' for the impulsive tension in the case of a fine chain. From either equa- tion deduce the result of the last question. 2563. A fine chain of variable density is placed on a smooth horizontal table in the form of a curve in which it would hang under the action of gravity and two impulsive tensions applied to its ends, which are to each other in the same ratio as the tensions at the same points in the hanging chain : prove that the whole will move without change of form parallel to the straight line which was vertical in the hanging chaiii. 2564. A heavy i^niform string PQ, of which P is the lower ex- tremity, is in motion on a smooth circular arc in a vertical plane, being the centre and OA the horizontal radius : prove that the tension at any jioint P of the string is -^yfsiny sin a , .,) W-\ — - cos (y -I- ^) cos (a + 6)} , a I, y a ) where 0, 2a, 2y are the angles AOP, POQ, POP respectively, and W the weight of the string. 2565. A poi'tion of a hea^y uniform string is placed on the arc of a four-cusped hy2:)Ocycloid, occupying the space between two ad- jacent cusps, and runs ofi'the curve at the lower cusp where the tangent is vertical : prove that the velocity which the string will have when just leaving the arc will be that due to a space of nine-tenths the length of the string. 2566. A uniform string is placed on the arc of a smooth curve in a vertical plane and moAes under the action of gravity : prove the equation of motion d^s g , . ^ = 7(^.-2/.), I being the length of the string, s the arc described by any point of it at a time t, and y^, y^ the depths of its ends below a fixed horizontal straight line. 2567. A uniform heavy string APP is in motion on a smooth curve in a vertical plane, and on the horizontal ordinate from a fixed vertical line to A, P, B are taken lengths equal to the arcs measui-ed DYNAMICS OF A PARTICLE. 439 from a fixed point of tlie curve to A , P, B respectively : prove that the ends of these lengths are the corners of a triangle whose area is always 2>i"oportional to the tension at r. 2568. A uniform heavy string is placed on the arc of a smooth cycloid whose axis is vertical and vertex upwards : determine the motion, and prove that, so long as the whole of the string is in contact with the cycloid, the tension at any given point of the string is constant throughout the motion and greatest at the middle point (measui-ed on the arc). 2569. A uniform heavy chain is in motion on the arc of a smooth curve in a vertical plane and the tangent at the point of greatest tension makes an angle with the vertical : prove that the diHerence between the depths of the extremities is I cos . 2570. A uniform inextensible string is at rest in a smooth groove, which it just hts, and a tangential impulse P is applied at one end : prove that the normal impulse per unit of length at a distance s (along the arc) from the other end is Ps-i-ap, where a is the whole length of the string and p the radius of curvature at the point considered. 2571. A straight tube of uniform bore is revolving uniformly in a horizontal plane about a vertical axis at a distance c from the tube, and within the tube is a smooth uniform chain of length 2a which is initially at rest with its middle point at the distance c from the axis of revolution : prove that the chain in a time t will describe a space ^c (€<"<-£-<"'!) along the tube, and that the tension of the chain at a point distant x from its middle point is where m is the mass of the chain and w the angular velocity. 2572. A circular tube of radius a revolving with uniform angular velocity to about a vertical diameter contains a heavy uniform rigid wire which just fits the tube and subtends an angle 2a at the centre: prove that the wire will be in relative equilibrium if the radius to its middle point make with the veriical an angle whose cosine is g -i- au>" cos a, and that the stress along the wii'e is a minimum at the lowest point of the tube (provided the wii-e pass through that point) and a maximum at the point whose projection on the axis bisects the distance between the projections of the ends of the wire. Discuss which position of ei^uilibrium is stable, proving the ecpiation of motion aa. -J-. + sin a sin 6 {(j — aco^cos a cos 6) = 0, where 6 is the angle which the radius to the middle point of the wii-e makes with the vertical. 410 DYNAMICS OF A PARTICLE. [Tlie highest position of equilibrium is always unstable; the oblique position is stable if it is possible, the time of a small oscillation being 9, y-T^ ^— , where cnJ cos a cos B ^ o : sin 2a ' 1^ J > 0) sin j3 ' and the lowest position is stable when acj" cos a <., ^ with the horizon and at the highest point, and p, p, r are the radii of curvatui-e at the same two points and the highest point respectively : prove that i 1 _ 2cos> 1 12^08^ V' v"' ic' ' p p r ' 2576. A heavy particle moves in a medium whose resistance varies as the 2 11^^ power of the velocity; v, v', u are the velocities of the particle when its direction of motion makes angles — , <^ with the horizon and at the highest point, and p, p', r are the radii of curvature at the same two points and the highest point respectively : prove that 112 cos'" , Clfp the equation of the hodograph will be r = a sec m9. 2585. A jioint describes half the arc of a cardioid, oscillating symmetrically about the vertex, in such a way that the hodograj^h is a circle with the pole in the circumference : prove that the ac- celeration of the point describing the cardioid varies as 2r — 3a, r being the distance from the cusp and 2a the length of the axis : also prove that the direction of accelei'ation changes at double the rate of the direction of motion. 2586. A heavy particle of weight TT is moving in a medium in which the resistance varies as the ?i"^ power of the velocity, and F is the resistance when the dii-ection of motion makes an angle ^ with the horizon ; prove that ^Y — 7iFcos" cj> J sec"'^^ (f> d<^. 2587. A heavy particle is projected so as to move on a rough plane inclined to the horizon at the angle of friction : prove that the hodo- graph of the path is a parabola and that the intrinsic equation of the path is V' f 3 sin \) s = 1 -■ — /. • — 772^^ + log tan - + ^ U ; where V is the velocity at the highest point and a the angle of friction. 2588. Two points P, Q describe two curves with eqiial velocities, and the radius vector of Q is always pjirallel to the direction of motion of P : shew how to find P's jjath when ^'s jiath is given ; and prove that (1) when Q desciibes a straight line P describes a catenary, (2) when Q describes the cii'cle r= a cos 6, P describes a circle of radius o, (3) when Q describes the cardioid r— a (1 -i- cos 6), P describes a two-cusped ejii- cycloid. 2589. A circle is described by a point in a given time under the action of a force tending to a fixed point within the cii-cle : prove that, for difi'erent positions of the centre of force, the action during a whole revolution varies inversely as the minimum chord which can be drawn through the point. DYNAMICS OF A PARTICLE. 443 [In any closed oval under a central force to a point within it the 2^ /•27r,.3 action during a whole revolution = -75 I -2 «^> where r is the radius A'ector from the centre of force, p the per])endicular from the centre of force on the tangent, the angle which the tangent makes with some fixed straight line, A the area of the oval, and F the periodic time.] 2590. A point describes a parabola under a central force in the vertex : prove that the hodograph is a parabola whose axis is at right angles to the axis of the described parabola. [In general if any conic be described under any central force the hodograph is another conic which will be a parabola when the described conic passes through the centre of force.] 2591. A point P describes a catenary in such a manner that a straight line drawn from a fixed point parallel and proportional to the velocity of F sweeps out equal areas in equal times : prove that the direction of P's acceleration makes with the normal at P an angle tan"' (f tan (^), where ^ is the angle through which the direction of motion has turned in passing from the vertex. 2592. A circle is described under a constant force not tending to the centre : prove that the hodograph is Bernoulli's lemniscate. 2593. A curve is desci-ibed with constant acceleration and its hodograph is a parabola with its pole at the focus : i)rove that the intrinsic equation of the described curve is ds . (b 2594. A point describes a curve so that the hodograph is a circle described with constant velocity and with the pole on its circumference : prove that the described curve is a cycloid described as if by a heavy particle falling from cusp to cusp. 2595. A point describes a certain curve with acceleration initially along the normal, and the direction of acceleration changes at double the rate of the direction of motion and in the same sense : prove that the hodograph will be a cii-cle with the pole on its cii-cumference. 2596. A particle is constrained to move in an elliptic tube under two forces to the foci, each varying inversely as the square of the distance and equal at equal distances, and is just displaced from the position of unstable equilibrium : prove that the hodograph is a cii'cle with the pole on its cix'cumference. [The particle will oscillate over a semi-ellipse bounded by the minor axis, and the hodograph corresponding to this will be a complete circle with the diameter through its pole parallel to the minor axis.] DYNAMICS OF A RIGID BODY. I. Moments of Inertia, Principal Axes. 2597. The density of an ellipsoid at any point is proportional to tlie product of the distances of the point from the principal planes : prove that the moments of inertia about the principal axes are \m{lf -{-G-), \m{c' + a^), \m{a- + Jf), where m is the mass and a, h, c the semi-axes. 2598. Prove the following construction for the principal axes at 0, the centroid of a triangular lamina ABC : draw the circle OBC, and in it the chords Ob, Oc parallel to AC, AB respectively, and let Bb, Cc meet in L ; then, if aa' be the diameter of the circle drawn through L, Oa, Oa will be the directions of the principal axes at 0. 2599. Pi'ove the following construction for the principal axes at the centre of a lamina bounded by a parallelogram ABGD : draw the circle OBG and in it chords Ob, Oc parallel to AB, BC, and let BC, be meet in L ; then, if aa be the diameter of this circle drawn through L, Oa, Oa will be the directions of the principal axes at 0. 2 GOO. Prove that any lamina is kinetically equivalent to three particles, each of one third the mass of the triangle, placed at the corners of a maximum triangle inscribed in the ellipse whose equation, refer-red to the jjrincipal axes at the centre of inertia, is Axr + By" — '2AB, where mA, rtiB are the princij)al moments of inertia and in the mass. 2601. Prove that any rigid body is kinetically equivalent to three equal unifoi-m spheres, each of one third the mass of the body, whose centres are corners of a maximum triangle inscribed in the ellipse ^' +X=2,.= 0, C-A C-B' and whose common radius is ^-| {A + B - G) ; the equation of the ellip- sc? 1^ ^ sold of gyration being -j- + ^ + -^ = 1, and A 20^, which could not be satisfied by a body of form approaching spherical. As the spheres need only be ideal for simplification of calculation, this condition is of no importance.] DYNAMICS OF A RIGID BODY. 445 2602. A straight line is at every point of its course a principal axis of a given rigid body : prove that it passes tlu-ough the centre of inertia. 2603. A tetrahedron is kinetically equivalent to six particles at the middle points of the edges, each y^ the mass of the tetrahedron, and one at the centroid of mass f the mass of the tetrahedron. 2604. Tlie principal moments of inertia of a rigid body, whose mass is unity, at the centre of inertia az-e A, B, C, and a' + b' + c' + r^ is a principal moment of inertia at the point (a, b, c), the principal axes at the centre of inertia being axes of co-ordinates : prove that b' c' + -r. :,-^^. ^, = 1. A-r JJ-r' C-r' 2605. The locus of the points at which two principal moments of inertia of a given rigid body are equal is the focal ciu'ves of the ellipsoid of gyration at the centre of inertia. 2606. The locus of the points at which one of the principal axes passes through a given point, which lies in one of the principal planes at the centre of inertia, is a circle. 2607. The locus of the points at which one of the principal axes of a given rigid body is in a given direction is a rectangular hyperbola with one asymptote in the given direction. 2608. In a triangular lamiua any one of the sides is a principal axis at the point bisecting the distance between its mid-point and the foot of the perpendicular from the opposite corner. 2609. In any uniform tetrahedron, if one edge be at any point a principal axis so also will the opposite edge ; the necessary condition is that the directions of the two edges shall be perpendicular ; and the point at which an edge is a principal axis divides the distance between the mid-point and the foot of the shortest distance between it and the opposite edge in the ratio 1 : 2. 2610. Straight lines are drawn in the plane of a given lamina through a given point ; the locus of the points at which they are princi- pal axes of the lamina is a circular cubic. 2611. The locus of the straight lines drawn through a given point, each of which is at some point of its course a principal axis of a given rigid body, is the cone a{B-C)7jz + b{C-A)zx + c{A-B)x^j = 0, A, B, C being the principal moments of inertia at the given point, a, b, c the co-ordinates of the centre of inei-tia and the principal axes at the given point the axes of reference. Also prove that the locus of the points at which these straight lines are principal axes is the curve '^ cy -bz az - ex \ by -ax )' 44G DYNAMICS OF A RIGID BODY. [The oqnation of the cone on wliich these sti'aight lines lie retains the same form when A, B, C denote the principal moments of inertia at the centre of inertia, and the co-ordinate axes are parallel to the principal axes at the centre of inertia.] 2012. The principal axes at a certain point are parallel to the principal axes at the centre of inertia : prove that the point must lie on one of the principal axes at the centre of inertia. 2G13. The different straight lines which can he drawn through the point {x, y, z), each of which is at some point of its course a principal axis of a given rigid body, will lie on a cone of revolution if x{B-C) = y{C-A) = z{A-B), the principal axes at the centre of inertia being co-ordinate axes and A, Bf C the principal moments of inertia. II. JTotlo7i about a fixed Axis. 2G14. A circular disc rolls in one plane on a fixed plane, its centre describing a straight line with uniform acceleration /"; find the magni- tude and position of the resultant of the impressed forces. [The resultant is a force Mf acting parallel to the plane at a distance from the centre of the disc of one half the radius on the side opposite to the plane.] 2615, A piece of uniform fine wire of given length is bent into the foi'm of an isosceles triangle and revolves about an axis through its vertex perpendicular to its plane : prove that the centre of oscillation will be at the least possible distance from the axis of revolution when the triangle is right-angled. 261 G. A hea-vy sphere of radius a and a heavy rod of length la swing, the one about a horizontal tangent, the other aboiit a horizontal axis perpendicular to its length through one end, each through a right angle to its lowest position, and the pressures on the axis in the lowest positions are equal : prove that the weights are as 35 : 34-. 26 17. The centre of percussion of a triangular lamina one of whose sides is the fixed axis bisects the straight line joining the opposite corner with the mid-point of the side. 2618. A lamina ABCD is moveable about AB which is parallel to CD : prove that its centre of percussion will be at the common point of AC and BD if AB' = WD\ 2619. In the motion of a rigid body about a horizontal axis under the action of gravity, prove that the pressure on the axis is reducible to a single force at every instant of the motion only when the axis of revolution is a principal axis at the point M which is nearest to the centre of inertia : and, if the axis be a principal axis at another point M and the forces be reduced to two acting at J/, N respectively, the former will be equal and opposite to the weight of the body. DYNAMICS OF A RIGID BODY. ^^^^^^^—:rzz^-^^^ 2G20. A rough uniform rod of leugtli 2« is placed with a longtli c (> (i) projecting over the edge of a liorizontal table, the rod being initially in contact with the table and perpendicular to the edge : prove that the rod will begin to slide over the edge when it has turned through an angle whose tangent is —, — r— ; , a being the coefficient ° ^ a^ + 9 (c - a)- '^ ° of friction. 2621. A uniform beam capable of motion about one end is in equilibrium ; find at what point a blow must be applied perpendicular to the rod in order that the impulse on the fixed end may be - th of the blow. [The distance of the point from the fixed end must be to the length of the rod in the ratio ,y?4— 1 : J'ijt.] 2622. A iinifomi beam moveable about its middle point is in equilibrium in a horizontal position, a particle whose mass is one-foui-th that of the beam and such that the coefficient of restitution is 1 is let fall upon one end and is afterwards grazed by the other end of the beam : prove that the height from which the particle is let fall bears to the circumference of the circle described by an end of the beam the ratio 49 (2/i+ 1) : 48, where n is a positive integer. 2623. A smooth uniform rod is i-evolving about its middle point, ■which is fixed on a horizontal table, when it strikes an inelastic particle at rest whose mass is one-sixth of its own, and the angular velocity of the rod is immediately reduced one-ninth : find the point of impact, and prove that, when the particle leaves the rod, the direction of motion of the particle will make wdth the rod an angle of 45". [The point of impact must bisect one of the halves of the rod, and during the subsequent motion /drV 16 where r is the distance of the particle from the centre of the rod, and w the angular velocity of the rod at any time, fi the angiilar velocity before impact.] 2624. A smooth unifoi-rri rod is moving on a horizontal table uniformly about one end and impinges on a particle of mass equal to its own, the distance of the particle from the fixed end being -th of the length of the rod : prove that the final velocity of the particle will be to its initial velocity in the ratio J{5n'-l){n'+3) : in. (In this case also c = 0.) 448 DYNAMICS OF A RIGID BODY. 2G'25. A uniform rod (mass ?«) is moving on a horizontal talJe al)Out one end and driving before it a smooth particle (mass p) -which starts from rest close to the axis of revolution : prove that, when the particle is at a distance r from the axis, its dii'ection of motion ■will make with the rod the angle cot"' - / 1 + ^^, , where mk' is the moment of inertia of the rod about the axis of revolution. 2626. A unifoi'm circular disc of mass m is capable of motion in a vertical plane about its centre and a rough particle of mass p is placed on it close to the highest point : prove that the angle 6 through which the disc will turn before the particle begins to slide is given by the equation ?>pa-\ 1 . . 2pa' 1 + -^, sm ^ — ^ •mJc' J fjL " vik' ' where a is the radius and oiikr the moment of inertia of the disc. 2627. A uniform rod, capable of motion in a vertical plane about its middle point, has attached to its ends by tine strings two particles which hang freely; when the rod is in equilibrium inclined at an angle a to the vertical one of the strings is cut : prove that the initial tension of the other string is mjyg -f- (m + 3^^ sin" a), and that the radius of curvatiire of the initial path of the particle is Qlj) sin^ a -h m cos a, m, p being the masses of the rod and of a particle, and I the length of the string. 2628. A uniform rod moA-eable about one end is held in a hori- zontal position, and to a point of the rod is attached a heavy particle by means of a string : prove that the initial tension of the string when the rod is allowed to fall freely is •nipga (la - 3c) ~ {ivia^ + ?>pc^), where m, p are the masses of the rod and particle, 2a the length of the rod, and c the distance of the string from the fixed end : also prove that the initial path of the particle referred to horizontal and vertical axes will be the curve ma (4a — 3e) y^ + 90c^Z [ma +pc)x = 0, where I denotes the length of the string. 2629. A unifonn rod moveable about one end has attached to the other end a heavy particle by a fine string ; initially the rod and string are in one horizontal sti'aight line without motion: prove that the radius of curvature of the initial path of the particle will be iab -^ {a+ 9b), DYNAMICS OF A RIGID BODY. 449 ■where a, b denote the lengths of the rod and string ; and explain ^vhy the result does not depend on the masses of the two. 2630. A imifoim rod, of length 2n and mass ???, capable of motion about one end, is held in a horizontal position and on tlie rod slides a small smooth ring of mass p : prove that, when the rod is set free, the radius of curvature of the initial path of the ring will be -^ (l + ^i -' 4« — 3c \ 7)1 a ■whei'e c is the initial distance of the ring fi'om the fixed end. 2631. A uniform rod capable of motion about one end has attached at the other end a particle by means of a fine string, and the system is abandoned freely to the action of gravity when the rod makes an angle a with the string which is vertical : prove that the radius of curvature of the initial path of the particle is 9/ ^1 + '^) siu^' a - cos a (2 - 3 sin- a); where 77i, p are the masses, and I the length of the string. 2632. A uniform rod is moveable about one end on a smooth horizontal table and to the other end is attached a particle by a tine string; at starting the rod and string are in one straight line, the particle is at rest, but the rod in motion : prove that when the rod and string are next in a straight line the angular velocities of the rod and string will be as 6 : a, or as h {^p (a - by - ma'} : a{Sp{a- bf + ma (a - 2b)}, where m, p are the masses, and a, b the lengths of the rod and string. III. Motion in Two Dimensions. 2033. Two equal uniform rods AB, BC, freely jointed at B and moveable about A , start from rest in a horizontal position, BG passiug over a smooth peg whose distance from ^4 is Aa sin a (where 3 sin a < 2): prove that, when BC leaves the peg, the angular velocity of -i^ is y ^9 2a 1 + sin' 2a ' where 2a is the length of either rod. 2634. A uniform rod of length 2a rests with its lower end at the vertex of a smooth surface of revolution whose axis is vertical and passes through a smooth fixed ring in the axis at a distance b from the vertex : the time of a small oscillation will be _ /~c ur + 3 (6 - ay ^''VS^ h'-ac 3^7 where c is the radius of curvature at the vei'tex. w. P. 20 4o() DYNAMICS OF A RIGID BODY. 2635. Two heavy particles are fixed to the ends of a fine wii-e in the form of a circular arc, which rests with its plane vertical on a rough horizontal plane, and a, /? are the angles which the radii through the particles make with the vertical : prove that the time of a small oscillation will be /: /^ n a + li "^ cos 2636, Two equal and similar uniform rods, freely jointed at a common extremity, rest symmetrically over two smooth pegs in the same horizontal plane so that each rod makes an angle a with the vertical : prove that the time of a small oscillation will be '-si- ^1 + 3 cos" a ' where 2a is the length of either rod. 2637. A lamina with its centre of inertia fixed is at rest, and is struck by a blow at the point (a, h) normally to its plane : prove that the equation of the instantaneous axis is Aax + Bhy = 0, the axes of co-ordinates being the principal axes at the centre of inertia and A, B being the principal moments of inertia; also that, if (a, h) lie on a certain straight line, thei'e will be no impulse at the fixed point. 2638. A uniform heavy rod revolves uniformly about one end in such a manner as to describe a cone of i-evolution: determine the pressure on the fixed jwint and the relation between the angle of the cone and the time of revolution; and prove that, if 6, cf)he the angles which the vertical makes with the rod and with the direction of pressure, 4 tan ^ = 3 tan $. 2639. A fine string of length 2b is attached to two points in the same horizontal plane at a distance 2a and can-ies a particle /> at its middle point; a uniform rod of length 2c and mass m has at each end a ring through which the string passes and is let fall from a symmetrical position in the same straight line as the two points: prove that the rod will not reach the particle if (a + b- 2c) (m + 2])) m<2 (2c - a) ^r. 2640. A heavy uniform cV.ain is collected into a heap and laid on a horizontal table and to one end is attached a fine string which, passing over a smooth fixed pulley vertically above the heap, is attached to a weight equal to the weight of a length a of the chain : prove that the Icnirth of the chain raised before the weight first comes to instantaneous DYNAMICS OF A lUGID BODY. 451 rest is a Jo, and that wlien the weight next comes to rest the length of chain which is vertical is ax, where x is given by the equation (^7^ ^V3- and that x is nearly equal to -j- . 2641. A uniform rod of length c has at its ends small smooth ring.s which slide on two fixed elliptic arcs whose planes are vertical and semi-axes are a, b; a + c, b + c respectively, and are inclined at angles a, -J + a to the horizon : determine the motion of the rod and the pressures on the arcs, the rod being initially vertical. 2642. A circular disc rolls on a rough cycloidal arc whose axis is vertical and vertex downwards, the length of the arc being such that the curvature at either end of the arc is equal to that of the circle : prove tliat, if the contact be initially at one end of the arc, the point on the auxiliary cii'cle of the cyoloid which corresponds to the point of contact will move with uniform velocity which is independent of the radius of the disc ; and that the normal pressure B and che force of fi'iction F in any position of the disc are given by the equations 3Ii = TT (5 cos ^ - 2 cos a), SF=W sin 9, where W is the weight of the disc, 6 the angle which the common normal makes with the vertical, and a the initial value of 0. 2643. A uniform sphere rolls from rest down a given length ^ of a rough inclined plane and then traverses a smooth portion of the plane of length ml ; find the impulse which takes place when perfect rolling again begins, and prove that the subsequent velocity is less than would have been the case if the whole plane had been rough; if 7)i = l'20, in the ratio 67 : 77. [The ratio in general is 2 + J'25 + 35»i ; 7 J.>n + 1. 2644. A straight tube AB of small bore, containing a smooth uniform rod of the same length, is closed at the end B and in motion about the fixed end A with angular velocity w : prove that, if the end Ji be opened, the initial sti-ess at a point F of the rod is equal to Mo)'AF.FB-^2AB, M being the mass of the rod. 2645. The ends of a uniform heavy rod are fixed by smooth rings to the arc of a circle which is made to revolve uniformly aboiit a fixed vertical diameter ; find the positions of relative equilibrium, and prove that any such position in which the rod is not horizontal will be stable. [If a be the radius of the circle, w its angular velocity, 2a the angle which the rod subtends at the centre, there will be no inclined positions of 29—2 452 DYNAMICS OF A RIGID BODY, equilibrium unless ao)' cos a> g: if ckJ cos a cos /? = (7, the time of a small oscillation about the inclined position w-ill be ^ ^-j^ -j^l + ^ tan*a ; the time of oscillation about the lowest position will be 2Tr Jl + I tan'-' a 4- ^ " - w" cos a ; and, when g - am" cos o, the equation of motion will be (1 + 1 tan^ a) -^'f + cu= sin ^ ( 1 - cos ^) = 0.] 2646. A smooth semicircular disc rests with its plane vertical and vertex upwards on a smooth horiz.'ntal table and on it rest two equal uniform rods, each of which passes through two smooth fixed rings in a vertical line ; the disc is slightly dis]>laced, and in the ensuing motion one rod leaves tlie disc when the other is at the vertex : prove that m 4 sin a — (1 + sin /?)* (2 — sin /8) p sin- /3 ' where m, p ai-e the masses of the disc and of either rod, a the angle which the i-adius to either point of contact initially makes with the horizon, and /? = cos"' (2 cos a). [When the one rod leaves the disc, the pressure of the other on the disc Ls 2^9 {^ — ^^^ (^)- ] 2647. A uniform rod moves with one end on a smooth horizontal plane and the other end attached to a string which is fixed to a point above the plane ; when the rod and string are in one straight line the rod is let go : prove that the angular velocity of the string when vertical will be *. / "^ I 1 ,) a^id its angular acceleration V ^ \ a + lj ° g la + I — h +1 V ^ a l + h' a, I, h being the lengths of the rod and string and the height of the fixed jioint above the plane respectively. 2648. A uniform beam rests with one end on a smooth horizontal table and has the other attached to a fixed point by means of a string of length I : prove that the time of a small oscillation in a vertical plane will be yf- 2649. A sphere rests on a rough horizontal plane with half its weight supported by an extensible string attached to the highest point, whose extended length is equal to the diameter of the sphere : prove that the time of small oscillations of the sphere parallel to a vertical plane is 2:ryil^ DYNAMICS OF A RIGID BODY. 453 2650. Two equal uniform rods AB, BC, freely jointed at Ji, are placed on a smooth liorizoutal table at right augles to each other and a blow is applied to A at right angles to AB : prove that the initial velocities of A, C are in the ratio 8:1. 26.51. Two equal uniform rods AB, BC, fi'eely joijited at B, are laid on a smooth horizontal table so as to include an angle a and a blow is a[)plied at ^1 at right angles to AB ; determine the initial velocity of C, and prove that it will begin to move jjarallel to yl^ if 9 cos 2a = 1. 2652. Five equal uniform rods, freely jointed at their extremities, are laid in one straight line on a horizontal table and a blow applied at the centre at right angles to the line : prove that, initially, V _ V, _ *'2 _ W, _ «o where v, f,, r„ are the velocities of the thi*ee rods, w,, w^ the angular velocities of the two pairs of rods, and '2a the length of each rod. 2653. Four equal uniform rods AB, BC, CD, DE, freely jointed at B, C, D, are laid on a horizontal table in the form of a square and a blow is applied at A at right angles to AB from the inside of the square: prove that the initial velocity of ^ is 79 times that of E. 2654. Two equal uniform rods siB, BC, freely jointed at B and moveable about ^1, are lying on a smooth horizontal table inclined to each other, at an angle a ; a blow is applied to C at right angles to BC in a direction tending to decrease the angle ABC : prove that the initial angular velocities of AB, BC will be in the ratio cos a : 8 - 3 cos"' a ; that 0, the least value of the angle ABC during the motion is given by the equation 8 (5 - 3 cos 0) (2 - cos^ a) ^ (1 - cos a)- (1 6 - 9 cos" a) : TT also pi-ove that, when a = ;; , the angular velocities of the rods when in a straight line will have one of the ratios - 1 : 3, or 3 : - 5. 2655. A heavy uniform rod resting in stable equilibrium within a smooth ellipsoid of revolution about its major axis, which is vertical, is slightly displaced in a vertical plane : prove that the length of the equivalent simple pendulum is «ce (3«" + 1) -f 6 (rt- c), where 2a is the length of the rod, 2c the latus rectum, and e the excentricity of the gene- rating ellipse. 2656. A uniform rod of length 2a rests in a horizontal position with its ends on a smooth curve which is symmetrical about a vertical axis : prove that the time of a small oscillation will be ^ /ar cos a ( 1 + 2 cos" a) "'"V ""^(a-rsin'a) ' r being the radius of curvature of the curve and a the angle which the normal makes with the vertical at either end of the rod. 454 DYNAMICS OF A RIGID BODY. 2057. Four equal rods of length a and mass «t are freely jointed so as to form a rli(jmbiis one of whose diagonals is vertical ; the ends of the other diagonal are joined by an extensible string at its natural length and the system falls through a height h on to a fixed horizontal plane : prove that, if $ be the angle which any rod makes with the vertical at a time t after the impact, o • /i\ /"^N i-oqk sura bg . .. (1+3 sm' ^) ( — ) = 4- „ ■ ., + — (cos a - cos 6) ^ ' \dtj a' \-¥ 3sm^a a ^ ' (sin Q — sin a)^ ; Maa siu a where a is the initial value of B and X the modulus of the stiing, IV. Miscellaiieous. 2658. A square is moving freely about a diagonal with angular velocity O, when one of the corners not in that diagonal becomes fixed. ; determine the impulse on the fixed point, and px'ove that the instantaneous angular velocity is iO. [If V be the previous velocity of the point which becomes fixed the impulse will be \MVi\ 2G59. A uniform heavy rod of length a, freely moveable about one end, is initially projected in a hoiizontal plane with angular velocity 12 : prove that the equations of motion are sin^6^ = 0, a\-T) = 3a cos ^ - «n- cot" ^ ; at \iioJ where 6, and the horizontal pressure is — (9 sin 29 H sin 6 J , in the vertical plane through the rod.] 2660. A uniform heavy rod moveable about one end moves in such a manner that the angle which it makes with the vertical never differs DYNAMICS OF A RIGID BODY. 455 mucli from a : prove that the time of its small oscillations will be ^Vlr 3^7 i + 3 cos' a ' where a is the length of the rod. 2661. A centre of force whose acceleration is [x (distance) is at a point 0, and from another point yi at a distance a are projected simul- taneously an infinite number of particles in a direction at right angles to OA and with velocities in arithmetical progression from f ajix to ~^ajix: prove that, when after any lapse of time all the particles become suddenly rigidly connected together, the system will revolve with angular velocity if JfjL. [If the limits of the velocity be n^a Jfi, n/i Jfx, and the time elapsed 6 -r Jim, the common angular velocity of the rigidly connected particles will be 3 (?ij + nj Jfx -^ 6 cos^ ^ + 2 [n' + «,?4, + n.^) sin" ^.] 2662. A uniform hea\'7 rod is suspended by two inextensible strings of equal lengths attached to its ends and to two fixed points whose distance is equal and parallel to the length of the rod ; an angular velocity about a vertical axis through its centre is suddenly communi- cated to the rod such that it just rises to the level of the fixed points : find the impulsive couple, and prove that the tension of either string is suddenly increased sevenfold. 2663. Two equal uniform heavy rods AB, BG, freely jointed at B, rotate uniformly about a vertical axis through A, which is fixed, with angular velocity Q : prove that the angles a, [i which the rods make with the vertical are given by the equations (8 sin a + 3 sin /?) cot a = (9 sin a + 6 sin jS) cot /S = , > where a is the length of each rod. 2664. A perfectly rough horizontal plane is made to revolve with uniform angular velocity about a vertical axis which meets the plane in ; a heavy sphere is projected on the plane at a point P so that its centre is initially in the same state of motion as if the si)here had been placed freely on the plane at a point Q and set in motion by the impulsive friction only : prove that the centre of the sphere will describe uniformly a circle of radius OQ, and whose centre A' is such that OR is equal and parallel to QP. 2665. A jierfectly rough plane inclined at an angle a to the horizon is made to i-evolve with uniform angular velocity Q. about a normal and a heavy motionless sphere is j)laced upon it and set in motion by the tangential impulse : prove that the ensuing path of the centre will be a prolate, a common, or a curtate cycloid, according as the initial point of contact is without, upon, or within the circle whose equation is 212" (x* + rf) = 2>ogx sin a, the axis of y being horizontal and the point where the axis of revolution meets the plane the origin. Also prove that, if the initial point of contact be the centre of this circle, the path will be a straight line. 450 DYNAMICS OF A RIGID BODY. 2GGG. A rough hollow cyliutler of revolution whose axis is vertical is made to' revolve with uniform angular velocity O about a fixed generator and a heavy uniform sphere i.s rolling on the concave surface : prove that the equation of motion is (f)'=^-^"-f'"--*^ where <^ is the angle which the common normal to the sphere and cylinder makes at a time t with the plane containing the fixed generator and the axis of the cylinder, and a + b,a are the radii of the cylinder and sphere respectively. 2GG7. A rough plane is made to revolve at a uniform rate Ci about a horizontal line in itself and a. sphere is set in motion U[)on it : deter- mine the motion, and prove that, if when the plane is horizontal the centre of the sphere is vertically above the axis of revolution and moving parallel to it, the contact will cease when the plane has turned through an angle $ given by the equation where a is the radius of the sphere. 2668. A uniform heavy rod is free to move about one end in a vertical plane which is itself constrained to revolve about a vertical axis through the fixed end at a imiform rate Q, and the greatest and least angles which the rod makes with the vertical during the motion are a, p : prove that aOr (cos a + cos ^) = 3^, where a is the length of the rod : also prove that, when 3^= 2af2^cos a, 27r the time of a small oscillation will be ----. — . il sin a 2669. Two heavy uniform rods of lengths 2a, 26 and masses A, B are freely jointed at a common end and are moveable about the other end oi A, and the rods fall from a horizontal position of instantaneous rest : prove that the radius of curvature of the initial path of the free end of B will be ^ah {A + Bf - [aA' + h{2A+ Bf]. 2670. A rigid body is in motion about its centi-e of inertia under no forces, and at a certain instant, when the instantaneous axis is the straight line whose equations are xJa{B-C) = zJC{A-B), y = 0, a point on the cylinder x' {A~B) + z' {B-G)+zx V^-iin^^K^^Z^) {C + A) = B (C-A) is suddenly fixed : prove that the new instantaneous axis will be perpen- dicular to the direction of the former. (The axes of co-ordinates are, as usual, the principal axes at the centre of inertia, and A, B, C the squares of the semi-axes of the principal ellipsoid of gyration.) DYNAMICS OF A RIGID BODY. 457 2G71. A number of concentric spherical sliells of equal indefinitely small thickness revolve about a common axis through the centre, each at a uniform rate 2)ro{)<)rtional to the ?i"' power of its radius ; the shells become suddenly rigidly united : prove that the subsequent angular velocity bears to the previous angular velocity of the outermost shell the ratio 5 : n + 5. 2672. An infinite number of concentric spherical shells of equal small thickness are revolving about diameters all m one [ilane with equal angular velocities, and the axis of revolution of the shell whose i-adius is r is inclined at an angle cos ' - to the axis of the outermost shell : prove a '■ that, when imited into a solid sphere, the axis of revolution will make an aiiijle tan"' -~~. with the former axis of the outer shell. ° lb 2G73. Prove that any possible given state of motion of a rigid straight rod may be represented by a single rotation about any one of an infinite number of axes lying in a certain plane. 2G74. A free rigid body is in motion about its centre of inertia when another point of the rigid body is suddenly fixed and the body then assumes a state of permanent rotation about an axis through that point : prove that the point must lie on a certain rectangular hyperbola. [With the notation of (26G9) the point to be fixed must satisfy the equations {B-C) J +{0-A) JL + ^A -B) -?- = 0, .dWj jDw^ Cw, (Jw.T + Bi^yv + C'o),^) (b^ - + C'^i ^ + A-B^] + {B-C){C-A){A-B) = 0, where w^ w^, w^ are the previous component angular velocities ; also the new axis of revolution must be parallel to the normal to the invariable plane of the previous motion.] 2675. A rigid body is in motion under the action of no foi'ces and its centre of inertia is at rest ; when the instantaneous axis is a certain given line of the body a point rigidly connected with the body is suddenly fixed, and the new instantaneous axis is parallel to one of the principal axes at the centre of inertia : prove that the point to be fixed must lie on a cei-tain hyperbola one asymptote of which is the given principal axis. 2676. A free rigid body is at a certain instant in a state of rotation about an axis through its centre of inertia when a given point of the body becomes suddenly fixed : determine the new instantaneous axis, and pi'ove that there are three directions of the former instantaneous axis for which the new axis will be in the same direction ; and these three directions are along conjugate diameters of the priucii»al ellipsoid of inertia. 458 DYNAMICS OF A RIGID BODY. 2G77. A rigid body is in motion under the action of no forces with its centre of inertia at rest and the instantaneous axis is describing a pUme in the body : prove tliat, if a point in that diameter of the ])i-incipal ellijjsoid of inertia wliicli is conjugate to this plane be suddenly fixed, the new instantaneous axis will be parallel to the former. 2G7S. Two equal uniform rods AB, BC, freely jointed at B and in one straight lino, are moving uniformly in a direction normal to their length on a smooth horizontal table when the poiut A becomes suddenly fixed : prove that the initial angular velocities of the rods will be in the ratio 3 : — 1, tliat the least subsequent obtuse angle between them will be cos"' (— ;i}), and that when next in one straight line their anerular velocities will be as 1 : 9. 2679. Three equal uniform rods AB, BC, CD, freely jointed at B and C, ai'e lying in one straight line on a smooth horizontal table when a blow is applied at their centre in a direction normal to the line of the rods : prove that ^^rTTit?^=0, where 6 is the angle through which the outer rods have turned in a time t and O their initial angular velocity. Pi-ove also that the velocity of BC will be ~'—t-\ 1 H — -, > , and that the direction of the stress at B or C will make with BC the angle tan"' (§ tan 6). 2680. Two equal uniform rods AB, BC, freely jointed at B, are in motion on a smooth horizontal table and their angular velocities are o)j, Wj when the angle between them is 6 : prove that (w^ + wj (5 — 3 cos B) and 5 (wj' + w/) — Gw^w^ cos are both constant throughout the motion. 2681. Three equal uniform rods (for all of which e = 0), freely jointed at common ends, are laid in one straight line on a siliooth horizontal table and the two outer are set in motion about the ends of the middle rod with equal angular velocities, (1) in the same sense, (2) in opposite senses : prove that, (1) when the outer rods make the gi'eatest angle with the direction of the middle rod produced on each side the common angular velocity of the three will be 4 to, and (2) that after the impact of the two outer rods the triangle formed by the three will move with velocity i aw, where a is the length of a rod. 2682. A uniform rod of length 2ft has attached to one end a particle by a string of length h and the rod and string placed in one straight line on a smooth horizontal table ; the particle is then projected at right angles to the string : prove that the greatest angle which the string can make with the rod (produced) will be ^^^-VmO-i). DYNAIMICS OF A RIGID BODY. 459 where m, p are the masses ; also that, if after a time t the rod and string make angles 0, with their initial directions, {k" + ah cos c^ - ^) -,- + {Ir + ah cos ^-B) -.- = (a + h) V, ('t etc where k' = ^ a" [ 4 + -- j and V is the initial velocity of the particle. 2G83. A circular disc capable of motion about a vertical axis through its centre normal to its [)lane is set in motion with angular velocity fl, and at a given point of it is placed fi'eely a rough uniform sphere : prove the equations of motion ^ dt ' o,'{r' + k'){b' + k') = k'n', cFr /Jd\' „ dd dt^ V /J6\' „ dd ^ r, 6 being the polar co-ordinates of the jDoint of contact at the time t, measured from the centre of the disc, co the angular velocity of the disc, h the initial value of r and 4//*^" = 7jjc^, where m, 2> are the masses of the sphere and disc and c the radius of the disc. [These equations are all satisfied by de_„ dt~''^'^~' k' + b , dO „ „ k'a T 2G84. A circular disc lies flat on a smooth horizontal table, on which it can move freely, and has wound round it a fine string carrying a particle which is projected with a velocity V from a point of the disc in a direction normal to the perimeter of the disc : prove that - ,~2 + 1 = sec y , (() = k tan , — u ; ale k k where 6, the angle through which the common normal has turned, and to the angular velocity of the sphere al)Out that normal, after a time ^.] 2G90. A sphere, radius a, is in motion on the surface of a cylinder of revolution of radius a + b whose axis makes an angle a with the vertical and is initially in contact with the lowest generator, its centre moving in a direction perpendicular to the genei'ator with such a velocity that the sphere just makes complete revolutions : prove the equations of motion '^'^y=:^sina(l7 + 10cos«^), doi dz d ^ dt^ Jt~di' 7 ( -7- ) + 2«-w" = 1 Ogz cos a ; z being the distance described by the centre of the sphere parallel to the axis of the cylinder, <^ the angle through which the common normal has turned, and w the angular velocity about the normal, after a time t. 2G91. A rough sphere of radius a rolls in a spherical bowl of radius a + b, the centre of the sphere being initially at the same height as the centre of the bowl and moving horizontally with velocity V : prove that, if 6 be the angle which the common normal makes with the vertical, and the angle through which the vertical plane con- taining the normal has turned at the end of a time t, . ,^d V /ddV 10.9 . P ,_ ''^^Tt = b' {dt)='7b'''^-V'^' and, if E, F, S he the reactions at the point of contact along the common normal, along tlie tangent which lies in the same vertical plane with the common normal, and at right angles to both these directions, that - +^"^cos0), F=-„''sin^, >y=0; b t J t also, if 0) , (i)„, W3 be the angular velocities of the sphere about these three directions, 7 (^^ (D,= 0, «(j)aSin = 5K, aw^ = -b-j-. R^mO 4(;2 DYNAMICS OF A RIGID BODY. 2692. A rough splierc of nidius a rolls in a si'lierical bo\vl of i-adius a + b in a stato of steady motion, the normal making an angle a with the vertical : prove that the time of small oscillations about this position is ^V 76 cos a 5^(1 + 3 COS' a) HYDEOSTATICS. [In the questions under this head, a fluid is supposed to be uniform, lieavy, and incompressible, unless otherwise stated : and all cones, cylinders, paraboloids, ttc. are supposed to be surfaces of revolution and their bases circles.] 2693. A cylinder is filled with equal volumes of n diflTerent fluids which do not mix ; the density of the uppermost is p, of the next 2p, and so on, that of the lowest being tip: prove that the mean pressures on the corresponding portions of the curve surfaces are in the ratios r : 2' : 3' : ... : n\ 2G94, A hollow cylinder containing a weight W of fluid is held so that its axis makes an angle a with the horizon : prove that the resultant pressure on its ciu-ve surface is IF cos a in a dii-ection making an angle a with the vertical. 2C95. Equal volumes of three fluids are mixed and the mixture separated into three parts; to each of these parts is then added its own volume of one of the original fluids, and the densities of the mixtures so foi-med are in the ratios 3:4:5: pi'ove that the densities of the fluids are as 1 : 2 : 3. 2G96. A thin tube in the form of an equilateral tnangle is filled with equal volumes of three fluids which do not mix and held with its plane vertical : prove that the straight lines joining the common surfaces of the fluids form an equilateral triangle whose sides are in fixed directions; and that, if the densities be in A. P., the straight line joining the surfaces of the fluid of mean density will be always verticaL 2G97. A thin tube in the form of a square is filled with equal volumes of four fluids which do not mix, whose densities are p^, p„, p^, p , and held with its plane vertical; straight lines are drawn joining adjacent points where two fluids meet so as to form another square : prove that, if p, + p^ = Pg + pg, the diagonals of this square will be vertical and horizontal respectively; but, if p^- p^ and p^ = p^, every position of the fluids will be one of equililnium. 464 HYDROSTATICS. 2698. A fino tul)e in tlie form of a rot,nilar pol^-gfon of n sides is fiUeil with equal volumes of n dirt'eieut fluids which do not mix and held with its ]»liuie vertical : prove that the sides of the polygon formed by joining adjacent points where two fluids meet will have its sides in fixed directions ; and, if the densities of the fluids satisfy two cei-tain conditions, every position will be one of equilibrium. [These conditions may be written p, cos a + p^ cos 2a + ... + p^ cos na = 0, Pi sin a + p^ sin 2t + . . . + p_ sin na = 0, wliei'e «a= 2- ] 2G99. A circular tube of fine uniform bore is half filled with equal volumes of four fluids which do not mix and whose densities are as 1:4 : 8 : 7, and held with its plane vertical : prove that the diameter joining the free surfaces will make an angle tan"' 2 with the vertical. 2700. A triangular lamina ABC, right-angled at (7, is attached to a string at A and rests with the side AC vertical and half its length immersed in fluid : prove that the density of the flviid is to that of the lamina as 8 : 7. 2701. A lamina in the form of an equilateral triangle, suspended freely from an angular point, rests with one side vertical and another side bisected by the sui-face of a fluid : prove that the density of the lamina is to that of the fluid as 15 : 16. 2702. A hollow cone, filled with fluid, is suspended freel}' from a point in the rim of the base : prove that the total pressures on the curve surface and on the base in the position of rest are in the ratio 1 + 11 sin^ a : 1 2 sin^ a, where 2a is the vertical angle of the cone. 2703. A tube of small bore, in the form of an ellipse, is half filled with equal volumes of two given fluids which do not mix : find the inclination of its axes to the vertical in order that the free surfaces of the fluids may be at the ends of the minor axis. 2704. A hemisphere is filled with fluid and the surface is divided by horizontal planes into n portions, on each of which the whole pressure is tlie same : prove that the depth of the r*^ of these planes is to the radius as Jr : Jn. 2705. A hemisphere is just filled with fluid and the surface is divided by horizontal planes into n portions, the whole pressures on which are in a geometrical progression of ratio k : prove that the dejith of the r'*" plane is to the radius as HYDROSTATICS. 4G5 2706. A lamina ABCD in the form of ^2 - 1 ; where a is the vertical angle. 2719. A cone with its axis vertical and vertex downwards is filled with two fluids which do not mix and their common surfece cuts off" one- fourth of the axis from the vertex: prove that, if the whole pressui'es of the fluids on the curve surfaces be equal, their densities will be as 45 : 1. 2720. A right cone just filled with fluid is attached to a fixed point by a fine extensible string attached to the vertex, and initially the string is of its natural length and the cone at rest : prove that the i)ressure of the fluid on the base of the cone in the lowest position is six times the weight of the fluid. 2721. A barometer stands at 29-88 inches and the thermometer is at the dew-})oiut ; a barometer and a cup of water are placed under a receiver from which the air is removed and the barometer then stands at -36 of an inch : find the space that would be occupied by a given volume of the atmosphere if it were deprived of its vapour without changing its pressure or temperature. 2722. In Hawksbee's air-pump, the machine is kept at rest when the n^^ stroke is half completed; find the difibrence of the tensions of the two piston rods. 2723. In Smeaton's aii'-pump, during the n^^ stroke, find the posi- tion of the piston at that instant of time when the upper valve begins to open. 2724. The volumes of the i-eceiver and barrel of an air-pump are A, B ; p, cr are the densities of atmospheric air and of the air in the receiver respectively, and 11 the atuiospheric pressure: prove that the work done in slowly raising the piston through one stroke is gravity being neglected. 2725. A portion of a cone cut ofi" by a plane through the axis and two planes perpendicular to the axis is immersed in fluid in such a manner that the axis of the cone is vertical and the vei-tex in the sur- face : prove that the resultant horizontal pressure on the curve surface passes through the c. G. of the body immersed. 30—2 468 HYDROSTATICS. 272G. Assuming that tho tompcriitiirc of the atmoi?phere in ascend- ing from the earth's surface decroasos slowly by an amount proportional to tlie height ascended, ])rove that tlie equation comiecting the pressure p and the density p at any height will be of the form p-kp^^'", where m is a small fraction. 2727. A cylinder floats in fluid with its axis inclined at an angle tan"' I to the vertical, its up])er circular boundary just out of the fluid and the lower one completely immersed: prove that the length of the axis is nine-eighths of the diameter of the generating circle. 2728. Two equal and similar rods AB, BC, fixed at an angle a at B, rest in a fluid of twice the specific gravity with the angle B out of the fluid, and the axis of the system makes an angle Q with the horizon: prove that cos 2^ = 2- sec a. 2729. A uniform solid tetrahedron has each edge equal to the opposite edge: prove that it can float partly immersed in fluid with any two opposite edges horizontal. 2730. A lamina in the form of a parabola bounded by a double ordinate rests in liquid with its plane vertical, its focus in the surface of the fluid, and its l)ase just out of the fluid: prove that the ratio of the densities of the solid and liquid is 1 : ( 1 + cos a)^, where a is the angle given by the equation 2 cos 2a = 3 (1 — cos a). 2731. A cone of density p floats with a generator vertical in a fluid of density o-, the base being just out of the fluid: prove that, if 2a be the vertical angle, ^ = (cos 2a)?, and that the length of the vertical side immersed is to the length of the axis as cos 2a : cos a. 2732. A cone is moveable about its vertex, which is fixed at a given distance c below the surface of a liquid, and rests with its axis, h in length, inclined at an angle Q to the vertical and its base completely out of the fluid : prove that cos B cos" a ah* ^ (cos*e-siira)^~PC*' 2a being the vertical angle and p, cr the densities of the liquid and cone. Also prove that this position will be stable, but that it cannot exist unless ah* COS" a > pc*. 2733. A homogeneous solid in the form of a cone rests with its axis vertical and its vertex at a depth c below the surface of a liquid whose density varies as the depth : prove that the condition for stable vj , where h is the length of the axis and 2a the vertical angle. Prove also that this is the condition that no positions of equilibrium in which the axis is not vertical can exist. HYDROSTATICS. 469 2734. An elliptic tube half full of lirpiitl revolves about a fixed vertical axis in its plane with angular velocity w : ])rove that the angle •which the straight line joining the free surfaces of the fluid makes with the vertical will be tan~'(-^, ), where » is the distance of the axis from the centre of the ellipse. 2735. A hollow cone very nearly filled with liquid revolves uniformly about a vertical generator : prove that the pressure on the base is •| W \ — (1+5 cos^ a) tan a + 8 sin a J- ; where W is the weight of the fluid, 2a the vertical angle, a the radius of the base, and o> the angular velocity. 2736. A hollow cone very nearly filled with liquid revolves about a horizontal generator with uniform angular velocity o) : prove that the whole pressure on the base in its highest or lowest position is o paV ( 1 + - , cos a + 5 cos* a ) ; o \ aw / where a is the radius of the base and 2a the vertical angle. 2737. A cone the length of whose axis is 7i and the radius of the base a floats in liquid with -^-Jj of its volume below the surface : prove that, when the liquid revolves about the axis of the cone with angular velocity * /flf"^, the cone will float with the length h or |/i of its axis immersed ; and investigate which of the two positions is stable. 2738. A sphere of radius a floats in liquid, which is revolving with uniform angular velocity w about a vertical axis, with its centre at the vertex of the free surface of the liquid : prove that 4 (p* + W) (a -pq) = a{p + iaqY; whei-e po}'' = 2g and 1 + (7 : 2 is the ratio of the densities of the sphere and liquid. 2739. A hollow paraboloid whose axis is equal to the latus rectum is placed with its axis vertical and vertex ujiwards and contains seven- eighths of its volume of liquid : find the angular velocity with which this liquid must revolve about the axis in order that its free surface may be confocal with the i)araboloid ; and prove that in this case the pressure on the base is gi-eater than when the lit^uid was at rest in the ratio 2^2 : 2^2-1. 2740. A liquid is acted on by two central forces, each vaiying as the distance from a fixed point and equal at equal distances from those points, one attractive and one repulsive : prove that the surfaces of equal pressure are planes. 470 HYDROSTATICS. 21 \\. A liquid is at. rest under tho action of two forces tending to two fixed points juid each varying inversely as the square of the distance, one attractive and one repulsive : prove that one surface of equal pressure is a sphere. 2742. A mass of elastic fluid is confined within a hollow sj)liere and rei)elk'd from tlie centre of the sjdiere by a force /u. -f- distance : ])rovo that the whole ])ressure on the sphere beai's to the whole pressure wliich would be exerted if no such force acted the ratio 3k + fj. : 3/i; ; where p = kp is the relation between the pressure and density. 274.3. A quantity of liquid not acted on by gravity just fills a liollow sphere and is re])el]ed from a point on the suiface of the sphere by a force equal to /x (distance) ; the liquid revolves aboxit the diameter tiii'ough the centre of force with uniform angular velocity w : find the whole pressure on the sphere, and prove that, if when the angular velocity is diminished one half the pressure is also diminished one half, (xf = 6/x. 2744. All s])ace being supposed filled with an elaf^tic fluid the total mass of which is known, which is attracted to a given point by a force varying as the distance ; find the pressure at any point. 2745. Water is contained in a vessel ha^dng a horizontal base and a cone is supported partly by the water and partly by the base on which the vertex rests : prove that, for stable equilibrium, the depth of the fluid must be greater than h J ni cos a, m^ being the S])ecific graA'ity of the cone, h the length of its axis, and 2a the vertical angle. 2746. A solid paraboloid is divided into two parts by a plane thi'ough the axis and the parts united by a hinge at the vertex ; the system is placed in liquid with its axis vertical and vertex downwards and floats without separation of the jiarts : ])rove tliat the ratio of the density of the solid to that of the liquid must be greater than x*, where X is given by the equation ?>hx^^ll{\-x), and I, h are the lengths of the latus rectum and axis respectively. 2747. A cone is floating with its axis vertical in a fluid whose density varies as the depth : prove that, for stable equilibrium, COS' a < i ' ^ :/] where 2a is the vertical angle, p the density of the cone, and a the density of the fluid at a depth equal to the height of the cone. 2748. A uniform rod rests in an oblique position with half its length immei-sed in liquid and can turn freely about a point in its length whose distance from the lower end is one-sixth of the length : compare the densities of the rod and liquid, and prove that the equilibrium is stable. HYDUOSTATICS. 471 2749. A uniform rod is moveable about one; end, which is fixed below the surface of a liquid and, when slightly displaced from its highest position, it sinks until just immersed : prove that, when at rest in the highest position, the pressure on the point of suppoi-t was zero. 2750. Two equal uniform rods AJi, EC, freely jointed at B, are cai)able of motion about A, which is fixed at a given dej)th below the surface of a liquid : find the position in which both rods rest partly immersed, and prove that for such a position to be possible the density of the rods must not exceed one-third the density of the liquid. 2751. A hemisphere, a point in the rim of whose base is attached to a fixed point hy a fine string, rests with the centre of the sphere in the surface of the liquid and the base inclined at an angle a to the horizon : prove that p 1 6 (tt — a) cos a — Stt sin a o- 27r (8 cos a - 3 sin a) ' where p, o- are the densities of the solid and liquid. 2752. A cone is floating with its axis vertical and vertex down- wards in fluid and - th of its axis is immersed : a weight equal to the weight of the cone is placed upon the base and the cone then sinks until just totally immersed before rising : prove that 11 + n" + n= 7. 2753. A hollow cylinder with its axis vertical contains liquid, and a solid in the form of an ellipsoid of revolution is allowed to sink freely in the liquid with axis also vertical : the solid just fits into the cylinder and sinks until just immersed before rising : prove that its density is one-half that of the liquid. 2754-. A hollow cylinder with its axis vertical contains liquid and a solid cylinder is allowed to sink freely in it with axis also vertical : prove that, if it sink until just immersed before rising, the densities of the solid and liquid must be in the ratio 1 : 2. Also, if the density of the liquid initially vary as the depth, prove that the density of the solid must be the initial density of the liquid at a depth of one-sixth of the whole distance sunk by the solid. 2755. A hollow cylinder with vertical axis contains a quantity of liquid and a solid of revolution (of the curve i/ oc x" about the vertical axis of x) is allowed to sink in the lifjuid, starting when its vertex is iu the surface and coming to instantaneous rest when just immersed : prove that the density of the solid must bear to the density of the liquid the ratio 1 : 2 (n + 2); and that, if a similar solid be allowed to sink in an unlimited mass of liquid of half the density of the former, this solid will also come to rest when just immersed. 2756. A cylinder whose axis is vertical contains a qiumtity of fluid ■whose density varies as the depth and into this is allowed to sink a solid of revolution whose base is equal to that of the cylinder, which sinks until just immersed befoi'e rising j iu the lowest position of this solid 472 HYDROSTATICS. the density of the suiTOundiiig fluid varies as the n^^ power of the depth : jirove tliat the weight of the solid is to the weiglit of the displaced Ihnd as ?i— 1 : 3 (2vi+ 1), whereas if the solid can rest in this position the ratio xnnst be yi - 1 : n + 1. Also prove that the generating curve of the solid will be where a is the radius of the base and h the height. [If ii = 2 the solid is a paraboloid, if ?i = 3, an ellipsoid.] 2757. A hollow cylinder with vertical axis contains a quantity of fluid whose density varies as the depth and into it is allowed to sink slowly, with vertex downwards, a solid cone the radius of whose base is equal to the radius of the cylinder ; the cone rests when just immersed : prove that the density of the cone is equal to the initial density of the fluid at a depth equal to one-twelfth of the length of the axis of the cone. If the cone be allowed to sink freely into the fluid, starting with its vertex at the surface and just sinking until totally immersed, the density of the cone will be to the density of the fluid at the vertex of the cone in its lowest j^osition as 1 : 30. 2758. A tube of fine bore whose plane is vertical contains a quantity of fluid which occupies a given length of the tube ; a given heavy particle just fitting the tube is let fall through a given vertical height : find the impulsive pressure at any point of the fluid ; and prove that the whole kinetic energy after the impact bears to the kinetic energy dissipated the I'atio of the mass of the particle to the mass of the fluid. [If «i, m' be the masses of the particle and fluid, V the velocity of the particle just before impact, the impulsive pressure at a point whose distance along the arc from the free end is s will be , -, , where I is ° 7)1 + m C the whole length of arc occupied by the fluid.] 2759. A flexible inextensible envelope vihen filled with fluid has the form of a paraboloid whose axis is vertical and vertex downwards and whose altitude is five-eighths of the latus rectum : prove that the tension of the envelope along the meridian will be gi-eatest at points where the tangent makes an angle -r with the vertical. [In general, if 4a be the latus rectum and h the altitude, the tension per unit of length at a ])oint whei'e the tangent makes an angle 6 with the vei-tical will be -tt ( "^ -"Ti ^^r^ I . where o- is the specific gravity 2 V sm 6 sin^ Oj l & j of the fluid.] 2760. Fluid without weight is contained in a thin flexible envelope in the form of a surface of revolution and the tensions of the envelope at any point along and perpendicular to the meridian are equal : pro^ e that the surface is a sphere. HYDROSTATICS. 473 2761. A quantity of liomoi;eneous fluid is contained between two parallel planes and is in equilibrium in the form of a cylinder of radius h under a pressure -ur ; that portion of the fluid which lies within a distance a of the axis being suddenly annihilated, prove that the initial pressure, at a point whose distance from the axis is r, is w log I - ) -^ log I I . 2762. A thin hollow cylinder of length h, closed at one end and fitted with an air-tight piston, is placed mouth downwards in fluid; the weight of the piston is equal to that of the cylinder, the height of a cylinder of equal weight and radius formed of the fluid is a, the height of fluid which measuies the atmospheric pressvii'e is c, and the air enclosed in the cylinder would just till it at atmospheric density : prove that, for small vertical oscillations, the distances of the piNton and of the top of the cylinder from their respective positions of equilibrium are of the foi-m ^-1 sin (Xt + a) + B sin {fit + (i), X, /u. being the positive roots of the equation x*-^(2w+l)x- + m^ = 0, and m={a-\- c)^ -^ ch. 2763. A filament of liquid PQR is in motion in a fixed tube of small uniform bore which lies in a vertical plane with its concavity always upwards ; on the horizontal oi'dinates to P, Q, li at any instant are taken points p, q, r, whose distances from the vertical axis of abscissae are equal to the arcs measured to P, Q, R from a fixed point of the tube : prove that the fluid pressure at Q is always proportional to the area of the triangle pqr. 2764. A centre of force, attracting inversely as the square of the distance, is at the centre of a spherical cavity witliin an infinite mass of li(|uid, the pressure in which at an infinite distance is ■u:, and is such that the work done by this pressure on a unit of area thi-ough a unit of length is one half the woi'k done by the attractive force on a particle whose mass is that of a unit of volume of the liquid as it moves from infinity to the initial boundary of the cavity : prove that the time of filling up the cavity will be Tra ./ — {2 — (y)*}; where a is the initial radius of the cavity and p the density of the fluid. GEOMETEICAL OPTICS. 27G5. Three plane mirrors are placed so that their intersections are parallel to each otiier and the section niade by a plane perpendicular to their intersections is an acute-angled triangle; a ray proceeding from a certain jioint of this plane after one reflexion at each mirror proceeds on its oi'iginal course : prove that the point must lie on the perimeter of a certain triangle. 2766. In the last question a ray starting from any point after one reflexion at each mirror proceeds in a direction parallel to its original direction: pi'ove that after another reflexion at each mirror it mil proceed on its original path, and that the whole length of its path between the first and third reflexions at any mirror is constant and equal to twice the peiimeter of the triangle formed by joining the feet of the perpendiculars. 2767. A ray of light whose direction touches a conicoid is reflected at any confocal conicoid: prove that the reflected ray also touches the first conicoid. 276S. In a hollow ellipsoidal shell small jiolished grooves are made coinciding with one series of circular sections and a bright point placed at one of the umbilics in which the series terminates: prove that the locus of the bright points seen by an eye in the opposite umljilic is a central section of the ellipsoid, and that the whole length of the path of any ray by which a bright point is seen is constant. 2769. A ray proceeding from a po'nt on the circumference of a circle is reflected n times at the circle : prove that its point of inter- section with the consecutive ray similarly reflected is at a distance from the centre equal to -rjl + 4:n{7i+\)&ux'd, where a is the radius and 6 the angle of incidence of the ray: also prove that the caustic surface generated by such rays is the surface of revolution generated by aii epicycloid in which the fixed circle has the radius ^^ — '—, and the 2n -(- 1 moving circle the radius 2n + 1 " 2770. A ray of light is reflected at two plane min'ors, its direction before incidence being })arallel to the plane bisecting the angle between the mirrors and making an angle with their line of intersection: prove that the deviation is 2 sin~' (sin ^ sin 2a), where 2a is the angle between GEOMETRICAL OPTICS. 475 the miri-ors. More generally, if />_. be the deviation after r successive reflexions, cos % -Oj_^_, = sin d sin [^In —I a — (f>), sin ^ Z)„_^ = sin 6 sin 2na, where (f> is the angle which a plane through the intersection of the mirrors parallel to the incident ray makes with the [)lane bisecting the angle between the mirrors. 2771. Two prisms of equal refracting angles are placed with one face of each in contact and their other faces pai*allel and a ray ])asse3 through the combination in a principal plane: prove that the deviation will be from the edge of the denser prism. 2772. The radii of the bounding surfaces of a lens are r, s, and its thickness is ( 1 + - ) (s — ?•) : prove that all i-ays incident on the lens from a certain point will pass tln-ough without aberration but also without deviation. 2773. Prove that a concave lens can be constnxcted such that the path of every ray of a pencil proceeding from a certain point after refraction at the first surface shall pass through the centre of the lens; that in this case there will be no aberration at the second refraction, and that the only effect of the lens is to throw back the origin of light a distance (fx — l)t, where t is the thickness of the lens. 2774. "What will be the centre of a lens whose bounding surfaces are confocal paraboloids on a common axis? Prove that the distance between the focal centres of such a lens is — r (a + h), ia, ib being the latera recta. 2775. The path of a ray through a medium of variable density is an arc of a cii'cle in the plane of xi/: pi-ove tliat the refractive index at any point (x, y) must be f { ) , where f is an arbitrary X -a \!J -uj function and {a, b) the centre of the circle. 2776. A ray of light is propagated through a medium of variable density in a plane which divides the medium symmetrically : prove that the path is such that when described bv a point with velocity always proportional to /u,, the index of refraction, the accelerations of the point parallel to the (rectangular) axes of x and y will be propoitional to -i^^' '^Irrespectively. 2777. A ray is propagated through a medium of variable density in a plane (:cy) which divides the medium symmetrically: prove that the projection of the radius of curv:itiire at any j)oiut of the path of the ray on the normal to the surface of erpuil density through the [loint ia equal to /x : J {^f^ -f {^[^ 476 GEOMETRICVX OPTICS. 2778. A small pencil of parallel rays of white light, after trans- mission in a principal jtlane tlirouyli a prism, is received on a screen whose )>lane is per])endicular to the direction of the pencil : pi'ove that the length of the spectrum will be jtroportional to {fx^ — fx.) sin i -^ cos* D cos (D + i — (f>) cos (j>' ; where i is the refracting angle, , ' the angles of incidence and re- fraction at the first surface, and D the deviation, of the mean ray, 2779. Prove that, when a ray of white light is refracted through a prism in a })rincipal plane so that the dispersion of two given colours is a minimum, sin(:5<^'-2(')_ 2 _ sin c^' ju." where „ I cos —^ cos - I 1 cos - - cos 1 - cos ^ = 2 a / b — c b + c cos - ( cos —J — h cos — suffice to determine b^c when a, E and & =f c are given.] 2792. The sum of the sides of a spherical triangle being given, prove that the area is greatest when the triangle is equilateral. 2793. In a spherical triangle ABC, a + b + c = tt, prove that B C cos A + cos B + cos C = 1 , cos a = tan ^ tan — , ^ 2 .A EC. and that 279-4. In a spherical triangle A + B -¥ C =27r : prove that b c cos A + cot - cot ^ = 0, tkc. 2795. In a spherical triangle ABC, A =B + C : prove that . o a . ob . „ c sm' ^ = sin" - + sin ^ . Ji Ji 'Ji 2796. The pole of the small circle circumscribing a sphei-ical triangle ABC is : prove that . ,6 . „c . ,rt ^ . i . c BOC sin" - + sua - - sm" - = 2 sin - sm — cos — _— ; and that, if P be any point on this circle, . a . PA . b . PB . c . PC ^ sm T. sm -— + sm - sm + sm - sm ^ — 0, that arc of the three PA, PB, PC being reckoned negative which cro.sses one of the sides. 2797. Prove that sin s > cos a sin (s - a) + cos b sin (s — b) + cos c sin (5 — c), an d cos *S' < cos A cos {S -A) + cos B cos (»S^- B) + cos C cos {S — C), where a, b, c are the sides and A, B, C the angles of a spherical triangle, and 2s=a + b + c, 2S=A+B+C. 2798. The centre of the sphere on which lies a spherical triangle ABC is 0, and forces act along OA, OB, DC proportional to sin«, sin 6, sin c respectively : prove that their resultant acts through the pole of the circle ABC. SPHERICAL TKTGONOMETRY AND ASTRONOMY. 479 2799. The great circle drawn througli a cnnicr of a spherical triangle perpendicular to the o])))osite side divid(!S the angle into ])arts whose cosines are as the cotangents of the adjacent sides, and divides the opposite side into parts whose sines are as the cotangents of the adjacent angles. 2800. Prove that a spherical triangle can be equal and similar to its i)olar triangle only when coincident with it, each side being a quadrant. 2801. In a spherical triangle A +« = 7r : prove that tan (J - ^ tan g - - - tan ( J - g = - tan (^ - ^ tan (^ - ^) . 2802. Prove the formula a+ h + c cos A + cos i> + cos C - 1 cos , . A . B . G 4 sin ^ sm ^^ sm ^- 280.3. Two sidef5 of a spherical trimgle ai'e given in position, and the included angle is equal to the spherical excess : prove that the middle point of the third side is fixed. 2804. Two sides of a Sjdierical triangle are given in position, including an angle 2a, and the spherical excess is 2/8 ; on the great circle bisecting the given angle are taken two points S, S', such that cos SA = - cos S'A = tan (a — jS) cot a : prove that, if F be the middle point of the base, sin a sin l (.S7' + ,S"7^) = sin (a - fi). 280.5. Two fixed points A, B ai'e taken on a sphere, and P is any point on a fixed small circle of which A is pole ; the great circle PB meets the great circle of which A is pole in Q : prove that the ratio cos PQ : cos BQ will be constant. 280G. Prove that, when the Sun rises in the N.E. at a place in latitude I, the hour angle at sunrise is cot"' (— sin^). 2807. Tn latitude i.")" the observed time of transit of a star in the equator is unaffected by the combined errors of level and of devia tion in a transit : prove that these errors must be very nearly equal to each othei'. 2808. The ratio of the radius of the Earth's oi-bit to that of an inferior planet is m : \, and the ratio of their motions in longitude (considered uniform) is n : \: prove that the elongation of the planet as seen from the Earth when the planet is stationary is tan" V 7, 4S0 SPUERICAL TRIGONOMETRY AND ASTRONOMY. 2809. The mean motions in longitudt^ of tlio Earth and of an inferior planet are m, m', and the difit'ience of their longitudes is ; prove that the planet's geocenti-ic longitude is increasing at the rate / nh I \ t\\ ininiy — \ni^ — (mm'r + *'^'''} cos d> m^ — 2 {mni'Y cos ^ + w'' and verify that the mean value of this during a synodic period is m. 2810. The maximum value of the aberration in declination of a given, stai' is 20"-5 J\- (cos 8 cos w 4- sin 8 sin w cos ay ; where a, S ai'e the right ascension and declination of the star, and to the obliquity of the ecliptic. 2811. Prove that all stars whose aberration in right ascension is a maximum at the same time that the aberration in declination vanishes lie either on a quadric cone whose circular sections are parallel to the ecliptic and equator, or on the solstitial colure. 2812. The right ascensions and declinations of two stars are a, a \ 8, 8' respectively, and A is the Sun's right ascension at a time when the aberrations in declination of both stars vanish : prove that tan 8 sin a — tan 8' sin a tan A = tan 8 cos a — tan 8' cos a ' 2813. In the Heliostat, if the diurnal change of the Sun's declina- tion be neglected, the normal to the mirror, and the intersection of the plane of the mirror with the plane of reflexion will each trace out a quadric cone whose circular sections are perpendicular to the axis of the Earth and to the reflected ray. 2814. The latitude of a place has been determined by observation of two zenith distances of the Sun and the time between them and each observed distance was too great by the same small quantity Az : prove that the consequent error in the latitude is Az cos (a + a') -^ cos (a — a') ; where 2a, 2a' are the azimuths at the times of observation. 2815. The hour angle is determined by observation of two zenith distances of a known star and the time between ; each observed zenith distance is too great by Az : prove that the consequent error in hour angle is Az sin (a + a') -f- cos I cos (a — a') ; where I is the latitude of the place and 2a, 2a! the azimuths of the star at the two observations. [See a paper by Mr Walton, Quarterly Journal, Vol. v., page 289.] THE END. CAMBRIDGE : PRINTED BY C. J. CLAY, M.A. AT THE UNIVERSITY PRESS. UNIVERSITY OP CALIFORNIA LIHKARY 13 "'°^»»""»»°"° below, or Rene-wed^DOO^^ RECD FEB 181 ^pff.i§5"n-'' U-C. Bm^EY '■'BRARi Sut'VC)