r,u .'irii\;(i, m m mm mm Cornell University Library The original of tliis book is in tine Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924031252020 Cornell University Library arV17952 A book of mathematical problems on sube 3 1924 031 252 020 olin,anx MATHEMATICAL WORKS PUBLISHED BY MACMILLAN AND CO. A Treatise on Solid Geometry. By the Rev. Pbr- civAL Frost, M.A., and the Rev. J. Wolstbnholmb, M.A. 8to. 18s. A Collection of Mathematical Problems and Exam- ples. By H. A. MoKQAN, M.A. Crown 8vo. 6s. 6d. Cambridge Course of Elementary Natural Philoso- phy, for the Degree of B.A. Originally compiled by J. C. Snowball, M.A. Fifth Edition, revised and enlarged, and adapted for the Middle Class Examinations, by Thomas Lund, B.D. Crown 8vo. 5s. 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Crowil 8vo. 10s. 6d. A Treatise on Conic Sections. With Examples. Fourth and Cheaper Edition, revised and enlarged. Crown 8vo. 7«. Sd. Algebra. Fourth Edition. Crown 8vo. 7s. 6d. Plane Trigonometry. Third Edition. Crown 8vo. 5s. A Treatise on Spherical Trigonometry. Second Edition, enlarged. Crown 8vo. 4s. 6d. Examples of Analytical Geometry of Three Dimen- sions. Second Edition, revised. Crown 8vo. 4s. An Elementary Treatise on the Theory of Equa- tions. Second Edition, revised. Crown 8vo. 7s. 6d. MACMILLAN AND CO. LOKDDN. A BOOK OF MATHEMATICAL PROBLEMS. PRINTED BY C. J. CLAY, M.A. AT THE UNrVEBSITT PRESS. A BOOK OF MATHEMATICAL PROBLEMS ON SUBJECTS INCLUDED IN THE CAMBRIDGE COURSE. DEVISED AND AEEANGED BY JOSEPH WOLSTENHOLME, FELLOW OF CHEIST'S COLLEQB; SOMETIME FELLOW OF ST JOHN'S COLLEGE; AND LATELY LEOTUBEB IN MATHEMATICS AT CHKI3T"B COLLEGE. " Deduct but what is Vanity or Dress, *' Or Learning's Luxuiy, or Idleness ; " Or tricks to sliew the stretch of human brain, " Mere curious pleasure, or ingenious pain ; * Then see how little the remaining sum." EonDon ant) d^ambiiDge: MACMILLAN AND CO. 1867. Y [All rights resei-ved.] PREFACE. This "Book of Mathematical Problems" consists, mainly, of questions 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 problems, 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 diflBculty. 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 infoimation as to those which still remain. I have, in some cases, where I thought I had anything serviceable to communicate, prefixed to .certain classes of problems fragmentary notes on the mathematical subjects VI PREFACE. 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. I have to express my best thanks to Mr B. Morton, Fellow of Christ's College, for his great kindness in reading over the proof sheets of this work, and correcting such errors as were thereby discoverable. JOSEPH WOLSTENHOLME. Wastdalk Eeap, July 31. CONTENTS. PROBLEMS PAGE 1—73, GEOMETRY (EucUd) 1 ALGEBRA. 74 — 77. I. Highest Common Divisor 11 78. II. Equations 12 79—88. III. Theory of Quadratic Equations 15 89—97. IV. Theory of Divisors 17 98—119. V. Identities and EquaKties 18 120—133. VI. Inequalities 23 ISi — 140. VII. Proportion, Variation, Scales of Notation 26 141 — ^155. VIII. Arithmetical, Geometrical, and Harmonical Pro- gressions 27 156 — 164. IX. Permutations and Combinations 29 165—175. X. Binomial Theorem 31 176 — 185. XI. Exponential and Logarithmic Series 33 186—192. XII. Summation of Series 36 193—200. XIIL Recurring Series 40 201—221. XIV. Convergent Fractions 42 222—231. XV. Properties of Numbers 50 232—251. XVL Probabilities 51 252—266. XVIL Miscellaneous 55 Till CONTENTS. PLANE TEIGONOMETRY, PKOBLEMa 267—278. I. 279—321. II. 322—329. III. 330—366. IV. 367-379. V. 380—397. VI. 398-400. VII. 401—410. VIII. FAQE 58 Equationa Identities and Equalities 61 Inequalities 69 Properties of Triangles 71 Heiglits and Distances, Polygons 77 Expansions of Trigonometrical Fractions 80 Series 82 Miscellaneoua Questions 86 CONIC SECTIONS, GEOMETRICAL. 411—444. I. Parabola 445—606. II. Central Conies 607 — 529. III. Rectangular Hyperbola . 92 530-548. -* S r I. 549— S88. ■589— j644. 645— JS57. 1 8 1 IL III. .658—675. V. 676—728. VI. 729-741. vn. 742— S05. VIIL 806—822. IX. .■823— .882. X. 883—922. 923—1012 D] 1013—1059 CONIC SECTIONS, ANALYTICAL. Straight Line, Linear Transformation, Circle'.. Parabola, referred to its axis EUipse, referred to its axes Hyperbola, referred to its axes, or asymptotes , Polar Coordinates General Equation of the Second Degree Envelopes Areal Coordinates Anharmonic Properties Reciprocal Polars, Projections THEORY OF EQUATIONS... DIFFERENTIAL CALCULUS , INTEGRAL CALCULUS..., 103 108 115 130 133 136 149 152 168 172 187 195 211 CONTENTS. IX SOLID GEOMETRY. PROBLEMS PAGB 1060—1088. I. Straight Line and Plane 221 1089 — ^1123. II. Linear Transformation. General Equation of the Second Degree 227 1124 — 1161. III. Conicoids referred to their Axea 235 1162—1177. rV. Tetrahedral Coordinates 242 1178—1187. V. Focal Curves, Eeciprocal Polars 245 1188 — 1193. VI. General Functional and Differential Equations ... 2i7 1194—1200. Vn. Envelopes 248 1201—1210. VIIL Curvature 249 STATICS. 1211 — 1233. I. Composition and Resolution of Forces 253 1234—1250. n. Centre of Gravity 257 1251 — 1254. III. Smooth Bodies under forces in one plane 260 1255—1267. IV". Friction 261 1268—1275. V. Elastic Strings 264 1276—1285. VL Catenaries 266 DYNAMICS, ELEMENTARY. 1286—1800. I. Rectilinear Motion, Impulses 269 1801-1315. IL Parabolic Motion 271 1316 — 1326, III. Motion on a Smooth Curve under the action of Gravity 274 1327—1354. NEWTON 277 DYNAMICS OF A POINT. 1355 — 1385. I. Rectilinear Motion, Kinematics 282 1386—1405. n. Central Forces 289 1406 — 1439. III. Constrained motion on Curves and Surfaces. Par- ticles joined by Strings 293 1440—1447. IV. Motion of Uniform Strings 301 1448—1462. V. Resisting Medium, Hodograph 303 X CONTENTS. DYNAMICS OF A KIGID BODY. PROBLEMS PAGE. 1463 — 1472. I. Moments of Inertia, Principal Axes 306 1473—1491. II. Motion about a Fixed Axis 307 1492—1510. III. Motion in Two Dimenmons 311 1511— 3S43. IV. Miscellaneous 316 1544—1594. HYDKOSTATICS 327 1595—1612. GEOMETRICAL OPTICS 338 1613—1628. SPHEBICAL TRIGONOMETRY AND PLANE ASTRONOMY 342 EEKATA. Paqe Peoblem 13 78 (17) /or - Tead+, 15 80 for Oj + 63 read a, + a^. 26 133 for positive read real. 31 167 in the last line for a^ read a, , and for a^ read o, . 33 176 (3) /oj-[m- re- 2" read! [« + «- 2". 39 188 in the first member of the equation for ra""""' read r3.2"-', and in the second member for r2"+^ read r3.2". — 191 for (m+n + l) (m + n-1) read (m + n-l) (m+re-2). 40 192 (2) for [3 read [2, and for [5 read [4. 41 194 (6) for integral part of read integer next greater than. , . „„„ MS ^ 1 1 1 1 ^1111 44 206 (1) for- r - - reader - . - t+.... ^ ' '' c+b + a+c+... b + a + o+b 45 211 for is a, read is , . ' 1 - a, 46 212 (5) for 6 read 3. ... (6) /or (2« - 1) ic reat? (2ji - 1) a', and for ( - a;)" read ( - l)''a:2«+i_ 47 213 (8) for [a^ - If read (a' - 1)». 53 245 for {e + d)^ read (c + d)^- c\ an noA ^ b-krO ,a+d , , d-a jb-c 62 284 for ; read j , and for — =- read -, . •^ a-d c-b -^ c+b a+d 290 for a-x read a+x. 65 301 in the result for sin read cos throughout. 304 /or a + g, jS+g, rea a;V24a; + 55 ' ^P> 4a;*+ 11*^+ 81 ' ■ a;° + nx''-54 a:°-209a;+ 5G ^" a;^ + lla; + 12' W 56a;'- 209a;V 1 ' 8a:'-377a:°+21 16a;»- a;°+ 16cc^ + 32 ^'' 21a;'-377^V.8' ^^^ 3lV + 16a;=-a;^ + 16 ' a;''+2a;V3a;''-2a;Vl (V) 6x'' + a;'+17a;'-7a;"-2' ,.v 1 +«:' ^ (5 + cxY +{a + bxf a + 2bx+cx' a{b + cxy-2b{a + bx){b+cx) + c{a+bxy' 75. Simplify the expressions x{l-y')(\-z')+y(l-z')0-x') + z(l~af)(l-p')-Ar.yz x + y + z — xyz ' a'{b'- c') + ¥ (c' - a') + c" (a" - V) (a) (y) a- {b-c) + b^ {c-a) + (? {a~b) ' 1 1 {a-b){a-o){a-d) {b -c){b- d) {b-a) 1 1 (S) {c~d){c~a)(c-b) (d-a){d-b){d-cy bed cda (a-b){a-c){a-d) {b- c){b-d){b-a) dab abc ■^ {c-d){c-a){o-b) "^ {d-a){d-U){d~^c) ' . {a + b){a + c) „ (& + c){b + a) {c+a)(c + b) ^' {a-b){a-c) (b-c){b-a) {c-a){c-b)- 12 BOOK OF MATHEMATICAL PEOBLEMS. 76. Prove that (ab + cd) (a" + P-c'- d') - {ao + bd) {a? -V+c^- d') {(f + b'-c'- d') {a' -b^ + c'-d') + i {ab + cd) {ac + bd) _ {b-c){a + d) ~ {a + df-ib-cf 77. Prove that (y-z) (1 +y'){\+z') + {z-x){l+z'){l +oi?) + {x-y) (1 +«=') (1 +;/°) x{y - «)(1 + /)(1 + z") + ?/(«:- a;) (1 + »^)(1 + a?) + zioa - y)(l + 3»)().+f) _ \ — yz — zx — !cy ~ x + y + z — xyz II. Equations. 78. Solve the equations (1) {x+ 1) (03 + 2) (x + 3)= (« + 4) {x + 5) (k- 3), (2) {x + V) (a;+ 2) («;+ 3) = (x-\) (»- 2)(a;-3) + 3 (4a;- l)(a; + l), (3) {x + a){x + a +b) = {x + b){x + 3a), /.N 1 7 5 3 a; + l a; + 5 a; + 3 a3 + 7' /KN 1 2 6a; + a?/= , ax' by' -^ c s' (40) x + = «+ =«+ ='/rcw2 ys: ^ aa; a;y ^ v «Jy«> (41) a; + jr + « = ma^jw, -j 2/8 + «a; + a;?/ = w, I (l + .x=)(l+y')(l + ^)=(l-n)v[ (42) 2x + ^' = 22/ + -==2« + ^' = a. V ALGEBRA. 15 III. Tlieory of Quadratic Equations. 79. In the equation a h c d :+- + -+ = 0, x — md x — mc x + mb x + ma prove that, if a + 6 + c + cZ = 0, the only finite value of x will be m {ac + hd) a + b 80. In the equation ''^l <^o C', "i x+b, ic + Oj, x + b^ x + b^ ' prove that, if '*i + ^2 + «3+«4 = 0> and afi^ + aj>^ + a^b^ + aj)t = 0, the only finite value of x will be 81 . Pind limits to the real values of x and y which can satisfy the equation x'+ 12a;y+42/'' + 4a; + 8y + 20 = 0. 82. If the roots of the equation ax' + 2bx + c = be possible and different, the roots of the equation (a + c) (ax' +2bx+c)-2 {ac - ¥) (x" + I) will be impossible, and vice versd. 83. Prove that the equations x + y + z = a + b + o, X y e ^ a c a'^b'^c'- ' .111 are equivalent only to two independent equations, if - + v- + -= 0. a c 16 BOOK OF MATHEMATICAl. PROBLEMS. 84. Obtain the several equations for determining o, /3, y so that the equations 03* + 1f^ + Cp? + ra + S = 0, (33^+^03 + a)" = (fix + y)' may coincide : and in this manner solve the equation 85. The roots of the equation (a; + a — c) (a; + 6 + c) (a3 + a — £?) (a; + 6 + (Z) = e will all be real, if 86. If a;,, x^ be the roots of the equation + 6+^ = 0, (1 — »i)(l — a;) Tnx then will 7= r-pi -+0 + = 0. (l-a;,)(l-a;^) x^x^ 87. If iji z be the roots of the equation (l-m^(l-a;°) 4m^ _ (1 + m°) (1 + x') b — c c — a a — b ' then will (WKlz:^),i^^_(l±l!)(l±i!). 0— c c— a a—b 88. Prove that the equation h' x^+ax'+bx + -— = (i ia can be solved directly, and that the complete cubic a? —px' + qx — r = may be reduced to this form by the substitution x=.y + h. Prove that the roots of the auxiliary quadratic are a{P-yr+fi{y~ar + y{a-l3Y^J{-Z){IS-y)(y-a){a-p ) {fi-yf + {y-c.Y+{a-py -' a, p, y being the roots of the cubic. ALGEBEA. 17 IV. Theory of Divisors. 89. Determine the condition necessary in order that 9? +2)x + q, and as' +p'x + q may have a common divisor of the form as + c, and prove that such a divisor will also be a divisor of fs? +{q—p')x — q. 90. The expression as" + Zax^ + 36a3* + ca^ + Mx' + Sex +f will be a complete cube if ^/V» = 5 = ^"='-"- 91. The expression x^ — baf + cx'+dx—e will be the product of a complete square and a complete cube, if 5 " b~ o~ c'' 92. Prove that csas' +hx + c, and a + bx* + cas" wiU have a common quadratic factor, if ¥c' = (c'-a'+b')(c'-a' + ab). 93. The expressions a^x* + a^af + a^x' + a^x + a^, and aja* + a^ + a^v? + a^x + a„ wiU have a common quadratic factor, if (a^ -a^- a„) (a^ - a„)' + {% - a,) {a^% - afi^ = 0. 94. The expression a? + px" + qx ■¥ r wUl be divisible by a;'' + aai + 6, if a'-2pa'' + {^' + q)a + r-pq='0, and W-qi" + rpb-r' = (i, w. 2 18 BOOK OF MATHEM-ATICAL PESBLEMS. 95. Prove tliat x*+px + q ■will be divisible by x'+ax+b, if a» - iqa' = p\ and {¥ + q) {V - q}' =/6». 96. The bigbest common divisor of p(x^—l) — q (af—V), and (q—p) a:'— qx^^'' +p is (x— 1)', p, q being numbers wbose greatest common measure is 1, and q>-p. 97: If n be any positive whole number not divisible by 3, the expression «''" + 1+ (as + l)™ -will "be divisible by k' + x + I; and, if w be of the form 3r + 2, by (»" + « + 1)^ V. Identities and Hqualitie^, 98. Prove tha-fe, (1) (a+J+c)'=a° + 6° + c'+3(6 + c)(c + a)(a + 6), ^ ' 6-c c — « 05 — (o-c)(c — a)(a — 6) (3) (6'-6») {S- c'^ + iS-c) {3- a'^ + iS- a=) {S-h') = As(s~a){s—b) {s — c), where 2/S'= a' + 6^+c^, and 2s = a + 6 + c, (4) 2yV{z + aiHa! 4- yf + 2«V (a; + y)' {y + z)' + 2xY(y + zY,(z + xf = x\y + z)* + y''{z + xy + s%x + yy'+lQii(?y^s?{yz + zx^-xy), (5) (6V + aV) (5 - c) (ft -.e?) + (cV + &'<^'') (c - ft) (6 - eZ) + K6VcW=)(a-5)(c-J) = {h-c){G-a){a-b){d-a){d-h){d-c), (6) (Sec? + c£?a + dab + abcf - abed {a+b + c+d)' = (bo - ad) (ca - bd) {ah — cd). ALGEBRA. 1 ,7^ (5 + cf+jc + ay+(a + 5)°- 3 (5 +<;) (c + g) (g + b ) ^ ^' a'+b'+c'-Sabc ~=^' (8) {x'-x + 1) {s:*^fe' + 1) . .. . .. (aj2" _ 3,2"- + 1) "" x' + x+l ' 99. If -+f =1, and— +^= ;: a b a b a + b' thenwiU ^H-C = f— T- as \a + bj 100. Having given . — r- =«, ^ =5, =c, y + « a + a; a; + 3/ ■ find the relation between u, h, -c; and prove that k" _ f g' a (1 — 6c) " 6 (1 - ca) ~ <; (1 - a6) ' 101. Having given the equations a! + y + s=l, 1 afl3 + Ji/ + c» = c?, > prove that dx + 6V + c'« = cZ" - (c?- a) (cZ -l){d- c). 102. If 1+1 + 1= ^ — ~ T ~ — — r^ > a 6 c a + b->ec the_n, ior fill integral values of tk. 1 1 «"+' 6™+' c="" (a + 6 + c)'»-''" 103. !£ x + y + z = xyz, then will 2a; 2y • 2a _ 2a! 2y 2g rr^^+ ir^^+r-^-T-^ r^Y i-«^' 2/ + S « + a; x + y _ y + z_ s + x x + y *° r^s'^T^^'^ l-xy~l-yzl-exl-xy' 2—2 20 BOOK OP MATHEMATICAL PEOBLEMS. 104 If x{b-c) + f/{c-a) + s(a-b) = 0, then will bz — cy ex — az ay — bx b — c - c — a a — b 105. li x,y, z,uhe all finite and satisfy the equations x=by + cz +du, y = ax + cz + du, z =ax + by+du, Vi = ax + by + cz, ,, „, a b c d ^ then will ^ — - + z. — j + t. — + ij ,= 1. !+.« 1 + 6 1+c \+d 106. If ^^M^and*-"^'-^ then will b — c, X c — a x^-y^ ^xy _ a — b z and if a + ^ = 6 + , b—c c—a then will each member of the equation be equal to c + ^ . yz zx ' '. a; — 2- y X It 107. If T = - ^ , and *, y be unequal, then will each member of this equation be equal to ^ , to x + y -yz, and to 1 1 1 -+ -+-. x y z 108. Having given the equations aU + bmy + cnz = aVx + brnSy + eriz = as(?+ by' + cz' = 0, prove that. X inin'- m'n) + y {nV -n'Fj+z {Im' - I'ni) = 0, ALGEBEA. 21 and that ^{{m-m')s-{n-n')yY + ^{{n-n')x~{l-l')zf + -[{l-r)y-{m-m')xY=0. c 109. Having given a + b + c + d= a - b' - c + d' = = aa' + bb' + cc' + dd', prove that aa'2 + bb" + cc" ^dd" ,,,a + d ,„ b+c , , =ba -j=ad •= . abed a a be a'^^b''^?'^! a'^l' b'""? 110. In the equations a; y z I {mb +nc — la) m (nc + la — mb) n {la + mb — nc) ' If m, n are all finite ; prove that I m n X {by + CZ — ax) j/{cz + ax — by) z {ax + by — cz) ' 111. If a, b, c, X, y, s be any six quantities, and a^ = bc— x', 6, = ca — 2/^ c^ = ab — ^, x^ = yz-ax, y^=sx-by, z^ = xy-cz; and a^, b^, c^, x^, y^, z^ be similarly formed from a,, 5,, c,, a;,, y^, «, and so on; then wUl a b c X y z = {ax^ + by" + c^- aba - 2xyz) ^ 112. Having given e X y+m y z+x l—x" m+ nyz ' 1 — 2/' m + nzaa ' prove that i£ x, yhe unequal, z x+y l — s? 'm + nxy I 22 BOOK OF MATHEMATICAL PKOBLEMS. 113. Prove the following equalities, having given that a+b + c = 0, a' + h'+c' _ a' + b' + c^ a°+5°+c° 5 ~ 3" 2 ' a' + h' + c' a' + ¥ + c' a' + b'+c^ a' +W + c^ a^ + V + c* 7 ~ 5 2 3 2 ■ a"+5"+c" _ a" + 5° + c° a^ + W + c'' (a° + 6^ + cT a°+ 5° + c^ 11 ~ 3 2 9 2 114. If a+6 + c + t? = 0, then will aS+5« + c» + tf a' + 6" + c' + c?' a^+¥ + c^ + d^ 5 3 2 ■ 115. Having given the equations (2/ + zf = 4a'^«, (a + a;)^ = ib'zx, (as + yY= 4c^a3y ; prove that a' + ¥ + -) + J(a;"- 2/'0 = 0. AIGEBEA. ^3 118. If the quaaitities x, y, z be all unequal and satisfy the equations ym «x ~ xy ' each member of the equations = »" — !, and ■ 1 1 1 yz + zx + !tm = — + — I- — . yz zx xy 119. If ^+f+!=^+,|^+i=.0; a c. a c then 'wiU a\o a J o\c a hj c\a o cJ VI. Inequalities. The symbols employed in the following questions are always supposed to denote real quantities. The fundamental proposition on which the solution generally depends is a" +¥ > 2ab. Limiting values of certain expressions involving an unknown quantity in the second degree only may be found from the con- dition that a quadratic equation shall have real roots : — e. g. " To X' — 4:X+ 3 find the greatest and least values of -j — ^ r ." Assuming the expression = y, we obtain the quadratic x'{l-y)-2x{2-y) + 3-4:y=Q, and if a; be a real quantity satisfying this equatiori we must have \%-yf^.{l-y){Z-iy), or Zf-Zy^l, or (22/ -!)='< J, shewing that y must lie between the values 24 BOOK OF MATHEMATICAL PEOBLEMS. 120. If X, y, z Ise three positiye quantities wliose sum is unity, then will / 121. Prove that \ 122. Prove that / 1- /w + lV \ ~n}\ ~n)' \ ~ n )'^\n' 123. If a, h, be three positive quantities of which any two are together greater than the third, then will 1 1 11119; - > - + r + - > - b+c—a c+a—b a+b—c aba a+b+c and, X, y, z being any real quantities, a^ (a; - y) (a; - «) + 6' (y - «) (y - a) + c' (« - aj) (« - y) cannot be negative. If x + y + z = 0, a'yz + Vzx + c'xy cannot be positive, 124. If a, 5, c be positive and not all equal, the expressions a(a — b) {a— c) + h {h — c) (b — a) + G{c — a){a — b), af{a-b){a-c) + b\b-c){b-a)+b\c-a){c-b), are positive. 125. Prove that {ax (6 + c) + by {c + a) + cz (a + b)Y > iabc (x + y + g) (ax +by + cz) ; a, b, c, X, y, z being all positive, and a,, b, c unequal. 126. If xyz = {l-x){l-y){l-z), 1 3' the greatest value of either of these equals is k > tii,y,!s being each positive and less than unity. ALGEBRA. 25 127. Find the greatest numerical values without regard to sign which the expression (x -8){x- 14) {x-16)(x- 22), can have for values of x between 8 and 22. 128. If a >5, and c be positive, the greatest value which the expression (x-a) (x-h)(x-a-c){x-h + c), can have for values of x between a and b is (a-hy{a-b + 2cy 16 129. If^>m, af — 2mx+p' p-m p + m x'' + 2mx+p'' p + m p~m' 130. The expression ax" + hx ■ ex' + bx + a' will be capable of all values whatever if 6'>(a + c)'j there wiU be two values between which it cannot lie if 6" <; {a + cf > 4ac ; and two values between which it must lie if V < icto. » 131. The expression {x — a)(x — ^) ■ {x—y){x — S)' can have any real value whatever if both, or neither, of the two a, P, lie between y and 8 : otherwise there will be two values between which it cannot lie. 26 BOOK OF MATHEMATICAL PE0ELEMI5, 132.. The expression a^ + 2 Ja; + c aV+26'a;+c" will be capable of all values, provided that {ad - a'cf < 4 {a'h - aj)') (b'c - bay Prove that this inequality involves the two b^ > ac, b" > a'o' ; and investigate the condition (1) that two limiting values exist between which the expression cannot lie, (2) that two limiting values exist between which the expression must lie. 133. If x^, x^, x^...x^ be positive, and if n— 1 „ < + a;/ +...+«!„»- a;,a, - 03,853 -...- fB^iS^ - a5„ + "2^ = 0, then will x^, x^, ... x^, I he in ascending order of magnitude. VII. Proportion, Variation, Scales of Rotation. 134. If b + c + d, c+d + a, d + a+b, a + b + c be propor- tionals, then wiU a — d b — c. 135. If 2/ vary as the sum of three quantities, of which the first is constant, the second varies as x, and the third as x^ : and if {a, 0), {2a, a), (3a, 4a) be three pairs of simultaneous values ot X and y, then when x = na, y = {n—lfa. 136. A triangle has two sides given in position and a given perimeter 2s : if c be the length of the side opposite to the given angle, the area of the triangle « s — c. ALGEBRA. 27 137. The raxiix of a scale in whicli 49 denotes a square number must be of the form (r +l)(r+ 4), where r is some whole number. 138. The radix of a scale being ir+2, prove that if the digit in the units' place of any number be either '2r + 1, or 2r + 2, the square of the number will have the same digit in the units' place. 139. Find a number of (1) three digits, (2) four digits, in the denary scale such that if the first and last digits be interchanged, the result represents the »ame number in the nonary scale, and prove that there is or^ly one solution in each case. 140. If the radix of any scale Lave more than one prime factor, there will exist at least one digit different from unity such that if any number have that digit in the units' place, its square will have the same digit ia the units' place. VIII. Arithmetical, Geometrical, and Han-monieal Progressions. 141. If the sum of m terms of an a. p. be to the sum of n terms as m' : n'; prove that the m!^ term will be to the w* term as 2m — 1 : 2n — l. 142. The series of natural numbers are divided into groups 1 ; 2, 3, 4 ; 5, 6, 7, 8, 9 ; and so on : prove that the sum of the numbers in the w* group is n^ + (n— ly. 143. 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)(3n-l)(a + 1)' + in{n + V)al 24(w-l) * 28 BOOK OP MATHEMATICAL PEOBLEMS. 144. Having given that r , -3- , —^i are in A. P. ; prove that a° + c° - 26' ' a + h + e a' + c'-ih'" 2 • 145. If a, I, c; b, c, a; or e, a, I be in A.P., then will and if in G. P., '1,1 1\ „3 -^*^''°(^^6'-^?) = '*'-^^^ + c°. 146. If a, I he the first and w"" terms of an A. P. the con- tinued product of all the n terms is 147. Tlie first term of a g.p. is ct, and the n^^ term l] prove that the r'" term is (a""' Z'"')"-'. 148. If a, b, c he in A. p., a, ;8, 7 in H.P., and aa, bp, cy in ». p., then will ■ A 1.1.1 a : : c :: - : -5 : -. y 13 a 149. The first term of an h.p. is a and the m"" term I, prove that the r'" term is " (to -I) a? (n-r)l+{r-l)a' Prove that the sum of these n terms <(» + ?)«, and their con- n tinued product <(aZ)', ALGEBRA. 29 150. If a, 6, c be in h. p., then will 1 4 111 + + 7= ■ o—c c—a a—o c a 151. li a,b, c, dhe four positive quantities in h.p., a + d>b + c. 152. Prove that h + c, c+a, a + b will be in h. p., if a', b', c^ be m. A. p. 153. If three numbers be in g.p. and the mean be added to each of the three, the three sums will be in h. p. 154. If n harmonic means be inserted between two positive quantities a and b, the diffetence between the first and last of these means bears to the difference between a and b a ratio less than n—1 : n+1. 155. If a^, ttj, fflj ... be an a.p., 6„, b^, b^ ... a g.p., and A, B, C, D be any four consecutive terms of the series a^^b^, a^^b^, ttj + 6j, . . ., then will Ab^ -B{b^ + 2,6 J + C (6, + 26„) - Db, = 0. IX. Permutations and Combinations. The number of permutations of n diferent things taken r together is denoted by .-P,, and the correspondiag number of com- binations by ,,(7.. 156. Prove a priori that P, = „_,P + 2r „.,P _. +r{r-l) „.,P _„ = n-A + 3»- „->P.., + 3r (r - 1) „_3/'._, + r{r- l)(r - 2) ^_,F^_,, + r(r-l)...{r-p + l)„_,P,_„ p being a whole- number < r. 30 BOOK OF MATHEMATICAL PROBLEMS. 157. In the expansion of {a^ + a^+ ... +«,)", wTiere w is a ■whole number not greater than p, prove that the coefficient of any term in which none of the quantities a^, a^ ,.. a^ appears more than once as a factor is [n. 158. The number of permiitations of n different letters taken all together, in which no letter occupies the same place as in a certain given permutation, is , (1 1 1 (-1)") 159. Prove that A =,.fi.-2„,A.. + 3,.,,C',_,- .„. +{- 1/ (r+ 1). 160. The number of combinations of 2w things taken n toge- ther, when n of the things and no more are alike, is 2" ; and the number of combinations of 3n things, n together, when n of the things and no more are alike, is ]2n 161. The number of ways in which mn different things can be distributed among m persons so that each person shall have n \mn of them is 7T=xh. 162. There are p suits of cards, each suit consisting of q cards numbered from 1 to g'; prove that the number of seta of q cards numbered from 1 to g' which can be made from all the suits is p^. 163. The number of ways in which p things may be dis- tributed among q persons, so that everybody may "have one at least, is ALGEBRA. 31 164. The numljer of ways in which r things may be distri- buted among w + p persons, so that certain :w of those persons may each have one at least, is {S^ (w+j>)'"-w(w + j?-l)''+ ■ \ — ' (w+p-2)'- Lf Hence prove that X. ^Binomial Theorem. 165. Prove that \ + x n{n-V) \ + 2x W I'-'^TT;^'^ U (1 +««!)' »(re-l)(w-2) l + 3ie ~ [3 (l+wa;)''^ ""' /„x , o /2w + 1\ - /2w + ly (2) -i^^(2;n)^%2^i)-^- ... + (2»-l)(g±l)""'=»*(2^-l). TC being a whole nnmber. 166. Determine a, 5, c, 2- To ^ + =W+1, If 1 _ („ _ 1) ^L^ H. ^LrMi:z^ _^' ^ ' {\+mf U (1+m)* (w-3)(?^-4)(w-5) m,' ^ m°+'-l 1 [3 (1 + m)""^ ~ m-\ (to + 1)"' (n -3)(n- 4) (n-5), ,„_, ._, J9"+' - g"" ~ 13 • ■ '{P+l) Pt+--^ p_l ■> n being a positive integer. 170. If ^ be nearly equal to q, then -will ^ — ■--- be nearly equal to /-. 171. If a^ denote the coefficient of aj' in the expansion of (1 + iC\" -j 1 in a series of ascending powers of asj the following rela- tion will hold among any three consecutive coefficients, ('• + 1) «,« - 2«« - (»• - 1) «,_, = 0. (1 + a;)" 172. If ^ a 1^6 expanded in ascending powers of x, the coefficient of te""^'"' is (w + 2r) 2""', m, r being positive integers (including zero|. ; ^ ALGEBRA. 35 173, The sum of the first n coefficients of the expansion, in ascending powers of x, of -f^^ is 2'-* !i(!i±^li^±I) . (V — x) 3 174.. Prove that ^1.2 12 *1.2" 'Is "^••• [2 "^1.2" [3 = 2'-»(w'' + 7w.+ 8X 175. Prove that. a''(6-c) + 6''(c-a)+c'(a-6) is equal to the sum of the homogeneous products of n dimensions of a, 6, c. XI. Exponential cmd Logarithmic Series. In the questions under this head, n always denotes a positive whole number. 176. Obtain the following equalities; — (1) n'-{n+l){n-\r + ^^^^{n-%Y- = 1, (2) (»i-l)»-«(«-2)"+^^^y^(»J-3)"-'... = [w-], (4) l"-«2" + '?^^^,3"-.. +(-l)"(«+l)"=(-l)"li». w. 3 54- BOOK OF MATHEMATICAL PEOBLEMS. 177. The eo'efficient of a;' in the (expansion of (1 + a;)?' feeing denoted by a,, prove that »„/'-«.(^-ir'+«.(p-2r'- +(-ir'«.-.. + (-1)—' «„_,.,, p being a whole niimber < n. , 178. Denoting by u^ the series 1" + 2° + Tg. + ., . + ^ / + ... to infinity^ prove that n(n—l) n(n—lL)(n — 2) «„-, + K = «*„+.- »»w„ + ij, «»-! - [3 ^ « -2 + ■ ■ • + (-l)V,; aud . ■"„+,- w„ = M„ + wM„_, and by means of either of these, prove that u., = 4140c. 179. If w„= i;-' -2-'+ -^ - j^+ ... to infinity, then will , n(n~l) 1 1 1 ^-IV*' 180. lfv,=^-^.^-....(^; then^ will 181. Having given ALGEBRA. 35 prove that the limit of j — '^ -when n is indefinitely increased is 182. If there be a series of terms «t„, Mj, u^ u^ ... m,^.,., of which any one is obtained from the preceding by the formula M„ = WM„_, + ( — 1)", and if w„ = 1 j then will , n{n—\) n{n — \) Prove also that _? tends to become equal to - , as n is indefi- nitely increased. 183. Prove that and that p' + q'^ip + qf- n{p + qf-'pq + ' (p + ?)"-y2= 5^ ^ -(p + qT V? +- 184. By means of the identity log (1 - a;') = log (1 - a;) + log (1 + a; + x'), prove that ^_, 3w(3w+.l) (3w-l)3w(3w+l)(3w + 2) ^ + Li the series being carried to 3n terms. 185. If there be n quantities a, b, c, ... and «__ represent their sum, s^_^ the sum of cwiy w — 1 of them, and so on, and if S^ = (O' - 2(«„_ J- + 2 (.„.,)'- ...+(- 1)"-' 2(«.)-; 3—2 36 BOOK OP MATHEMATICAL PEOBLEMS. prove that 'S'l = ■^s = 'S'a = • • • = 'S',,-, = 0, iS', = [nalo ... , 2/S'„^, = \n+l aba ... {a + b + c + ...), and 12 *S'„^,= \n + 2 aba... {2 2(a') + 3 2(o6)}. If a be any other quantity, and if S^ now denote (a + 0'--2(a+s„./ + 2(a + O'--+(-ir'2(a+«0', then will S^ = JS^ = S^= ...=S^_^ = 0, S^ = [nabc..., 2S^^j = \n+l abc ... {2a + a + b + c ...), and 12 »S„^, = \ n + 2 a6c . . . {2 2 (a') + 3 S (ab) + 6a 2 (a) + 60"=}. XII. Summation of Series. If u denote a certain function of n, and *S'„ = M, + ltj+ +M„, the summation of the series means expressing S„ as a function of w involving only a fixed number of terms. The usual artifice by which this is efiected consists in expressing u^ as the difference of two quantities, one of which is the same function of n as the other is ofw — 1, {U,,— U„_^. This being done, we have at once s,MV,- u,) + {u^~ u,)+ +{u^- u^_,)= u„-u,. Thus, if u„ be the product of r consecutive terms of a given A. p., beginning with the «."', u^^{a + {n-l)b}{a+nb) ...... {a + {n + r-2)b} _ \a+(n-l)b](a+nb)...{a+(n+r-l)b}-{a+(n-2)b}...{a+(n+r-2)b } {r + l]b ' ALaEBRA. 37 whence U =.{« + ('»- 1) 5}(» + '»*) {a + (M + r-l)6} Whence 6/„_ ^— -^^^ , S'ld -^n = 7^7:^77^ [{« + ('»-l) 6} (» + wS) {a+(re + r-l)6} - (a - 6) (a) (a + 6) {«+ (r - 1) J}]. The sums of many series can also be expressed in a finite form by equating the coeflB.cients of x" in the expansions of the same function of x effected by two different methods, of which examples have been given in the Binomial, Exponential, and Logarithmic Series. In the following examples, n always means a positive whole number. 186. Sum the series : 1 _3_ _5_ 2w - 1 ' ^ 172 "^2. o "^5.10"^ ■^{l+(»-in(l + «=)' 1 1.3 1. 3.5 ^^) 274 "^270 "^2. 4. 6. 8 ■^■■" 1.3.5 (2ji-1) ■jj. 4. 6 2?i(2ji + 2)' ,„x 1-2 2.2' . n.r (^) 17273+1^^ I2i±i' 3 2.3' W.3" r 2 . r^ wr' h ' \r+\ | r + 2 (5) ^3TT+^^-^ + ^^^ {1 + a;) (1 + 2a;) "^ (1 + 2a;) (1 + 3a;) 1 + ... ... + (l+M*){l + (w+l)a;}' 38 BOOK OP MATHEMATICAL PEOBLEMS. L, 1 _2_ n ^' T73"^1.3.5'*' ■^1..3.5...(2»i+l)' 1 3 5 2n-l ^^^,3* 3.7"^ 3. 7. 11"*" "^3.7.11...(4w-l)' 1 5 n' + n~l (9) [3-^14-^ ^l^IT' ,,„, 4 9 2n'+3n~l (10) ro + n+ + — I — T^ — ' ,„, 5 11 , w' + w-l (12) 5.10 10.17 ■ {l + n'){l + {n + iy\' 1 X (1 + «) (1 + a;") "^ (1 + a;^) (1 + a;'*) "*" "' 3!°-' ... +- (l + aj'Kl + a;"-"')' a; a; a: x x x* ^^' iT^'^iT^ rt^'^rr^^ iT? TT^""^' ... + (14) l + a;" 1+a;* 1 + ai^" ' X (1 — ax) (1 + a;) (1 + aa;) (1 + a'x) ax (1 — a'x) (1 + aa;) (1 + a^x) (I + a^x) a"~^x (1 - a'x) '" (1 + o"-'a;) (1 + a"x) (1 + a"^'x) ' (15) -L. + r + 1 ^ (rH-l)(2r+l) ' p + r (^ + r) (p + 2r) (p + r) (^ + 2r) (p -h 3r) (r + l)(2r + l)...{(w-l)r + l} ■'" {p+r}{p + 2r) ... (p'+nr) AUGEBEA. 39 187. Prove that ^_w(w — 1) „w(w — 1)(m— 2) ^~[2 [3 , ^ n(n-l)(n~2){n-3) 188. Prove that — l-r-r' + r" 189. Prove that ^. n-l ,,.. (n-2){n-S) , _ 4"-'-2°-' |2_ J^ ji + 2 190. Prove that ' 1 I ■■ I njn + l) n{n + l){n+2) \2_ [3 ■*■■■•■, n{n+l)...{2n-l) in _ {n + l){n + 2) ... 2n [n 191. Prove that 1 1.3 1.3.. 5 4 4.64.6.8 ' 1 m ... m + w (m + w)(»4 + M — 1) wi (m — 1) , 1 , 1 ^ ' + ... to wi + 1 terms = - , {m+n){m + n + l){m + n-l) '" n + 7 h — rs + — *° °°= i » wi+p (jw+/))(»»+^+l) p-i p being any positive quantity > 1, 40 BOOK OF MATHEMATICAL PROBLEMS. 192. From the equality log (1 - k') = log (1 - «) + log (1 +X+ x% prove that 3ji(3j» + 1) (3w-l)3w(3 w + l)(37» + 2) , \ 3ji + 1 6^rr2^^ ^^"Isw + l [3 3w (3w + 1) (3w + 2) I XIII. Recwrring Series. The series m„, Mj, m^, m„ is a recurring series if any fixed number (r) of -consecutive terms are connected by a relation, of the form «» +:pi <»«-i + i'i^.-i, + • ■ ■ + p,-i»„-,+i = 0, U) in ■which m may have any integral value, but p^, p^ •••Pr-, ^re constant. It follows that the series d^ '+ «! a; + a^af + . . . + ajx^ + ... is the expansion in ascending powers of a; of a function of as of the form -= — — -^ ^^=^- — ;:=-! (the generating jfimction of 1 H- p^SC + p^ + . . . + Pi^i^ the series); and if the scale of relation (A) and the first r— 1 terms of the series be given this function will be completely deter- mined; when, by separating this function into its partial fractions , — 1 1-= — s — + and expanding each, we obtain the «"" 1 — OiS; 1 - ajO! ^ o ■ ' term of the series and the sum of n terms. If the scale of relation is not given, we shall require 2(r— 1) terms of the series to be known to determine all the constants ; thus, if four terms ai-e. given we can determine a recurring series with a scale of relation between any three consecutive terms, and whose first four terms are the giyea. quantities. ' ALGEBEA* 41 193. Prove that every A. p. is a" recurring Series iand that its generating function is ...' .J , a being the first term and b the common difference. 194. Find the generating functions of the following series (1) l+3x + 5x' + 7x^+. ...... (2) 2 + 6a; + 1 3a:' + 35a!' + (3) 2 + 4a;+ 14a;'' + 52a!' + (4) 4 + 5a; + 7a!'+lla!»+ (5) 2+2a:+8a;' + 20a!»+ (6) 1 + 3a! + 12a;» + 54a!' + and employ the last to prove that the integral part of (^3 + 1)'" is divisible by 2"'^', ** being any integer. • 195. The generating function of the recurring series whose first four terms are a, b, c, d, is aV -ca' + x (a^d - 2abo + b') ¥-ac+x{ad-bc)+x'{c'-bd) ' 196. If the scale of relation of a recurring series be and if m„ = 2, u^ = 7, find m„ and the sum of the series U^+Ui+ ... M„_i. 197. Prove that, if ^ »o> »i> «a-"*n te an A. p. and b„ b^..., 5„ a g.p., the series a„ + 6„, a, + 6,, ... a„ + 6„, .... %\> "'A' ••■ "»*»' are recurring semes. 42 BOOK OF MATHEMATICAL PEOBLEMS. 198. Prove that the series l'-+2'-+ 3'+...+n' are recurring series, the scales of relation being between 4, 5, ... r + 2 terms respectively. 199. Find the generating functions of the recurring series (1) l+2a: + 5a^ + 10a;' + 17£t;* + 26a!'+..., (2) l + 3a! + 4a;''+ 8a;= + 12a!* + 20a!'+ ..., (3) 3+6x + Ux' + 36a!' + 98a!* + 276a!'' + . . ., (4) 3-a!+13ai''-9a!' + 41a!*-53a!'+..., and the w"" term of each. 200. If the terms of the series a^, ,a„ a^... be derived, each from the preceding, by the formula pq '""-'' p + q-a^' prove that XIV. Convergent Fractions,^ 201. If — be the w'" convergent to the continued fraction »! «2 «, b,+ b,+ b\+..., we have the equations Pn = hPn-l + <^nPn-U and the like law for q : and for the fraction a. »s ^3 6,-6,-63--..., ALGEBRA. 43 and the like law for q. The solution of this equation, «„, 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 ^j, p^; g,, q^ respectively. The fraction — thus determined will not iisually be in its lowest terms. We will take as an example the question "To find the w" convergent to the continued fraction 1 1 4 12 2w(w-l) 1- 3-6- 9 - ...- Zn -.... Take m„ to represent either p^ or q^ (since the same law holds for both), then M„^, = 3wM„ -2n{n- 1) m„_, ; or u„^i-2nu„ = n{u^-2{n-l)u„_i}. So u„-2{n-l)u^ = {n-l) {u,_^ -2{n-2) m„_ J, M3- 4m, = 2(1*2 -2m,), ( = 2 or as u=p or q). Hence m„^.i - 2«m„ = \n or 0, !;-'_2-^=lor0. [n \n~l So 2«*„ -2=M„_, 1 T i o = 2 or 0, \n—l \n — 2 2"-X - 2"M^ = 2"- ■' or 0, LI 2X :.2". i^2»+'. -1 or 2", and the «" convergent is 2"-l 44 BOOK OP MATHEMATICAL PEOBLEMS. Ill • ' 202. The w* convergent to -5 ^ jr ^ -\- 2 -i- J + .. t i, (i+V2r-(i-v2)" (i+V2r'-(i-x/2r'" 203. If — be the w*'' convergent to the infinite continued fraction - - — \ v , q "will be coeflScients of a;""' and a;" respectively ia the expansion of ^ —2 , in 204. Prove that, — being the w"" convergent to the infinite continued fraction — -r - t- , a+ b + a+b + ... Pn^2 - (2 + ab)p^ + p„_^ = 0, q^^^ - (2 + ab) q„ + q„_, = 0. 205. Prove that the products of the infinite continued frac- tions -,,1111 111 (1)-=--,- ,C + Y-- , ^' a+b + c + a+... b + a + c+... ,„, 1 1 1 1 1 ,1111 ^' a+ b + c + d+ a+ ...' c + b + a+d+... ,,. l + bc ,-. b + d+bcd are, (1) .j =-, (2) =-. ^ ' 1+ab ^ ' a + e + abo 206. Prove that the difiierences of the infinite continued frac- tions J, 1111 ilil ^ ■' a+ b + c + d + a+ ...' c + b + a+d+c + ...'. """'(^^rr^' ^^ K + c + aba ' ALGEBRA. 45 207. The continued fractions 4+i if 2+11^ ^8+8+8+...' ^ + 4+4+4+...' each to n quotients, are in the ratio 2:1. 208. Having expressed ^{n' + a) as a continued fraction in the form n + ^ jr- , -^ is the r"" convergent: 2n+2n+... q^ ° ' prove that Pr' - K + «) 9-/ = (-«)', ?,+! + «?,_, = 2;?^, 209. Prove that w n—1 n—2 2 1 1 n+l n+ n-l + n-2+ ...+2+ 1 + 2 n + 2' 210. Prove that the value of the infinite continued fraction 12 3 .1 is 1 + 2 + 3 + .. . £-1" 211. Any two consecutive tenns of the series a„a„... a„, ... satisfy the equation n(n + l) » find a^ in terms of 0^; and prove that, when n is indefinitely increased, the limit of ■ ' " "" — '^^ is a,. [n ' 212. Prove that .^. r(r + l)r{r+l) ^ ' 1+ 1 + ... tow terms _ r(r + ir' + (y + l)(-r)"^' 46 BOOK OP MATHEMATICAL PROBLEMS. 1 1 3^ 5^ (2^-1)' '' > 1 + 2+2+2+ + 2 1 1 1 (-1)"" = 1 i + * '- — - 3 5 2n-l ' 1 2 6 12 n(n + l) ^' 2 + 2+2+ 2 + + " 2 ^J^ 1_ J_ (-1)° -1.2 2.3"^ 3.4— + {n+l){n + 2)' 1 1 r+1 r+2 ^ ' i+r + r + l + r + 2 + ...ton quotients 1_ 1 1 - r+l"^ (»-+l)(r + 2) (r + l)(r+2)(»- + 3) + . . . to w terms, XX ix n'x '' 1 + 2-£c+ 3-2a; + +n + l-7ix x'_ a;' _ (-l)°a!"^' 2"^ "6 "*■ w + 1 ' = 33- X _^ (3^)' {{2n-l)xY ^ ^ l + 3-a;'+6-3a;'' + + 2w +l-(2w-l)a; 83° 83° (-XY = X— — +—- +> ' 3 5 "2»i+l' . . 1 r r{r + l) r{r + n — l) '' 1+1+ ~~2 + + n = 1- '■ r + 1 (r + 1) (r + 2) " ... + (r + l)(r + 2)...(r + w)' . 1 1 16 81 n\ ^^ *- ' 1+3+ 5 + 7 + + 2»i + l + ...tooo~12' (^) T+r + I+l + +T+... tooc = ^°S2' ALGEBRA. 47 ^ ' 1 + 2+3+ +W-1 + ... tooo~ £=+1' nn 3^ M i:^ _^ 2€°+l ^ ' 1+ 2 + a + +71-2+ ... to oo~ 5£°-2 ■ 213. Prove that ,j> 1 1 1 1 251 (2} 1 — 4 — 1 — 4- ... to n quotients n+\' 1111 n 4 — 1 — 4— 1 — ...to w quotieiits 2 (ji + 1) ' ,„, 1 1 4 9 w' _, 1 1 1 (■^^ 1-3-5-7-. ..-2;^^ = ^ + 2+3+- + ^rrr ,,,119 25 (2w-l)^ .,11 1 1-4-8-12-...- in - 3 5 2^1+1' .„, 1 1^ (r+iy (r + n-iy r-2r+l- 2r+3 -...-2(r+w)-l 1 1 1 = - + =• + ... +- (6) r r+l '" r'+n' 1 a° . (a + Vf {a + {n-l)hY a-2a + 6-2a + 36-...-2a+(2M-l)6 1 = -+ i- + ,..+- (^ a a + h '" a + 9i6' a' (a-1)' o' (a-l)° 1— 1 —1—1 — ...ton terms a (n. + 2a — 1) na . . , s — ^^ — ^ — 5 or ■ jr jr , as w IS odd or even, n+l n — 2a + 2 ,„. a(a + l) a(a-l) ffl(a + l) ^^ 1 - i - I -... to 2n terms _ a(a + l){{a'-lY-a'"} ~ (a + l).(aV !)"-»'"■" ' 48 BOOK OF MATHEMATICAL PEOBLEMS. X 3c 4x n^x (9) 'i.-2+x-3 + 2x-...—n+l+nx . x' x' af^ = X+-^ + -S- + ... +- 2 3 n+l' (10) i_iJ_..._^=i + i+L2+L3 + ... + L-, nn i 1 — K-1)' _ n{n+l}{2n+l) ^ ' 1-5-13-.. .-n' + in+iy- Q 2 3 8 n'-l _n(n+3) (13) 1-5-7-.. .-2»i + l- 2 ' 1 1* 2* n* l_r+2'-2' + 3'-...-n' + {n + l)' 1 — 12 "^ 02 T'" "•" P 2'' "(n+l)" nA.\ I I I ^ -Ji_ -1 V^*^ l_3_4_5-...-w + 2-... tooo ~*~ ' ,, _. , r r 2r 3r _ , \^^) ^ + l_^ + 2-r+3-r+4-... tooo"*' r' (r+1)^ (r + 2)' ^ ' 2r+ 1 — 2r4-3 - 2r + 5 -... ' (17) - ^ to 00 = ^. "■ ' r — r+\—r + 2-... r — 2 214. The numerator and denominator of any convergent to ! fri unity. 7* T T' the fraction =■ differ from each other by r-1 +r— 1 + r-l + ... 215. If ^ be the w* convergent to ~ -Xr -^ g„ r-l+r+l + r-l+... then -will ALGEBEA, 49 prove that aj - 26,/ = (- 1)""'. 2 — x 217. If = — 3 = = 2 + ax + ay+ ... + ax'+ .... 1-403 + 03" ' ^ and ,;— i 2= l+5,a3 + 6„a;'+ ... + 6a;°+ ..., 1-4:03 + 33^ 12 prove that a/ - SbJ = 1. 218. If ., 'I — . = r + ax + aoa' + ... + ax" + ... 1 - 2ra3 + x' 12. l-2ra5 + a3* prove that a/ - {r' - 1) 6„^ = 1. 219. The w'" convergent to 1- j_i is equal to the 1111 (2m - 1)"" convergent to 1^.2 + 1+2+...' 220. In the equation x" - 2nx^ -a; + M(»-l) = 0, prove that a; = ± 'V «, ± Vw + as, and find all the roots of the equation. Prove that Vt-Vt + v/T-... to 00 = 2, and express the other roots of -the biquadratic in the same form. W. 50 BOOK OP MATHEMATICAL PEOBLEMS. 221. Prove that / ,/ /- If? — TOW + W° V^_V^_V^_...= _^_ , / ./ / — =^ n? — mn — r? . . 'Jp-Mp + Jp-... = ^_ , {m^n), I ./ /^ ^= n'-mn — m' ~'Jp + 'yp-»/p+ ... = - 2mn m* + mV + n' where p = -. — ft XV. Properties of Nwrnhers. V 222. If m be a positive whole number, 2'" + 15w - 1 is divisible by 9, (2»j + l)''-2w-l 120, 3«"+2_8w-9 64, 3=»+»+40w-27 64, S^"-^" + 160m' -56«- 243 512. 223. If 2p+ 1 be a prime number (b)*+(-l)'' is divisible by 2p+l. 224. If ^ be a prime number, j,_i(7„ + (— 1)""' is divisible by y. 225. If w — 1, w + 1 be both prime numbers > 5, m must be of one of the forms 30«, or 30«±12; and w'(«'+16) will be divisible by 720. 226. If w - 2, w + 2 be both prime numbers >5, n must be of one of the forms 30i+ 15 or 30<±9. ALGEBRA, 51 227. If w be a whole number, w + 1 and n'-n + l cannot both be square numbers. 228. The -whole number next greater than (3 + ;^5)" is divi- sible by 2", 229. The integral part of -^ {^3 + ^5)"-' is divisible by 2». 230. The equation af-2^ = d=l cannot be satisfied by any integral values of x and y different from unity. 231. The sum of the squares of all the numbers less than a given number iV and prime to it is and the sum of the cubes is a, b, c being the different prime factors of If. XVI, Probabilities. 232. A and B throw for a certain stake, each one throw with one die. J.'s die is marked 2, 3, 4, 5, 6, 7 and Bs 1, 2, 3, 4, 5, 6; •and equal throws divide the stake. Prove that ^'s expectation is 47 =^ of the stake. What will A's expectation be if equal throws go for nothing 1 233. 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; sup- posing them to throw in the order named until the event happen, prove that A's chance of winning is ij^ , and Cs (y^J . 4-8 52 BOOK OF MATHEMATICAL PROBLEMS. 234. Ten persons each write down one of the digits 0, 1, 2. ..9 at random; find the probability of all ten digits being written. 235. A throws a pair of dice each of which is a cube; B throws a pair one of which js a regular octahedron and the other a regular tetrahedron whose faces are marked from 1 to 8, and from 1 to 4 respectively; which throw is likely to be the higher, the number on the lowest face being taken in the case of the tetrahe- dron 1 If A throws 6, what is the chance that JB wUl throw higher 1 236. The sum of two positive quantities is known, prove that it is an even chance that their product will be not less than three fourths of their greatest possible product. 237. Two points are taken at random on a given straight line of length a: prove that the probability of their distance (yv y.\ 2 j . 238. If three points be taken at random on the circum- ference of a circle the probability of their lying on the same semi- circle is T . 4 239. If q things be distributed among p persons, the chance that every one of the persons will have at least one is the coeffi- cient of of in the expansion of I j (e^ — l^. 240. 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 - th of the rod is —i. n n 241. There are 2m black balls and m white balls from which 6 balls are drawn at random; prove that when m is very large A.LGEBEA. 53 the chance of drawing 4 black and 2 white = — — , and the chance of drawing 2 black and 4 white = ^-^ . 242. If n whole numbers taken at random be multiplied together, the chance of the last digit in the product being 1, 3, 7, or 9 is ( g j ; of its being 2, 4, 6, or 8 is ; of its being 5 is 5° -4° n „.^ , . „. 10"-8"-5" + 4" j^Q. ; and of its bemg is ^^ . 243. If ten things be distributed among three persons, the chance of a particular person having more than 5 of them ^ is 1507 19683" 244. If on a straight line of length a + h be measured at random two lengths a, b, the probability that the common part of these lengths shall not- exceed c is — = , (c < a or 6) : and the probability of the smaller b lying entirely within the larger a is a— b 245. If on a straight line of length a+b + c be measured at random lengths a, b, the chance of their having a common part which shall not be greater than d is -; — ^ — r-r^ — r; • (d < either * {c + a){o + b) ^ a or 6). 246. There are m+p + q coins in a bag each of which is equally likely- to be-a shilling or a sovereign; p + q being drawn, p are shillings, and q sovereigns : prove that the value of the expectation of the remaining sovereigns in the bag is — — ^ £. l{m = 5, p = 2, q=l, find the chance that if two more coins be drawn they will be a shilling and a sovereign, (1) when the coins previously drawn are not replaced, (2) when they are replaced. 54 BOOK OP MATHEMATICAL PROBLEMS. 247. A bag contains ten balls each equally likely to be wMte or black : thi-ee balls being drawn turn out two wMte and one black ; these are replaced and five are then drawn, two white and three black : prove that the chance of a draw from the remaining .71 five giving a white ball is ^ks • 248. From a very large number of balls, each of which is equally likely to be white or black, a ball is drawn and replaced p times and each drawing gives a white ball : prove that the chance . » + 1 of drawing a white ball at the next draw is . 249. A bag contains four white and four black balls ; from these four are drawn at random and placed in another bag j three draws are made from the latter the ball being replaced after each draw, and each gives a white ball : prove that the chance of the next draw giving a black ball is "3 3. 250. A bag contains m white balls and n black balls, and from it balls are drawn one by one until a white ball is drawn. A bets J3 at each draw x : y that a black ball is drawn : prove that the value of .4's expectation at the beginning of the drawing is ny ■as. 251. Prom an unknown 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 2« more draws are made, the balls being not replaced. Prove that the probability that in the 2n+l draws more white balls are drawn than black is -^ — = . 4« + 2 ALaEBEA. 55 XVIT. Miscellaneous Questions. 252. By performing tte operation, for tlie extraction of the square root find a value of x which will make a;* + 6a;' + 110^ + 3a; + 31 a perfect square. 253. If lj,^ + m/n^=lj,^ + mjp^^ = l^l^+m^m^ = \, then will K-"».)K-«^.)K-"^.) ^ JAL^O. 254. Prove that {ac-l^{x(x-h) + y{y-k)Y-c{x-hY+2h{x-h){y-h)-a(y-h)' will be divisible by (a; - hf + {y- ^)' if h'-¥ _hk_ 1 a — c b b'— ac' 255. Having given the equations lx+my'+n^ = 0, (6-c)| + (c-a)|+ (a-6)- = 0, .^ .,,. (b-cY (o-ay {a-by. prove that Pyz {mz — ny) + m?zx (mo; - ?«) + t^xy (ly — mx) = Imn (&-<')("-«)(«-&)• 256. 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. 56- BOOK OF MATHEMATICAL PEOBLEMS. 257. The equation "''^ + -^ + ••■+ '^" , =0 will re- ^ x + b^ x + b^ x + b^ duce to a simple equation if =0, afi;-^ + aj>;-' + ...+ «&„"-» = 0, and the value of as will in that case be "'"'-"" a b ""+■■■+ '^b'-"' 11 » n 258. Prove that (a + b + c+dy—i{a + b + c + d){bo + ca + ab + ad+bd + cd) + 8 (bed + cda + dab + abc) = {b + c — a — d){G + a—b—d)(a + b-c-d). 259. Having given , a + b + c + a' + b' + c' = 0, a" + b^ + c' + a'= + 6'" + c" = 0, , , (6' + c' - a?) (c' + a'- b") (a' + b'- c'), unless a, b, c are all equal. 261. Prove that pCp+i) C,.,+ ... +(- iy^(^ + l)-;-(^ + ^-l). j2 n+j-^r-s^ ••• ^v ■'y rr ALGEBRA. 57 262. Prove that , 2w 2ji-2 4 2 „ 263. If ^^^'^ n+1 n^ ^^ ^ n + n+l+n + 2+ ... +1, ^ 1 1 1 1 prove that = ; — + —. ^.-, ^- ... to oo . n + x n n{n + l) n{n + l){n + 2) 264. The 2w + 1* convergent to ^ i^ ^ -^r- is equal " l + 2p+l+2p + ... ^ to the w"" convergent to 1 ■ 20' + l)-2(^ + l)-...- and ^„ = „_!i(!^|)(^l^)+!L(!^zI)^>zi)_..., then will u^ + v^^^= 2 (u^u^_^ + v^v^^_^) ; w being a positive whole number. 266. if.ai+^^^).'^ "^^-^^<:-^)(-^) .-4-... [2 , [4 and v^ = nx+~ ry -x^+..., then will u^u^_^ — v^v^_^= (1 — as')""'; n being a positive whole number. ( 58 ) PLANE TRIGONOMETEY. I. Equations, In the solution of Trigonometrical Equations, it must be remembered that when an equation has been reduced to the forms (1) sin x = sin a, (2) cos x = cos a, (3) tan x = tan a, the so- lutions are (1) a; = wir + (— 1)" a, (2) a;=2w7r=i=a, (3) x = nTr + a, n denoting an integer positive or negative. The formulae most useful in Trigonometrical reductions are 2 sin 4 cos 5 = sin (^ + -B) + sin (A - £), ' 2 cos A cosB = cos {A + £) + cos {A — B), 2smA mxB=: cos {A -■ B) - coa {A +B);^ and . , . „ „ . A + B A-B sin 4 + sm ^ = 2 sin — = — cos — ^— , Z Z , _ - A+B A-B cos A + cos B= 2 cos — = — cos cos A — cos B= 2 sin B 2 ' -A . A+B which enables us to transform products of Trigonometrical func- tions, (sines or cosines) into sums of such functions and con- versely. Thus, to transform sin 2 {(yS-y)} + sin 2 (y -a) + sin 2 (a -^) ; sin2(y-o) + sin2(a-/3) = 2 sin (y - ;8) cos (j3 + y - 2a), sin2(^-y) = 2sin(;8-y) cos(^-y); .-. sin2(^-y) + sin2(y-a) +sin2(a-/3) = 2 sin (j8 - y) {cos (/3 - y) - cos (/3 + y - 2a)} = 2 sin (^ - y) {2 sin (^ - a) sin (y - o)} S -4sin()8-y) sin(y-a) sin(a-y8). PLANE TEiaONOMETRT. 59 So to transform cos (j8 — y) cos (y — a) cos (a — ;8), cos (y - a) cos {a-13) = ^ {cos (y - /3) + cos (;8 + y — 2a)} ; .•. cos(j8 — y) cos(y — a) cos (a — y3) S :i {1 + cos 2 (y-/3) + cos 2 (y-a) + C0S 2 (a-jS)}. 267. Solve the equations ; (1) 2 sin as sin 3a! = 1, (2) cos 33 cos 3x = cos 2a! cos 6a!, (3) sin 5a! cos 3a! = sin 9a: cos 7a!, (4) sin 9aj + sin 5a; + 2 sin^ x = l, (5) cos ax cos Ja! = cos (a + c)x cos (h + c) x, (6) 4 sin a sin ;8 sin a! = sin 2a + sia 2y8 + sin 2a;, (7) cos a! + cos (a! — a) = cos (x — P) + cos (a: + j8 - a), (8) 2 sin' 2a! cos 2a: = sin' 3a!, (9) 2 cot 2a! — tan 2x=3 cot 3a:, (10) 8cosa: = -^+-^, ^ ' sm a: cos x (11) sin 2a3 + cos 2a3 + sin a: — cos a: = 0, (12) (l + sina)(l-2sina!)'' = (l-cosa)(l + 2cosa)', . sin a cos (P + x) _ tan /3 ^ ' sin /3 cos (a + x) tan a ' (14) cos 2a: + 2 cos a: cos a — 2 cos 2a = 1, (15) sin a cos 3a! — 3 sin 3a cos a: + sin 4a + 2 sin 2a = 0, (16) <'°«'_^ + «_^=l. ^ ' cos a: sma: 268. If and then will cos {x + 33/) = sin (2a! + 2y), sin (3a3 + y)= cos (2a: + 2y) . a! = (3»i-3w)| + Y^ 8 16 2/ = (5«-3w)^ + ^ m, u, T being integers. >; or x-y = 2rir + ^ ; 60 BOOK OF MATHEMATICAL PiROBLEMS. 269. The real roots of the equation, tan' x tan s = 1 satisfy the equation cos 2a; = 2 — ^5. 3 /3 270. Given cos 3a3 = — 2"^ J prove that the three values of /3 . TT /3 . IT / 3 . 3 ITT cos X are , / s sm ^tt; > . / ?; sm 7; , — . / ^ sm y^r . nhTi TP ,1 i- J a! tanaj + a— 1 , , 271. If the equation tanK = r have real roots, ^ 2 tana; + a + .l a'>l. 272. Find the limits of the value of - — ) -, for possible tan (as — a) values of x. 273. If j8, y be different values of x given by the equation sin {a + x)=m sin 2a, cos ., i m sin (^ + -y) = 0. 274. The real values of x which satisfy by the equation 2' sin ( ■= cos xj= cos ( ^ sin a; J are 2mr or 2w7r ± n being an integer. 275. If X, y be real, and if sin' X sin" y + sin' (x + y) = (sin a; + sin yf, X, or y, must be a multiple of tt. 276. If a, /3, -y be three angles, unequal and less than 2ir, which satisfy the equation a i .. +-. — + c=0, cos X sin a; then will sin (j8 + y) + sin (y + o) + sin (o + ^) = 0. PLANE TEIGONOMETEY. 61 277. If )3, y be angles, unequal and less than tt, 'whicli satisfy the equation cos a cos X sin a sin x 1 + =^ =-, a c then -will (6''+ c^- a^) cosjS cos y + {c'+ a^- V) sin ^ sin y = a°+ V- c". 278. If a, )8 be angles, unequal and less than it, which satisfy the equation a cos 2a; + 6 sin 2a! = 1, and if (I cos° 2a + m sin^ 2a) (I cos'' 2/3 + m sin" 2/3) = {I cos' (a + /3) + m sin' (a + P)}", then will either l = m, or a'—b^= , . m + l II. Identities and Equalities. 279. If tan'^ =1 + 2 tan'-B, then will cos 25 = 1 + 2 cos 2A. 280. Having given that sin {B + G- A), sin (0 + A- B), sin (-4 + -B — (7) are in a. p. ; prove that tan -4, tan B, tan G are in A. p. 281. Having given that 1 + cos (;8 - y) + cos (y - a) + cos (a — j8) = j prove that (j8 — y), (y — a), or (a — j8) is an odd multiple of ir. 282. If cot a, cot p, cot y be in A. P., so also will cot (/3 — a), cot P, cot {fi - y). 6 6' S cos ^ = cos a CCS j8, cos 6'= cos a' cos P, tan - tan jr = tan ^ ; 283. Having given COS ^ = cos a ccsjS, c( prove that sin'^ = (sec a - 1) (sec a -1). 62 BOOK OF MATHEMATICAL FEOBLEMS. 284 If tan2a = 2 - , „ „ „ , taii2^ = 2- then -will tan (a - fl) be equal to = or to ^ . ^ '^' ^ a-d h + o 285. If tan (<^ + a), tari~^, tan (^ + P) be in A. p. then will cot a, tan <^, cot j8 be in a. p. 286. If a, p, y be all unequal and less than 27r, and if cos a + cos )8 + cos y = sin a + sin /3 + sin 7=0; 3 then will cos'' a + cos'' )8 + cos'' y = sin" a + sin" j3 + sin" y = „ . ■ ..o), Tx • • tan2i8 tan2'y ,, , , 287. Having given - — -. — - — r = - — ; — '-^ : prove that each ° " tan (y + a) tan (a + /3) ^ member of the equation is equal to r?; r , ^ ^ tan (j8 + y) and that sin 2(/3 + y) + sin 2 (y + a) + sin 2 (a + /8) = 0, unless tan (^S — y) = 0. 288. Prove that (l -tan"|) (l-tan" J) (l tan^j) to tana 289; If C =2 cos 61-5 coa'^ + 4 cos'^, /S' = 2 sine-5 sin=^ + 4 sin' 6; then will C cos 361 + (S' sin 30 = cos 2^, and >S' cos 30 - C sin 36 = - ^ sin 20. 290. Having given as cos <^ + 2/ sin <^ = a; cos (j^'+y sin ^' = 2a, and 2 cos ^ cos ^ = 1 ; prove that y^=4:a(a-x); <^, ^' being unequal and less than 27r. PLANE TEiaONOMETET. 63 291. Having given (a; — a) cos 6 + y sin 6=.{x — a) cos ff + ya\nff = a, 6 & and tan - — tan — = ± 2e j prove that y'^= lax - (1 -e") x^, 6, ff being unequal and less than 2n-. 292. If (1 + sin 6) (1 + sin <^) (1 + sin \^) = cos 6 cos ^ cos i/', then will sec° 6 + sec^ ^ + sec°i/r - 2 sec ^ sec <^ sec i/f = 1. 293. Having given cos ^ + cos 5 + cos C+ cos .4 cos 5 cos (7 = 0; prove that cosec°.i + cosec'2? + cosec' C ± 2 cosec A cosec B cosec (7=1. 294. If tan"^ tan J! = tan''^ tan F = tan^C tan C = tan j1 tan B tan C, and cosec 2 A + cosec 25 + cosec 2C = ; then win tan {A-A') = tan (5 - £') = tan (C - C") = tan ^ + tan 5 + tan C. 295. Having given a; sin 3 (^ — y) + 2/ sin 3 (y - o) + » sin 3 (a — j8) = ; prove that X sin (/3 — y) + 2/ sin (y — a) + « sin (a — 18) 03 cos (/8 - y) + 2/ cos (y - a) + 2; cos (a — )8) sin 2(;8-y) + sin2(y-a)+ sin2(a-^) cos 2 (j8 - y) + cos 2 (y - a) + cos 2 (a - ^) ~ 296. Having given the equations a;= = /3'' + y'-2/8ycos(9) x + y+m^O' y^ = -f + a^- 2ya cos <^ »"=a'+)8'-2a^cosi/', prove that )8y sin 6 + ya sin ^ + a;8 sin i/r = 0. ia; + 2/+8 = 0'| , + <^ + i/f = oJ' 64 BOOK OF MATHEMATICAL PROBLEMS. 297. If COS A + COS J3 cos C + sin ^ sin CcobA= 0, then wUl tan ^45° =<= i) tan ^45° - |) tan Ud" --)=-!; and if cos 4 = cos 5 cos C =fe sin 5 sin C cos A, then "wiU cos B = gosG cos -4 ± sin (7 sin A cos 5. 298. Reduce to its simplest form the equation {sc cos (a + j8) + 2/ sin (a + yS) - cos (a - ^)} {x cos (y + S) + 2/ sin (y + 8) — cos (y - S)} = {« cos (a + y) + 2/ sin (a + y) - cos (a - y)} {jc cos (/3 + 8) + 2/ sin (;8 + S) - cos (;8 - 8)}. 299. Given the equations yz-y'z + zx~zx + xy'-xy = 0, A + £ + C=180'', xa/ sivc'A + yy sin^ B + zz sin^ G = (y«' + 2/'«) sin B sin (7 cos A + (aas' + s'a;) sin C sin ^ cos 5 + {xy + sc'y) sin ^ , sin 5 cos G; prove that, for real values, either x = y = z; or x' =y' ^z'. 300. If ^ + 5 + C = 180", and if y' +: + f- sin = 0, A cos {a-.p) + £ (cos a + cos ;8) + C (sin a + sin /3) + Z> = j prove that Ll^ + £» + C"'=2^i?, unless sin^ sin^lZ^ sia^=.0. 310. Having given the equation A cos {fi-y) + B cos (j8 +y) + C sin (/3 + y) + .4' (cos j8 + cos y) + B' (sin |3 + sin y) + C = 0, PLANE TRIGONOMETRY. 67 imd tlie two like equations between y, a, and a, j8 respectively; prove that if be finite. • P — y ■ y-a . a- 3 sin — ^ sm '— — sin -~- •i 2 A B' + G'-A'-A"-B"' + 2AC'=0. 311. Eliminate ^ from the equations X cos 6 y sinO . . . „ »..^,. ^ +^-^=1, a!sin(9-ycos^ = ^(a'sin'(9 + 6''cos'e). 312. Having given the equations P+y+e y+a+e a+^+O, m tan ^ tana=tan-f — tany8 = tan ^5 tany = — j prove that sin (y8 + y) + sin (y + a) + sin (a + ;8) = " cos P cos y sin ^ sin y _ 1 _ a, /3, y being unequal^ n m m—n ' and < tt. &c. / 313. Prove that cos^ 6 — 2 cos 6 cos a cos (a + 6) + cos" (a + &) is independent of 6. 314. Prove that sin 2a cos;8 cos y sin (fi — y)+ sin 2;S cos y cos a sin (y — a) + sin 2y cos a cos j3 sin (a — 13) = 0, cos 2a cos/3 cosy sin(;8-y) + + E sin (/8 — y) sin (y - a) sin (a - j8). 315. Prove that »f . ,561 . ,35) |sin'--sin»^| s cos' 6 + cos' 29 + cos' 36-3 cos0 cos 28 cos 36, , „ . 36f .36 ,56'1 and 2 sm -„- -j cos -„- - cos -^ > = sin' 6 + sin' 26 + sin' 36-3 sin 6 sin 26 sin 36. 5—2 68 BOOK OF MATHEMATICAL PBOBLEMS. 316. Prove that sin 2a sin ()8 - y) + sin 2/3 sin (y - a) + sin 2y sin (a - ^) = {sin (y - /3) + sin (a - y) + sin-(/3 - a)} {sin (/3 + y) + sin (y + o) + sin (a + /8)}, and cos 2a sin (^ - y) + cos 2/3 sin (y - a) + cos 2y sin (a - ^) = {sin (y - /3) + sin (a - y) + sin (/8 - a)} {cos (/3 + y) + cos (y + a) + cos (a + ;8)}. 317. Prove that sin 3a sin (P--y) + sin 3/8 sin (y — a) + sin 3y sin (a - /3) '' ' Cos^3a sin (;8 - y) + cos 3^ sin (y - a) + cos 3y sin (a - fi) = tan (a + ;8 + y), sin 5a sin (/3 - y) + sin 5/8 sin (y - a) + sin 5y sin (a - /3) ^ ^ cos 5a sin (/i - y) + cos 5/8 sin (y — a) + cos 5y sin (a - jS) _ sin (3a + j8 + y)+ ... + ... ~ cos(3a + j8 + y)+ ... + ...' and that .„, sin 7a sin (j8 - 7) + sin 7;8 sin (y — a") + sin 7y sin (a — /8) ^ ' cos 7a sin (/8 — y) + cos 7/3 sin (y - a) + cos 7y sin (a - /}) _sin (a + 3/3 + 3y) + ...+...+ sin (5a + /3+y)4-... +... ~ cos (a + 3j8 + 3y) + . . . + . . . + cos (5a + /3 + y) + ...+... ' 318. Prove that cos' a sin (/3 — y) + cos' /3 sin (y — a) + cos' y sin (a — /8) sin" a sin (/i - y) + sin" /3 sin (y — a) + sin" y sin {a — ^) + cot (a + j8 + y) = 0. 319. "Prove that J 1 1 _sinwa+asin(w-l)a 2 cos a — 2 cos a — 2 cos a — ... — 2 cos a + a sin(w+ l)a +a sinwa ' there being n quotients in the left hand member. PLANE TEIGONOMETRY. 69 320. From the identity {x-h){x-c) (x-h)(x-a) , (a!-a)(a;-&) _ „ {a-b){a-c} {b-c){b-a) {c~a){c-b) " ' deduce the identities „ , . , sin (^ - a) sin (5 - y) ,. cos 2 (^ + a) -^ ' . -. +... + ...= cos i9, ^ ' sm (a — p) sin (a — -f) . „,^ , sin (61 -/3) sin (61 -y) . ,„ sm2(e + a)^-^ ^' . ) 4+... + ... =sm4A ^ ' sm (a — p) sin (a — y) 321. Prove the identity COS (/3 + y - a - 8) sin ^-^ sin — ^ .7 — a.j8— 8 + COS (y + a - (8 - 8) sm ^-g— sin — 2" ., .a — /8.y— S + cos (a + /3 — y - 6) sia sm ^— fl_y . Y_a . a-;8 . 8-a . 8-^ . S-y = 8 sintl— -' sin ^-— sm -— - sm -^ sm -— - sm -—- . 2i Z ^ ^ ^ ^ III. Inequalities, 322. Prove that cot ^ > 1 + cot 6 for values of $ between and IT. -r. . ,. • . „ x' — 2x cos a + 1 , . 323. Prove that, for real values 01 x, -» — 7: 5 — r lies ' a; — 2a3 cos /i + 1 1 — cos a J 1 + cos a between ^j 5 , and = -: . l-cos;8 1+cos/i 324. If X, y, zhe any real quantities, and A, B, G the angles of a triangle, gf + y' + g^> 2yz cos ^ + 2sa; cos -S + 2a!y cos G, X _ y «_ sin- "^^^'^^ dO-ihr:e~^i^'' 70 BOOK OP MATHEMATICAL PEOBLEMS. 325. Having given the equation sec )8 sec y + tan /8 tan y = tan a, prove that, for real values of ^ and y, cos 2a must be negative; and that tan /3 + tan a tan y cos yS _ . tan y + tan a tan /i cos y 32&. li A+B + G=\ 80", prove that > cos A- cos B cos C, sin -jr sin -^ sin „ > (1 — cos A) (1 — cos jB) (1 ^ cos C) ^ ^ ^ unless A = B=C. Also ' sin A sxa.B miG > sin 2^ sin 25 sin 2C. 327. Prove that f co3'(a-e) sin'(a-e) ) f cos°(a + e) sin'(a + 6l)| sin°2 a and that the two cannot be equal unless tan' o lie between -= alid ^ . 328. If tan a tan )8 tan y = 1, a, ^, y being angles between IT . . 1 and ^ , sm a sin ^ sin y < ^ — ^ , unless o = /8 = y. 329. If a + -4, P + B, y + G be the angles subtended at any point by the sides of a triangle ABC, sin'' a sin' /3 sin* y sin a sin ;8 sin y im7 "^ sin 5''" sin C^ ^ . A . B . C 2 sin jr- sin — sin jr- J ^ J unless the point be the centre of the inscribed circle. PLANE TKIGONOMETEY. 71 IV. Properties of Triangles. In these questions a, b, c denote the sides, A, B, G the re- spectively opposite angles of any triangle; B the radius of the circumscribed, circle, r, r^, r^, r^ the radii of the inscribed circle and of the escribed circles respectively opposite A, B, C. 330. If 0, ^, ^ . ^. B^ G and tan _ tan ^ tan i = =fc tan -^ tan -^ tan ^ . AAA AAA 331. If sin j4, sin-B, sin C be in harmonical progression, so also are 1 — cos A, 1 — cos B, \ — cos G. 332. Prove that sin A sin {A - B) sin {A-G) + sin B sin {B - C) sin {B - A) + sin G sin {G — A) sin (G—B) = sin A sin B sin G -sin2^sin2.Ssin2C. 333. From the tliree relations between the sides and angles given in the form a' = 6' + c' - 2Sc cos A, &c., deduce the equations sin A sin B sin G a ^ b c assuming that each angle is < 180°; , 4 + 5 + (7=180"; 72 BOOK OP MATHEMATICAL PROBLEMS. 334. In the side BG, 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, C are all acute and tan ^ < tan -B + tan C, or when B ov C is obtuse. These conditions being satisfied, prove that there are in geiieral two such points which lie both between B and G, one between and one beyond, or both beyond, according as A is the greatest, the mean, or the least angle of the triangle. 335. P, Q are two points on the circumscribed circle, the distance of either from A being a mean proportional between its distances from B and G ; prove that aBAP~iGAQ = ^^. 336. The line joining the middle points of BG and of the perpendicular from A on BG makes with BG an angle cot"' (cot B ~ cot C). 337. The line joining the centres of the inscribed and cir- cumscribed circles makes with BG an angle sin B ~ sin G , ( sin 5 ~ sin G \ \cos5 + cos(7— ij ' 338. The line joining the centre of the circumscribed circle and the centre of perpendiculars makes with BG an angle , f tan5~tan<7 ) \tan^tanC-3j" 339. The perpendicular from A on BG is a harmonic mean between r^ and r.^. 340. If be the centre of the circumscribed circle and AO meet BG in B, DO : AO :: cos^ ; cos(5-C). PLANE TEIGONOMETEY. 73 341. The perimeter of a triangle : perimeter of the inscribed circle : : the area of the triangle : area of the circle ^^ ^^ .G : : cot -=- cot -^ cot -jr- : tt. ja Ji Z 342. A triangle is formed by joining the feet of the per- pendiculars of the triangle ABC; and the circle inscribed in this triangle touches the sides in A!, F, C ; prove that FC G'A' A'£' „ , „ ^ -d77 = -7TT — 'T~D = ^ COS A COS B COS C. 343. A circle is drawn to touch the circumscribed circle and A the sides A£, AG ; prove that its radius is rsec^-^ : and if it touch the circumscribed circle and the sides AB, AC produced, its A 7 radius is r^ sec^ -^. I£B=G and the latter radius = Ii, coa A = . 344. Prove the formulse, 6' sin 2C - 26c sin (B-G)- c" sin 2JS = 0, 6' cos 2C+ 26c cos {B-G) + c' cos 25 = a\ 345. Having given y sin^ G + z sin^ B = z s,WA + x sin^ G = x sin^ B + y sin^ A ; X y s prove that sin 24 sin 25 sin2C' 346. 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 of the vertical angle, prove that cot ^, + cot 0^ — — r-j — . G SIJlL CL Prove that only one of these values corresponds to a true solution ; and, if this be 6^ , that $, Jh' + c^ sin^ a - h *^"ir = ^ — M ^ — • z c (1 —cos a) 74 BOOK OP MATHEMATICAL PROBLEMS. 347. 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 suqh triangle exists it 2m sin -^ > a. 348. A triangle A'B'G' has its angles respectively supple- mentary to the half angles of the triangle ABC; and its side B'C equal to BG ; prove that aA'FG' ®"^ 2 ^ABG - ^ . B . G- 2 sm -^ sin -jr- 349. If aj, y, z be perpendiculars from the angular points on any straight line ; prove that a? {x-y){x- z)\-V {y -z){y -x)^ c^ {z-x){z^y) = (^ i.ABG)\ any perpendicular being reckoned negative which is drawn in the opposite direction to the other two. 350. If p^, p^, p^ be the perpendiculars of the triangle, 1111 cos A cos B cos G 1 Pi P2 Pz »• fx P2 Pz ^ 351. The distances between the centres of the escribed circles being respectively a, p, y ; prove that ;8^ / 4J? = 2 7 ^ Li ""■■• to (w + 1) terms, 384. If n be an odd integer, , w-1 »i-l 2m-1 1 + cos B + ; — cos 25 + ... to 00 n n 2n n-1 = («-!) Within what limits of 6 is this true ? ,. ' PLANE TBIGONOMETET. 81 385. Prom the equivalence log (1 +.xf^^^) + log (1 - a!«*^^) = log {1 - SB (a! + 2 J^ sin 6)}, obtain the expansions of cos 2n6, and sin (2n + 1)9 in terms of sin 0. 386. ■ From the equivalence £ - 2 cos 6 + £ = 4 sm ^ sin ^ , resolve the former into its real quadratic factors. 387. If tan (a + P'J^^) = J^, «, P l^eing real, then will u, be indeterminate and j8 infinite. 388. If cos (a + ^ n/ -^) = cos ^ + 7^ sin <^, where a, /?, ^ are real; then will sin^ = ±sin'a. Find also the relation be- tween a and ;8. 389. If tan (a + )8 J^) = cos ^ + ^ - 1 sin ^, a, ^, , K. - 1 cos 35 1.3 cos 55 (15) cos5 + ^-g— +2y|— g— +...tooo, /ic\ a • a. 2 a sin 25 ..sin 35 . (16) cos5 sin5+coa 5— s— ri-cos'fl — 5-- + ... toooj .a O /i»T\ 0/1 /I cos''25 cos 35 1.3 ,„.cos55 (17) cos25cos5 + — ^ — — 5 — +.s— rcos'25 — = — +...tooo. ^' 2 32.4 5 399. Prove that sec 5 + sec(^ + 5) + sec(^ + 5) + ... +3ec/2(«i-l)^ + 5l is equal' to 0, or to (— 1) ^ rn, sec m5, according as m is even or odd : also that sec (9 + sec=f^ + 5) + ...+sec'|2(»i-l)^ + 5| JW" is equal to -^ , or to m'sec^mO, according as » is 1 _(_1)^ cos m5 even or odd. . , , . , 400. Prove that - - n sin n4> -^ sitK^ _^ , sin<^ cos»»<^-cos«5 cos .^- cos 5 cos.^-cos(- +5) ' '" sin^ , sin^ cos4,-coa(— + 0\ cos.^-eosi2(M-l)^ + 5J 8^ BOOK 01' MATHEMATICAL >EdiBLEMS. VIII. Mit cos 401. Qie ambiguities in the equations A . A „. . .. A . A g- + sin2- = ±^(l+sin-(l), cos -g- - sin -s- = ± ^(1 -sinvl) may be replaced by (— l)", (— 1)", wbere m, n denote the greatest . . . ^ + 90° .. , ^-+270° ^. , ^ ^ integers in -ggQO- > and g^^, respectively. 402. 1£ a + b (tan 6 + tan ^) + c tan 6 tan = 0, prove that 1 1 a cos" ^ +-26sin 6 cqs^+csin'^ ■ a cos' ^ + 26 sin ^ cos^ + trsin'i^ , . a+ c is equal to =^. ^ ao—o 403.- AB is a £xed straight line, C a point in it such that SAG = A£, and F a point bet>f een- JB and :. if GF^ GB sin ^, then -vrill BP . ^P' ocl + sin ZO. Hence prgve that £P ^AF' is greatest -when F bisects BG: 404. .Find x from the equation ^ tan"' - + tan"' -= =■ = tan"' as n'-x+l n-l' and find the tangent of the aiigle tan"' ^ + 3 tan"' = + tan"' ^ - f . o 7 26 4 405. Three parallel straight lines are drawn through the angular points of a triangle ABG to meet the opposite sides in D,E,F; prove that I>B.J)G EG.E'A FA.FB DA' '*' EB" "*■ FG" ~ ' the segments of a side being affected "with opposite signs when they fall on opposite sides of the point of section. PLANE 'TEIGON6meTET. 87 406. Reduce to its simplest form {x cos 2a+y sin 2a-l) {x oos2j8+2/siii 2;8-l)-{a!cos {a+^)+y siii(a+;8)-cos (a-;8)}' sin" (a-;8) 407. Prove that (1) tan-l=taii-'^ + tan-jl„ (2) tan-^=tau-jA_+tan-'A, (3) = tan- ^ - tan- ^ - tan"' 1 4- ten- ±-^, (4) -7=tan"' 7 +tan-' 5 +tan~'=+2tan"'TlI+^aIl-'T^r- V 4 4 5 7 : 13 21, = tan"' TT + tan"' ^ + tan"' -^ + tan"' jr — tan"' -ttt o o o 43 = 3 tan"' T + tan"' KTT + tan"' - 4 ^ 20 1985 * 408. Prove tliat ' sin 4a sin (y — j8) + sin 4/8 sin (a — y) + sin 4y sin (;ff — o) sin (j8 - y) + sin (y — a) + sin (a — j8) ; =2Ssin(2a + ^ + y) + 2sin2(/3 + y) + 2sin(3j8 + y). 409. Prove that ,,,. . no? n (5x — 4) . . smsm... sm a; = a; - -j-g- + .. - x li Li -g(175M'-336M + 164)Y+- the sine being taken n times. 410. If P be any point on the Nine Points' Circle of the triangle ABC, FA' sin A nos {B-C) + PE' smBcos{G-A)^ PC sin Ccos (C- A) , . =.B'(4sinil 8in5 sinj(7 + sin2jl sin.2^sia2C), ( 88 •) CONIC SECTIONS, GEOMETRICAL. I. Pp,rahola. •411. Two parabolas having the same focus intersect ; prove that the angles between, their tangents at the points of intersec- . tion are either equal or supplementary. 412. A chord PQ of a "parabola is a normal &% P and sub- tends a right angle at the focus S; prpve that SQ is twice SP. 413. A chord PQ of a parabola normal at P subtends a right angle at the vertex j prove that SQ is three times SP. 414. 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 with the parabola = 120°. 415. Two parabolas have a common focus and their axes in opposite directions ; prove that if a circle be drawn through the focus touching both the parabolas the line joining the points, of contact subtends at the focus an angle of 60°. 416. In a parabola AQ is drawn through the vertex A at right angles to a chord ^P to meet the diameter through P mQ; prove that Q lies on a fixed straight line. 417. Through any point P of a parabola a straight line QPQf is drawn perpendicular to the axis and terminated by the tangents at the- extremities of the latus rectum ; prove that the distance of P from the latus rectum is a mean proportioned be- tween QP, pq. ' 418. The locus of a point, dividing in a given ratio, a chord of a parabola which is parallel to a given line is a parabola. ; 419. Prom any point on the tangent at any point of a parar bola perpendiculars are let fall on the focal distance and on the CONIC gECtlONS, GEOMETtelCAL. 89* axis ; prove that the sum, or the difference, of the focal distances of the feet of these perpendiculars is equal to half the latus rectum. 420. T-wo points are 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 nor- mals; prove that the difference of the normals is equal to the latus rectum. 421. ST is the perpendicular from the focus of a parabola oi^ any tangent, a straight line is drawn through Z" parallel to the axis to meet in § a straight line through S at right angles to ST I prove that the locus of ^ is a parabola. 422. At one extremity of a given finite straight line is drawn any circle touching the line, and from th« other extremity is drawn a tangent to the circle ; prove that the point of intersec- tion of this tangent with the tangent parallel to the given straight line lies on a fixed parabola. 423. Two parabolas have a common focus; from any point on their common tangent are drawn the other tangents to the' two ; prove that the distances of these from the common focus are in a constant ratio. 424. Two tangents are drarwn to a parabola making equa^ angles with a given straight line ; prove that their poiut of inter- section lies on a fixed straight line passing through the focus. 425. Two parabolas have their axes -parallel and two parallel tangents, are drawn to them ; prove that the straight line joining the points of contact passes through a fixed point. 426. Two parabolas have a common focus S^ parallel tangents drawn to them at P, Q meet their common tangent in i*", Qf ; prove that the angle PSQ is equal to the angle between the axes of the parabolas, and the angle P'SQ' supplementary. 9Q BOQK OP MATHEMATICAL PEOBLEMS. 42T. If on a tangent to a parabola be taken two points equi- distant from the focus, the two other tangents drawn to the para- bola from these points will intersect on the axis, 428. A. circle is described on the latus rectum of a parabola as diameter, and a straight line drawn through the focus meets the curves in P, Q ; prove that the tangents at P, Q intersect pither on the latus rectum, or on a straight line parallel to the latus rectum at a distance from it equal to the latus rectum. 429. A chord of a parabola is drawn parallel to a given straight line and on this chord as diameter a circle is described ; prove that the distance between the middle points of this chord &,nd of the chord joining the other two points of intersection of the circle and parabola "is of constant length. .; 430.' On any chord' of a parabola as diameter is described a circle cuttiag the parabola again in two points ; if these points be joined the portion of the axis of the parabola intercepted between the two chords is, equal to the latus rectum. 431. 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. 432. Circles are described having for diameters a series of parallel chords of a given parabola; proye that they will all touch another parabola related to the given one in the manner described in the last question. 433. The locus of the centre of the circle circumscribing the triangle formed by two fixed tangents to a parabola and any other tangent is a straight line. 434. Two equal parabolas, A and B, have a common vertex and axes in the -same straight line ; prove that the locus of the poles with respect to B of tangents to ^4 is ^. 435. Three common tangents PP', QQ', BR are drawn to two parabolas and PQ, P'Q intersect in Z ; prove that LB, LE are parallel to the axes of the two parabolas, , . ' CONIC SECtldNS, GEOMETEICAL. 9l 436. Two equal parabolas have a common focus and axes opposite j two circles are described touching each other, each with its centre on one parabola and touching the tangent at the vertex of that parabola: prove that the rectangle under their radii is constant whether the circles touch internally or externally, but in the former case is four times as great as in the latter. 437. 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 twdi points : prove tha,t the circle on this chord as diameter will touch the parabola of which this is the chord. ; 438. Two equal parabolas have their axes parallel and oppo* site, and one passes through the centre of curvature at the vertex of the other; prove that this relation is reciprocal and that the parabolas intersect at right angles. 439. PP' is any chord of a parabola, PM, P'M' are drawn perpendicular to the tangent at the vertex; prove that the circle on MM' as diameter, and the circle of curvature at the vertex wiU have PP" for their radical axis. 440. A parabola touches the sides of a triangle ABC in A', S, C, B'C meets BG in P, another parabola is drawn touching the sides and P is its point of contact with BO; prove that its axis is parallel to SC. 441. The directrix of a parabola and one point of the curve being given ; prove that the parabola will touch a fixed parabola to which the given straight line is the tangent at the vertex. 442. If a triangle be self-conjugate to a parabola, the lines joining the middle point of its sides will touch the parabola ; and ihe lines joining any angular point of the triangle to the point of contact of the corresponding tangent will be parallel to the axis. 443. If a complete quadrilateral be formed by four tangents to a parabola, the common radical axis of the three circles on the diagonals as diameters will be the directrix of the parabola. 92 BOOK: OP MATHEMATICAL PEOBLEMS. 444. A circle and parabola meet in four points and tangents are drawn to the parabola at these points; prove that the axis of the parabola will bisect the three diagonals of the quadrilateral formed by these tangents. II. Central Conies, 445. If ST be perpendicular on the tangent, SZ on the pormal, S being the focus, Y£i will pass through the centre. 446. A common tangent is drawn to a conic and to the circle whose diameter is a latus rectum; prove that the latus rectum, bisects the angle between the focal distances of the points of contact. 447. A perpendicular from the centre on the tangent meets the focal distances of the point' of contact in two points ; prove that either of these points is at a constant distance from the feet of the perpendiculars from the foci on the tangent. 448. The tangent at a point P meets the major axis in T; prove that SP : ST ■.: AN : AT, JHf hemg the foot of the ordinate and J. the nearer vertex. 449. The ciffcle 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 extremity of the major axis by the distance between the feet of the perpendiculars is equal or supplementary to the angl'e which either focal distance makes with the corresponding perpendicular. 400. A series of conies having a common focus S and major axes equal and in the same straight line will all touch the two parabolas having the same focus S, aiid latus rectum a line coinci- dent with the major axes in direction and of double the length. 451. A conic is described harving the same focus as a parabola- and major axis coincident in. direction with, the latus rectum of . CONIC SECTIONS, GEOMETEICAL. 93 the parabola and equal to half the latus rectum; prove that it will touch the parabola. 452. In a conic PG is the normal at P, S the focus ; prove that if SG = PG, SP is equal to the latus rectum. ■453. CPQ is a common radius to the circles on the miaor and major axes of an ellipse, and tangents to the circles at P, Q meet these axes in U, T ; prove that TV will touch the ellipse. 454. S, S' are the foci of a conic, SPY, S'P'T' perpendicu- lars on a tangent to the auxiliary circle meeting the conic in P, P' ; prove that the rectangle ST, (S'jP'=the rectangle S'Y', SP = BG\ 455. Given the foci and the length of the major axis; obtain by a geometrical construction the points in which the conic meets a given straight line drawn through one of the foci. 456. A tangent to a conic at P meets the minor axis in. T, and TQ is drawn perpendicular to SP one of the focal distances ; prove that SQ is of constant length : and, PM being drawn perpen- dicular to the minor axis, that QM will pass through a fixed point. 457. One focus of a conic, a tangent line, and the length of the major axis is given; prove that the locus of the second focus is a circle. Determine the portions of the loeus which correspond; to an ellipse, and to a hyperbola of which the given point is an interior focus to the branch touched by the given straight line. 458. PCG is a diameter of a conic, Q7^ a parallel chord bisected in Y, P7 intersects CQ, or C§' in R ; prove that the locus of ^ is a parabola, 459. If CP, CD be conjugate radii of an ellipse, and if through C a straight line be drawn parallel to either focal distance of P,^ the distance of D from this straight line will be equal to half the minor axis. ■ . 94 BOOK OF MATHEMATICAL PEOBLEMS. 460. If three; tangents to a conic be such that their points of intersection are at equal distances from one, of the foci, each distance will be equal to the major axis : and the second focus will be the centre of perpendiculars of the triangle formed by the tangents. 461. A straight line is drawn touching the circle on the minor axis of an ellipse, meeting the ellipse in P, and the direotof circle of the ellipse in Q, Q'; prove that the focal distances of P are equal to QP, Q'P. 462. A conic is inscribed in a triangle and is concentric with the Nine Points' Circle ; prove that it wiU have double contact with the Nine Points' Circle. 463. EFis a chord of a circle, S is its middle point; con- struct a conic of which M is one point, S one focus, and the given; circle the circle of curvature at JE, , 464. If P-be a point on an ellipse equidistant from the minpr axis and from one of the directrices, the circle of curvature at P, will pass through one of the foci. 465. If S be the focus of a conic, K the foot of the directrix, Q a point on the tangent at P, QB, QR perpendiculars on SP, jSZ" respectively j then. will SB bear to KB! & constant ratio. 466. If an equilateral triangle PQR be inscribed in the auxiliary circle of an ellipse, and P, Q, S be the corresponding points on the ellipse, the circles of curvature at P', Qf, B' meet in one point lying on the ellipse and on the circle circumscribing 467. Prom a point on an ellipse perpendiculars are drawn to the axes and produced to meet the circles on these axes re^ spectively ; prove that the line joining the points of intersection passes through the centre. ,468. ^P, TQ are tangents to a conic, Qq, Pp chords parallel; to TP, TQ respectively; prove that pq is parallel to PQ,- :-. i . CONIC SECTIONS, GEOMETRICAL. 95 469. QQf is a chord of an ellipse parallel to one of the equi- conjugate diameters, QR, QN' are drawn perpendicular to the major axis; prove that the triangles. $(7iT, ^CiV^' are equal j also that the normals at QQ' intersect on the diameter which is per- pendicular to the other equi-conjugate. 470. Any ordinate NP 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; RK, RL are drawn perpendicular to the axes; prove that E, P, L lie on one straight line and that EP, PL are equal respectively to the semi axes, 471. Two conies are described having a common minor axis, and such that the outer touches the directrices of the inner ; MPP' is a common ordinate; prove that MP" is equal to the normal at P. 472. QPP" drawn perpendicular to- the major axis of an ellipse meets the ellipse in P, P' and the auxiliary circle in Q ; prove that the part of the normal to the circle at Q intercepted between the normals to the ellipse at P and P' is equal to the minor axis. 473. The perpendicular from the focus of a conic on any tangent and the central radius to the point of contact will in- tersect on the directrix. 474. On the normal to an ellipse at P are taken two points Q, Q', such that QP=QP = CD ; prove that the cosine of the angle QGQ is ^q2_^q2 - 4:75. A hyperbola is described through the focus of a para- bola and with its foci lying on the parabola; prove that one of its asymptotes is parallel to the axis of the parabola. 476; A parabola passes through two given points and its axis is parallel to a given line ; prove that the locus of its focus is a hyperbola. ' 96 BOOK OF MATHEMATICAL PROBLEMS. 477. If two tangents of a hyperbola be tbe asymptotes of another hyperbola and that other touch one of the asymptotes of the former it ■will touch both. 478. Two similar conies A, B are placed with their major axes in the same straight liiie, and the focus of A is the centre of B ; if a common tangent be drawn the focal distance of its point of contact with A will be equal to th« semi major axis of B. 479. A series of similar conies are described having the same focus and direction of major axis, and tangents are draw.n to them at points where they meet a fix;ed circle having its centre at the common focus ; prove that these tangents will aU touch a similar fixed conic whose major axis is a diameter of the circle. 480. If a chord of a conic subtend a right angle at each of the foci, it must be either parallel to the major axis or a diameter. 481. Trom the foci S, S' of an ellipse perpendiculars SY, S'T are let fall on any tangent j prove that the perimeter of the quadrilateral SYT'S' will be the greatest possible when YY' subtends a right angle at the centre. 482. The angle which a diameter of an ellipse subtends at the extremity of the axis major is supplementary to that which its conjugate subtends at the extremity of the axis minor. 483. From the focus of an ellipse is drawn a straight line perpendicular to the tangent at a point of the auxiliary circle ; prove that this perpendicular is equal to the focal distance of the corresponding point of the ellipse. 484. If on any tangent to a conic be taken two points equi- distant from one focus and subtending a right angle at the other focus ; their distance from the former focus is constant. 485. If a come be described having one side of a triangle for directrix, the opposite angle for centre, and the centre of perpen- diculars for focus ; the sides of the triangle which meet in the centre will be conjugate diameters. CONIC SECTIONS, GEOMETRICAL, 97 486. An ellipse is described touching two confocal ellipses, and having the same centre ; prove that the tangents to the two ellipses at two points of contact wiU be perpendicular to each other. 487. An ellipse is described having double contact with each of two confocal ellipses ; prove that the sum of the squares on its axes is constant. 488. If SY, SZ be perpendiculars from the focus S on two tangents drawn from T to a conic, the perpendicular from T on TZ will pass through the other focus. 489. The tangent to a conic at P meets the axes in T, t, and the central radius at right angles to CP in Q ; prove that QT bears to Qt a constant ratio. 490. The foot of the perpendicular from the focus of a conic on the tangent at the extremity of the farther latus rectum lies on the minor axis. 491. The tangents and normals drawn to a series of confocal conies at the extremities of their latera recta will touch two para- bolas having their foci at the given foci and touching each other at the centre. 492. Through a given point on a given conic are drawn two chords OP, OQ, equally inclined to a given straight line ; prove that PQ passes through a fixed point. 493. A chord PQ of a conic is normal at P, and a diameter LL' is drawn bisecting the chord ; prove that PQ makes equal angles with LP, L'P, and that LP ± L'P is constant. 494. A given finite straight line is an equi-conjugate dia- meter of an ellip.se; prove that the locus of its foci is a lem- niscate. 495. A parallelogram is inscribed in a conic, and from any point on this conic are drawn two straight lines eaoh parallel to w. 7 98 BOOK OP MATHEMATICAL PKOBLEMS. two sides of the parallelogram ; prove that tlie rectangle under the segments of these lines made by the parallelogram are in a constant ratio. 496. Any two central conies in the same plane have two con- jugate diameters of the one parallel respectively to two conjugate diameters of the other ; and in general no more. 497. In two similar and similarly situated ellipses are taken two parallel chords PP', QQ ; PQ, PQ' meet the two conies in B, S; K, S' respectively ; prove that RB', SS' are parallel to each other. Also QQ', BB', and PP", SS' intersect in points lying on a fixed straight line. 498. A circle is described touching the focal distances of any point on a given conic, and passing through a given point on the major axis; prove that it will meet the major axis in another fixed point.' The given point must be between the foci for a hyperbola, and beyond them for an ellipse'. 499. A circle described on the part of the tangenii at P intercepted between the tangents at the ends of the major axis meets the conic again in Q ; prove that the ordinate of Q is to the ordinate of P as the minor axis to the sum of the minor axis and the diameter conjugate to P. 500. If a conic be inscribed in a triangle JJBC and have its focus at ; and if the angles £00, CO A, AOB be denoted by A!, S, C; OAsmA _ OBsmS _ OOsmG . s.m(A-A)~ sin {F - B) ~ sin {G'~C)~ ™^-*°'" *^'®- With what convention wUl this be true if be a point without the triangle ? 501. OA, OB are tangents to a conic, a straight line is drawn meeting OA, OB in Q, Q, AB in B, and the conic in P, P; prove that QP.PQ' : QF.FQ' :: BP' : EF' ; and that, for a COKIC SECTIONS, GEOMETRICAL. 99 series of parallel straight lines the ratio QP.PQ' : RP^ is con- stant. 602. S, S' are foci of an ellipse whose minor axis is equal to SS', P any point on the ellipse, the centre of the circle circum- scribed to SPS' ; prove that the circle on OP as diameter will touch the major axis at the foot of the normal at P. 503. Through different points of a given straight line are drawn chords of a given conic, bisected respectively at the points; prove that they will touch a fixed parabola. 504. With a fixed point on a conic as focus is described a parabola touching any pair of conjugate diameters of the conic ; prove that this parabola will have a fixed tangent parallel to the tangent at 0, and that this tangent divides CO in the ratio CO" : G0'\ CO, Cff being conjugate radii. 505. Through a point are drawn two straight lines, each passing through the pole of the other with respect to a given conic ; any tangent to the conic meets them in P, Q ; prove that the other tangents drawn from P, Q to the conic intersect on the polar of 0. 506. A parabola is described having the focus *S^ of a given conic for its focus and touching the minor axis ; prove that a common tangent to the two curves will subtend a right angle at S, and that its point of contact with either conic lies on the directrix of the other. III. Rectwngula/r Hyperbola. 507. Ay B,C, D are four points on a rectangular hyperbola and BG is perpendicular to AD; prove that CA is perpendicular to BD and AB to CD. 608. The angle between two diameters of a rectangular hy- perbola is equal to the aaigle between the conjugate diameters. 7-2. 100 BOOK OP MATHEMATICAL PROBLEMS. 509. AA! is tte transverse axis, P any point on tlie curve, FK, PL are drawn at right angles to AP, A'P to meet the axis ; prove tha,t PK=A'P and PL = AP, and that the normal at P bisects KL. 510. The foci of an ellipse are the extremities of a diameter of a rectangular hyperbola ; prove that the tangent and normal to the ellipse at any one of the points where it meets the hyperbola are parallel to the asymptotes of the hyperbola. 511. On a series of parallel chords of a rectangular hyperbola as diameters are described a series of circles ; prove that they will have a common radical axis. 512. A circle and a rectangular hyperbola intersect in four points, two of which are the extremities of a diameter of the hyperbola ; prove that the other two will be the extremities of a diameter of the circle. . 513. Any chord of a rectangular hyperbola subtends at the extremities of any diameter angles which are either equal or sup- plementary : equal if the extremities of the chord be on the same branch and on the same side of the diameter, or on opposite branches and on opposite sides : otherwise supplementary. 514. AB is a chord of a circle and a diameter of a rect- angular hyperbola, P aia.j point on the circle ; PA, PB meet the hyperbola again in Q, B; prove that BQ, AR will intersect on the circle. 515. Two points are taken on a rectangular hyperbola and its conjugate, the tangents at which are at riglit angles to each other; prove that the central radii to the point are also at right an'glefe to each other. 516. GP, CQ are radii of a rectangular hyperbola, tangents at P, Q meet in ^and intersect GQ, CP respectively in P'jQ'; prove that a circle can be described about CFTQ'. '' ,; CONIC SECTIONS, GEOMETRICAL. 101 517. A paraUelogi-am has its angular points on the arc of a rectangular hyperbola and from any point on the hyperbola are drawn two straight lines parallel to the sides ; prove that the four points in which these straight lines meet the sides of the paralle- logram lie on a circle. 518. The tangent at a point P of a rectangular hyperbola meets a diameter QGQ in T ; prove that CQ, TQf subtend equal angles at P. 519. Through any point on a rectangular hyperbola are drawn two chords at right angles to each other; jirove that the circle passing through the point and bisecting the chords will pass through the centre. 520. If PG be a fixed diameter and Q any point on the curve, the angles QPG, QGP will differ by a constant angle. 521. A, B are two fixed points, P a point such that AP, BP make equal angles with a given straight line ; prove that the locus of P is a rectangular hyperbola. ' 522. QR is any chord of a rectangular hyperbola, CP a radius perpendicular to it ; prove that the distance of P from either asymptote is a mean proportional between the distances of Q, R from the other. 523. Two circles touch the same branch of a rectangular hyperbola in the points P, Q, and touch each other at the centre G ; prove that the angle PGQ = 60°. 524. On opposite sides of any chord of a rectangular hyper- bola are described equal segments of circles ; prove that the four points in which the completed circles again meet the hyperbola are the angular points of a parallelogram. 525. A circle and rectangular hyperbola intersect in four points ; prove that the diameter of the hyperbola which is per- pendicular to the chord joining any two of the points will bisect the chord joining the other two. 102 BOOK OP MATHEMATICAL PEOBLEMS. 526. PJ*' is a diameter of a rectangular hyperbola, Q^, BK two ordinates to it on opposite brandies ; prove that a common tangent to the circles whose diameters are QQ', MR will subtend a right angle at P and at P'. 527. Circles are drawn through two given points, and diame- ters drawn parallel to a given straight line ; prove that the locus of the extremities of these diameters is a reotaiDgular hyperbola which asymptotes make equal angles with the line of centres ) — {ax' + } separate into factors," the corresponding transformed expression must also separate into factors. Hence the two equations — c{h — a){h — b) + 2a'b' (h cos -cy = Q, and -c{h-A){h-£)+ = 0, must coincide : and thus all the invariants may be deduced. One form of the equation of the circle may be mentioned : if (a;,, y,), {x^, y^ be the extremities of a diameter, the equation of the circle ia {x-x;){x-x^) + {y-y^){y-y^) = 0. 530. The equation of the two straight lines which pass through the origin and make an angle a with the straight line x + y = is aj" + 2xy sec 2o + y' = 0. CONIC SECTIONS, ANALYTICAL. 105 531. If the straight lines represented by the equation a;' (tan" 4> + cos' = make angles a, /S with the axis of x, tan a ~ tan ^=2. 532. Form the equation of the straight lines joining the origin to the points given by the equations {x-Kf + {y-kY=c\ hx + hy=2hk; and prove that they will be at right angles if A' + A' = c'. 533. The locus of the equation „ x'-l x'-l ^ 2 + 2+ ... to 00 is the part of two straight lines at right angles to each other which include one quadrant. 5M. If the formulae for effecting any transformation of co- ordinates be x = aX+lY+c, y = a'X+h'7+c', then will {ah - ab) {aV - a'h) = lb' - aa. 535. The expression aaf + hy' + c + 2a'y + 2b'x + 2c'xy is transformed to AX' + BT' + G + 2A'X+ 2ST + 2G'X7, the origin being unaltered ; prove that a'" + 1" _ 2a'b' cos o, _ !"'+£"- 2A'F cos Q sin'o) ~ sin' (2 2ab'c'-aa:'-bh'' ^ 2A'B'C' -AA"-Bff' . sin" o> sin' il 0), 12 being the angles between the axes. 106 BOOK OP MATHEMATICAL PROBLEMS. 536. If ABC be an acute-angled tmngle, P any point in its plane ; the three circular loci PA'=^PB^ + PU', FB'=PC' + PA, PA' = PF' + PC' will have their radical centre at the centre of the circle ciroum- sciibing the triangle. 537. The radii of two circles are a, b, and the distance between their centres ,^{2 (a' + b')} ; prove that any common tangent subtends a right angle at the point bisecting the distance between their centres. 538. A certain point has the same polar with respect to each of two circles; prove that any common tangent subtends a right angle at that point. 539. AJB is a diameter of a circle, a point in a fixed straight line passing through A, from two tangents are drawn to the circle meeting the tangent at £ ia. P, Q ; prove that BP + BQ is constant. 540. AB is a diameter of a circle, a chord through A meets the tangent at B in P, and from any point in the chord produced are drawn two tangents to the circle ; prove that the lines joining A to the points of contact will meet the tangent at B in points equidistant from P. 541. Three circles A, B, G have a common radical axis, and from any point on C two tangents are drawn to A, B respect- ively ; prove that the ratio between the squares on these tangents is equal to the ratio between the distances of the centres of A, B from the centre of 0. 642. On two circles are taken two points such that the tangents drawn each from one point to the other circle are equal; prove that the points are equidistant from the radical axis. 543. There are two systems of circles such that any circle of one system cuts any circle of the other system at right angles > COISIIC SECTIONS, ANALYTICAL. 107 prove tliat tlie circles of either system have a common radical axis •which is the line of centres of the circles of the other system. 544. Given two circles, a tangent to one at P meets the polar of P with respect to the other in P' ; prove that the circle on PP' as diameter wiU pass through two fixed points^ which will be imaginary or real, according as the circles intersect in real or imaginary points. 545. One circle lies entirely within another, a tangent to the inner meets the outer in P, P ', and the radical axis va.Q: if aS^ be the internal vanishing circle which has the same radical axis, the ^. . PSP' SQP . ratio sin — = — : cos — = — is constant. 546. Prove that the equation {xcos(a + ^) +2/sin(a + ^)-acos(a-^)} {x cos (v + 8) + 2/ sin {y+h)-a cos (y - 8)} = {a; cos {a.+ y) + y sin (a + -y) - a cos (a - y)] {a; cos (|8 + 8) + 2/ sin (;8 + 8) -a cos (^-8)} is equivalent to the equation a^ + y^ = a^; and state the property of the circle expressed by the equation in this form. 547. The radii of two circles are B, r, the distance between tiieir centres is J{E'+-2r'), and r<2P; prove that an infinite number of triangles can be inscribed in the first, which are self- conjugate with respect to the second : and that an infinite number of triangles can be circumscribed to the second which are self- conjugate with respect to the first. 548. A triangle is inscribed in the circle x'+y' = B' and two of its sides touch the circle {x - Bf + y' = r^ i prove that the third side will touch the circle Prove that this circle coincides with the second if 8^ = ^^ ± 2Br. 108 BOQK OF MATHEMATICAL PEOBLEMS. II. Parabola referred to Us axis. The equation of the parabola being taken y' — iax, the co- ordinates of any point on it may be represented by (— ,, — J; and with this notation, the equation of the tangent is y = mx -i — ; of the normal my +x = 2a-\ — ^; and of the chord through two points (m^, m^, 2m^m^x — y(m^ + m^ + 2a = 0. The equation of the two tangents drawn through a point (X, T) is ( 7' - 4aX) (2/= - iax) = {Ty-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, F) be a point on the locus, the points on the parabola to which the normals are drawn from (X, Y) are given by the equation m''T+ m'iX- 2a)-a=0; and if m^, m^, m^ be the three roots of this equation, 2a-X . a also since two normals meet at right angles in the point, the pro- duct of two of the roots is — 1. Let then m^m^ = — 1. Then -a • 3a-X -T or the locus is the parabola T' = a{X-3a). Again, " The sides of a triangle touch a parabola, and two of its angular points lie on another parabola having the same axis, to. find the locus of the third. angular point." . CONIC SECTIONS, ANALYTICAL. 109 Let tlie equations of the parabolas be ^^ = 4ax, a m, and the equation of the locus is a \ a ) ' a parabola which coincides with the second if a = 4a'. 549. Two parabolas have a common vertex A and a common axis, an ordinate NFQ meets them, and a tangent at F meets the outer parabola in R, R ; AB, AR' meet the ordinate in L, M ; prove that NP, NQ are respebtively harmonic and geo- pietric means between NL, NM. 550. A triangle is inscribed in a parabola and another tri- angle similar and similarly situated circumscribes it ; prove that the sides of the former triangle are respectively four times the corresponding sides of the, latter. 110 BOOK OP MATHEMATICAL PKOBLEMS. 551. TP, TQ are two tangents to a parabola ; prove tliat tte perpendiculars let Ml from P, T, Q on any other tangent are in geometric progression. 552. 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^ on any other tangent j prove that PjPa = PiPi- 553. The distance of the middle point of any one of the three diagonals of a quadrilateral from the axis of the parabola which touches the sides is one fourth of the sum of the distances of the four points of contact from the axis. •554. Through the point T, where the tangent to a given parabola at P meets the axis, is drawn a straight line TQQ' meeting the parabola in Q, Q', and dividing the ordinate of P in a given ratio; prove that PQ, PQ will both touch a fixed parabola having the same vertex and axis as the given one. 5^^. Two equal parabolas have their axes in the same straight line, and from any point on the outer tangents are drawn to the inner ; prove that these tangents will intercept a constant length on the tangent at the vertex of the inner. 556. If p, q, r be the perpendiculars from the angular points of a triangle AJBG, whose sides touch a parabola, on the directrix, and X, y, z perpendiculars from the same points on any other tangent, p tan .4+2 tan B + r tan (7=0, , p tan A a tan B r tan C _ and ^-~ — - + + = 0, X y z 557. A tangent is drawn to the circle of curvature at the vertex of a parabola, and the ordinates of the points where it meets the parabola are y,,y„\ prove that = - , 2c being the latus rectum. CONIC SECTIONS, ANALYTICAL. Ill 558. OA, OS are two tangents to a parabola meeting the tangent at the vertex in F, Q ; prove that d>.PQ=^OA cos QPA = OB cos FQB. 569. Two parabolas have a common focus and direction, of axis, QPQ' a chord of the outer is bisected by the inner in P, PP' parallel to the axis meets the outer ia P' ; prove that QP is a mean proportional between the tangents drawn from P' to the inner. 560. The locus of the centre of the Nine Points' Circle of a triangle, formed by three tangents to a parabola, of which two are fixed, is a straight line. 661. Prove that the parabolas y^ - iax, y' + 2cy + iax = Sa", cut each other at right angles in two points, 562. Through each point of the straight line - + ^ = I is h k drawn a chord of the parabola y'^ = iax bisected in the point j prove that this chord touches the parabola {y-'2.a^ = %a(x-K). 563. Two equal parabolas have a common focus and axes in the same straight line ; 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 tan- gents and their chord of contact lie on the parab.ola to which the tangents are drawn. 564. The two parabolas y"^ = ax, y^ :^4:a{x + a) are so re- lated that if a normaFto the latter meet the former in P, P', and A be the vertex of the former, A P or AP' is perpendicular to the normal. 665. The. normals at three points of the parabola y' = icas meet in the point {h, k) ; prove that the equation of the circle through the 'three points is 2 (a;' + y") -2x(h + 2a) - % = 0. 112 BOOK OP MATHEMATICAL PEOBLEMS, 566. A straight line parallel to the directrix of a parabola meets the axis produced at a distance from the vertex equal to the latus rectum, and a point P on this straight line is joined to the vertex A hj & straight line meeting the directrix in Q ; with centre Q and radius QA is described a circle meeting the pa- rabola again in £; prove that FB will be normal to the parabola at B. 567. A chord of a parabola passes through the centre of curvature at the vertex ; prove that the normals to the parabola at the extremities of the chord intersect on the parabola. 568. Prom any point of a straight line drawn through the focus of a parabola, and making an angle a with the axis, three normals are drawn ; prove that the sum of the angles which they make with the axis exceeds a by a multiple of tt. 569. Normals are drawn at the extremities of any chord passing through a fixed point on the axis of a parabola ; prove that their point of intersection lies on a fixed parabola. 570. Two normals to a parabola meet at right angles, jand 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 will also be a normal. 571. Two equal parabolas have their axes coincident but their vertices separated by a distance equal to the latus rectum; through the centres of curvature at the vertices are drawn chords PQ, P'Q'i equally inclined in opposite directions to the axis, F, P being on the same side of the axis ; prove that (1) FQ', FQ are normals to the outer parabola ; (2) their intersection S lies on the inner parabola; (3) the normals to the inner' parabola at P, Q', R meet in a point which lieS on a third equal parabola. CONIC SECTIONS, ANALYTICAL. 113 572. From a point are drawn three normals OP, OQ, OR, and two tangents OL, OM, to a parabola ; prove that the latus OP .OQ. OR rectum = 4 - OL.OM 573. The normals to the parabola y" — iax at points P, Q, R, meet in the point (X, Y) ; prove that the co-ordinates of the centre of perpendiculars of the triangle PQR are X-Qa, -IT. 574. A circle and parabola intersect in four points A, B,C,D; AB, CD produced meet in P ; BG, AD in Q; both points being without the parabola ; and from any point on the parabola per- pendiculars are let fall on these lines j prove that the rectangle contained by the perpendiculars on the two former : that con- tained by the perpendiculars on the two latter : : 1 ± cos P : 1 =1= cos Q. 575. In the two parabolas y^ = 2c (a; ± c), a tangent drawn to one meets the other in two points, and on the part intercepted is described a circle ; prove that this circle will touch the second 576. 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 chord with respect to the parabola ; prove that each chord will touch a fixed parabola equal to the given one. 577. Two parabolas have a common focus S, and axes in the same straight line, and from a point P on the outer are drawn two tangents PQ, PQ to the inner ; prove that the ratio QPq PSA cos -y- : cos -j- is constant, A being the vertex of either parabola. 578. If a parabola circumscribe a triangle ABC, and its axis make with BG an angle 6 (measured from G^ towards GA), its 114 BOOK OP MATHEMATICAL PROBLEMS. latus rectum is 2Ii sin 6 sin (0 — 6) sin (£ + 0), R being the radius of the circumscribed circle of the triangle ; and if a parabola touch the sides of the triangle, its latus rectum is &R sin 6 sin (C - 6) sin {B + 6). 579. A triangle ABO is inscribed in a parabola, and the focus is the centre of perpendiculars of the triangle ; prove that (1 — cos ^) (1 — cos B) (1 — cos (7) = 2 cos A cos B cos G ; and that each side of the triangle will touch a fixed circle which passes through the focus, and whose diameter is equal to the latus rectum. 580. Through a fixed point within a parabola is drawn any chord PP', and the diameters through P and P' are drawn ; prove that there are two fixed straight lines perpendicular to the axis, the part of either of which intercepted between the dia- meters subtends a right angle at- 0. 581. 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 is parallel to the axis and bisected by the directrix. 582. A triangle is inscribed in a parabola, its sides are at distances x, y, a from the focus, and subtend angles 6, ^, ij/ at the focus ; prove that . o • J • I sin ^ + sin d) + sin i/r + 2 tan ^ tan ^ tan ^ sm 6 siVL *^ .—V, 122 BOOK OP MATHEMATICAL PROBLEMS. 597. In an ellipse whose axes are in the ratio jj2 + 1 : L a circle whose diameter joins the extremities of two conjugate dia- meters of the ellipse, will touch the ellipse. 598. Two circles have eaeh double contact with an ellipse and touch each other; prove that r, r' being the radii ; also that the point of contact of the two circles is equidistant from their two chords of contact with the ellipse. 599. Two ellipses have common foci S, S', and from a point P on the outer are drawn two tangents FQ, PQ to the inner; prove that cos — ^ — '> cos — ^ — is a constant ratio. A Z 600. The sides of a parallelogram circumscribing an ellipse are parallel to conjugate diameters; prove that the rectangle under the perpendiculars let fall from two opposite angles on any tangent is equal to the rectangle under those from the other two angles. 601. Prove that the equation ||cos{a-/8)+|sin(a-^)-l|gcos(a + ^) + |sin(a + ^)-lJ fa! y ■ „V = < - cos a + f sm d - cos fi / is true at any point of the ellipse a;" / , — + ^ = 1 • and hence that the locus of a point, from which if two tangents be drawn to the ellipse the centre of the circle inscribed in the Jriaingle formed by the two tangents and the chord of contact shall lie on the ellipse, is the curve CONIC SECTIONS, ANALYTICAL. 123 602. The three points whose eccentric angles are a, /S, y are the angular points of a triangle; prove that the co-ordinates of the centre of perpendiculars of this triangle are a' + b' . - . a° - 6' , „ , — = (cos a + cos p + cos y) = cos (a + p + y), a' + SV. . o . X w'-S" . , a s — nv— (sin a + sin ^ + sin y) ^r— sm (o + ^ + y) ; and "of the centre of the circle circumscribing the triangle are — -j {cos a + cos /8 + cos y + cos (a + yS + y)}, T9 2 , {sin a + sin /8 + sin y — sin (a + y8 + y)}. 46 Find the loci of each of these points -when the triangle is a maximum. 603. If a chord of a parabola be drawn through a fixed point on the axis, and an ellipse be described with the extremi- ties of this chord for foci, and passing through the focus of the parabola ; the minor axis of this ellipse will be constant. 604. Two tangents to a given circle intersect a constant length on a fixed tangent ; prove that the locus of their point of intersection is a conic which the given circle osculates at a vertex. 605. If a tangent be drawn to an ellipse, and with the point of contact as centre another ellipse be described similar and simUarly situated to the former, but of three times the area ; then if from any point of this latter ellipse two other tangents be drawn to the former, the triangle formed by the three tange^its will be double of the triangle formed by joining their points of contact, 606. TP TQ are tangents to an ellipse at points whose eccentric angles are a, P, another tangent meets them inF, Q' ; prove that _ PF . Q^ = TP' . TO! COS" —^ . 124 BOOK OP MATHEMATICAL PEOBLEMS. 607. If two sides of a triangle be given in position and the third in magnitude, 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 sides be 60°. If c be the given side, and 2a the given angle, the axes of the c sin 3a ccos 3a ellipse are equal to -. — ;— = , -. — ; 5— . ■^ ^ 4 sin a cos a 4 sin a cos a 608. If FM, FJV be perpendiculars from any point of an ellipse on the axes, and the tangent at P meet the equi-conjugates in ^, ^ j the tangents from Q, E to the ellipse will be parallel to MN. 609. If a right-angled maximum triangle can be inscribed in an ellipse, the eccentric angle of the point at which is the right angle, is 1 mT 1 a' + vy 610. GP, CD are conjugate radii, and FQ is measured along the normal at P equal to m times CJ) ; prove that the locus of Q is the ellipse PQ being measured inwards or outwards as m is positive or negative. 611. If a triangle circumscribe a given ellipse,, and its centre of gravity lie in the axis of aJ, at a distance c from the centre, its angular points will lie on the fixed conic , (sB - 3c)' y' (g' - 9c') a'V' - = 4. - 612. If two tangents be drawn to an ellipse from a point P, the cosine of the angle between them is CP'-AG'-£C.^ ' .. 2sp:s'f ' . CONIC SECTIONS, ANALYTICAL, 125 613. Two points H, H' are conjugate witli respect to an ellipse, P is any point on the ellipse, and PH, PH' meet the ellipse again in QQ' ; prove that Q^ passes through the pole of HE'. 614. A triangle circumscribes the circle a? + if' = a", and two of its angular points lie on the circle {x — cf + y' = h^ ; prove that the locus of the third angular point is a conic touching the common tangents of the two circles; and that this conic becomes a parabola if (c =t a)° = 6' — a^. 615. 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 ellipse, and that the peri- meter of the triangle is constant. 616. The lines T^B V 1 T ^ y 1 7* y 1 Z,- + m,f=l, ?„-+m^| = l, l^-+m^l = \, ^ a ' b 'a 'o "a ^o form a self-conjugate triangle to the ellipse ; prove that ^3 + '^2"*3 = y. + ™3™1 = hh + ^I'^a = 1' and that the co-ordinates of the centre of perpendiculars . of the triangle are —^Khk^ — J— '^i™a'"3- 617. A triangle is self-conjugate to a given ellipse, and one angular point is fixed ; prove that the circle circumscribing the triangle passes through another fixed point Q ; that G, Q, are in one straight line, and that GQ.CO = a^ + b\ 618. In the ellipses a tangent to the former meets the latter in P, ^ ; prove that the tangents at P, Q are at right angles to each other. 126 BOOK OP MATHEMATICAL PROBLEMS. 619. Two tangents OP, OQ are drawn at the points o, ^ ; prove that the co-ordinates of the centre of the circle circumscrib- ing the triangle OPQ are a + j8 cos ; cos 2 «' + (a' - h') cos g cos /S . a + fi sin 2 6= + (5" -a'), sin a sin /8 a-B 26 cos -2- If this point lie on the axis of x, the locus of is a circle. 620. Two points P, Q are taken on an ellipse, such that the perpendiculars from Q, P on the tangents at P, Q intersect on the ellipse ; prove that the locus of the pole of PQ is the ellipse If R be another point similarly related to P, the same relation win hold between Q, R : and the centre of perpendiculars of the triangle PQR will be the centre of the ellipse. 621. Perpendiculars jp,, p^ are let fall from extremities of two conjugate diameters on any tangent, and^^ is the perpendicular from the pole of the line joining the two former points : prove that ^3' = 2^,^,. 622. The normal at a point P of an ellipse meets the curve in Q, and any other chord PP' is drawn; QP' siaA. the line through P at right angles to PP' meet in R : prove that the locus of R is the straight line X , y . , a^ + b* -cos,^-^sm<^=^-^, being the eccentric angle of P. The part of any tangent inter- cepted between this straight line and the tangent at P will be divided by the point of contact into two parts which subtend equal or supplementary angles at P. ' CONIC SECTIONS, ANALYTICAL. 127 623. Two normals are drawn to an ellipse at the extremi- ties of a chord parallel to the tangent at the point a ; prove that the locus of their intersection is the curve 2 (ax sin a + 6y cos a) (ax cos a + by sin a) = (a^ — Vy sin 2a cos'' 2a. 624. The normals at three points of an ellipse whose eccen- tric angles, are a, j8, y will meet in a point if sin (/8 + y) + sin (y + a) + sin (a + P) = 0. 625. If four normals to an ellipse meet in a point, the sutn of the corresponding eccentric angles will be an odd multiple of IT. Prove also that two tangents drawn to the ellipse parallel to two chords passing through the four points will intersect on one of the equi-conjugate diameters. 626. If (a;,, yj, (x^, y^, (x^, y^) be three points on an ellipse, such that x^ + x^ + x^ = 0, 2/; + y^ + 2/^ = 0> ^^^ circles of curvature at these points will pass through a point on the ellipse, whose co-ordinates are = — , ■ ,„ . Of 627. If four normals be drawn from a given point to any one of a series of confocal ellipses, the sum of the angles made hj them with the axis is constant. 628. The normals to an ellipse at the points where it is met by the straight lines X y . o ^ ^ y -1 — cos a + T sin a = cos p, • + t — ; — - == 5 , ^ ° "■ ^ J I"' a cos a 6 sin a cos ^ will aU intersect in a point. 629. Pp is a diameter of an ellipse, PM, PN perpendiculars on the axes, MR produced meets the ellipse in Q, q; prove that the normals at Q, q, intersect in the centre of curvature at p, 630. If fi'om a point be drawn OP, OQ, OR, OS normals to an ellipse, and p, q, r, s be taken such that their co-ordinates are 128 BOOK OP MATHEMATICAL PROBLEMS. the intercepts on the- axes of the tangents at P, Q, R, S respect- ively ; p, q, r, s ■will lie on one straight line. Also, if through the centre C be drawn straight lines at right angles to CP, CQ, GR, OS to meet the tangents at P, Q, R, S respectively, the four points so determined will lie on one straight line. 631. If the normals to an ellipse at P, Q, R, S meet in a point, and circles described about QRS, RSP, SPQ, PQR meet the ellipse again in P', Q', R', S', the normals at P', Q\ R', S' will meet in a point. 632. PQ is a chord of an ellipse, normal at P, PP' a chord perpendicular to the axis, the tangent at P' meets the axes in T, T', the rectangle TCT'R is completed, and GR meets PQ in U ; prove that CR.GU = a'-h'. 633. Normals drawn to an ellipse at the extremity of a chord passing through a given point on the stxis will intersect in another ellipse whose axes are a \ cj \ c, the distance of the given point from the centre being c times the semi major axis. 634. A triangle A'B'C is inscribed in the ellipse -^ + ^ = 1, x" y" . and its sides touch the ellipse — + ,-2=1 in the points A, B, G ; prove that, if the relation which must exist between the axes be -; + -J, = 1, the eccentric angles of A and A' will differ by tt : but a if the relation be -; ~ 77 = 1 , the sum of these eccentric angles ah . ' will be either jr or 37r, 635. Two ellipses are concentric and similarly situated, and triangles can be inscribed and circumscribed ; prove that the normals to one ellipse at the angular points of any such triangle meet in a point ; and also the noriuals to the other at the points of contact. CONIC SECTIONS, ANALYTICAL. 129 636. The elUpses _2 + I,a ~ ■'j „6 + 16 ~ a= ¥ ;' a' b' (a' + by are so related that (1) an infinite number of triangles can be inscribed in the former whose sides shall touch the latter j (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 point. 637. The ellipses a'^b'-' "^= ¥~ {a'-b'f ^ "^'^ are such that the normals to the latter at the angular points of any inscribed triangle which circumscribes the former, meet on the latter. 638. Two ellipses are confocal, and are such that triangles can be inscribed in one whose sides touch the other ; prove that the perimeters of all such triangles are equal. 639. Triangles are circumscribed to an ellipse, such that the normal at each point of contact passes through the opposite angular point ; prove that the anguiar points lie on the ellipse k being the positive root of the equation n' {a'-by + 2aW (a' + b") z = 3a*b\ Also the perimeter of the triangle formed by joining the points of contact is constant. 640. The two similar and similarly situated conies , <^,y\, {x-Kf (y~hY_ w. 9 130 BOOK OP MATHEMATICAL PEOBLEMS. will be capable of having triangles circumscribing the first and inscribed in the second, if a 641. Two straight lines are drawn parallel to the major axis at a distance - from it ; prove that the part of any tangent inter- e cepted between them will be divided by the point of contact into two parts subtending equal angles at the centre. 642. 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 a;' + (y ± 6)' = a". 643. Two tangents are drawn to an ellipse from a point (X, 7) ; prove that the rectangle under the perpendiculars from any point of the ellipse on the tangents bears to the square on the perpendicular to the chord of contact the ratio 1 : X; where 644. If a triangle be inscribed in an ellipse, and the centre of perpendiculars of the triangle be one of the foci, the sides of the triangle will touch one of the circles IV. Hyperbola, re/erred to its axes, or asymptotes. 645, Prove that the four equations b {x^J{o^ - a')} =^a{y^ ^{f + b')} represent respectively the portions of the hyperbola bV-ay = aV which lie in the four quadrants. CONIC SECTIONS, ANALYTICAL. 181 646. Two points (x^, y,), (x^, y^ are taken on the hyperbola xy = (?; prove that the equation of the line joining them is 647. The equation of the chord of the hyperbola -^ — |j = 1 which is bisected at the point (X, T) is a? ¥ ~ a' b^ ' 648. The locus of points whose polars with respect to a given parabola touch the circle of curvature at the vertex is a rectangular hyperbola. x' v' 649. A double ordinate PP' is drawn to the ellipse -^ + ti = 1, and the tangent at P meets the hjrperbola — — ^= 1 in Q, Q' ; prove that P'Q, P'Q' are tangents to the hyperbola ; and, if S, R' be the points in which these lines again meet the ellipse, that BR divides PP' into two parts in the ratio of 2 : 1. 650. A circle is drawn to touch the asymptotes of a hyper- bola ; prove that the tangents drawn to it at the points where it meets the hyperbola will also touch the auxiliary circle of the hyperbola. 651. Two hyperbolas have the same asymptotes, and NPQ is drawn parallel to one asymptote meeting the other in N, and the curves in P, ^ ; a tangent at Q meets the outer hyperbola in two points, and the lines joining these to the centre meet the ordinate NQ in L, M; prove that NQ is a geometric mean between NL, NM, and that NP is a harmonic mean between NQ and the har- monic mean between NL, NM. 652. The locus of a point from which can be drawn two straight lines at right angles to each other, each of which touches .9—2 132 BOOK OF MATHEMATICAL PBOBLEMS. one of the rectangular hyperbolas xy = ^c^ is also the locus of the feet of the perpendiculars let fall from the origin on tangents to the hyperbolas Qi? — y'^ = ^^:C^. 653. The axes of an ellipse are the asymptotes of a rectan- gular hyperbola which does not meet it in real points ; prove that if two tangents be drawn to the ellipse from a point on the hyper- bola, the difference of the eccentric angles of the points of contact will be least when the point lies on an equi-conjugate of the ellipse. 654. The locus of the equation 2 2 c c iB + as + . . . to 00 , is that part of the hyperbola if — y? = The foci may be determined from the condition that the rect- angle 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 ax" + by' + 2c'xy =/, and {X, Y) be one focus, (- X, - T) the other, we must have, in order that the straight line Ix + my — 1 may be a tangent, (l-lX-mT)(l + lX+mY) ^ ^ =5^^-^^ — 3 = a constant = ii, or, I' {ix + X')+m' (ii + r')+2lniXY- 1 = (A). But the condition that this line may be a tangent gives also the condition that ax'' + by" + 2cxy -/{Ix + my)' must be a perfect square, or {a-/P){b-M) = (o'-flmy; whence l'b/+ m'af- 2lmc'f+ c'"-ab = (B). 'Now (A) (B) expressing the same condition must give the same relation between I and m; hence Ix + X' ,x+T' XT 1 _ X'-T' {b-a)f ¥ - af -of- ab-c" The equations are then X"-T' a — b XT f 'c"-ab 138 BOOK OF MATHEMATICAL PROBLEMS. It may be noticed also that we obtain the following equation for /x \r ab-oVy^ ah-c")~ {ab-cj equivalent to (^ ~a\(—-h\ = c'" ; whose roots are the squares of the semi axes. To each root correspond two foci, which are real for one and unreal for the other. The same method wiU apply to all cases ; and, th? foci being found, the directrices are their polars. The more useful forms of the equation are (1)^,.2X.,4,-.(14)-,,(1.^,).1 = 0, which, for different values of X, represents a series of conies passing through four given points ; two of the joining lines being taken as axes : (^) M->-^^©' the equation of a conic touching the axes at distances h, Je re- spectively from the origin. It is sometimes convenient to use this as the equation of the conic touching four given straight lines ; h,h,\ being then parameters connected by the equations ' \h d)\k h)~hk~\h a')\h h'J' the equations of the other two given straight lines being When (2) represents a parabola, X = 1 ; and the equation may be written The equation of the polar of (X, Y) to the conic in the most general form is X {aX+ c'T+ h')+y{oX+h7 + a') + b'X+ a'T+ c = 0, CONIC SECTIONS, ANALYTICAL. 139 and this may be adapted to all the special cases. The equation of the tangent at a point (X, T) to the parabola the signs of the radicals in the equation of the tangent being determined by the corresponding signs in the equation of the curve at the point {X, Y). The equation of the polar of course cannot be expressed in this form. The equation of two tangents from (X, T) is {aX' + 67=' + c + 2a 7+ 26'X + 2cXY) {aac' + Sy" + c + 2a'y + 2b'x + 2c'xy) = {x {aX+ c'Y+ V) + y (c'X+ 6 F + a ) + bX + a'Y+ c}^ 676. The equation Ix {hx + cy) — my (ax + iy) = represents a pair of conjugate diameters of the conic oaf + 2hxy + cy' = d. " 677. If — 3 + ^ = 1 be an ellipse referred to conjugate dia- meters iaclined at an angle co, the condition that the circle of + 2xy cos u} + y^ = r' may touch the ellipse is V a") V 1\/1 iN cos'o) " Hence determine the relations bet-ween any conjugate diameters and the axes. 678. The axes of the conic aici' -i- 2hxy + cy' = d make with the lines bisecting the angles between the axes of co-ordinates angles 6; prove that . (a- c) sin a tan 29 = -^ — --^ ^, (a + c) cos u) — 26 = and < 1. 692. Trace the conies 2a? - 2xy-2ay + a^ = 0, 2/ - 2xy-2ai/ + a^ = 0, 2xy-2ay + a' = 0; shewing that they touch each other two and two. 693. Two parabolas are so situated that a circle can be de- scribed through their four points of intersection ; prove that the distance of the centre of this circle from the axis of either para- bola is equal to half the latus rectum of the other. 694. A hyperbola is drawn touching the axes of an ellipse and the asymptotes of the hyperbola touch the ellipse ; prove that the centre of the hyperbola lies on one of the equi-conjugates of the ellipse. 695. Five fixed points are taken, no three of which are in one straight line, and five conies are described, each bisecting aU the lines joining four of the points, two and two : prove that these conies wiU have one common point. 696. A, F and B, Q are points taken respectively on two parallel fixed straight lines, A, B being fixed points and P, Q variable points, subject to the condition that AP . BQ is constant; prove that PQ touches a fixed conic which will be an ellipse or hyperbola according as P, Q are on the same or opposite sides of AB. 697. One side AB of a rectangle ABGD slides between two rectangular axes ; prove that the loci of C, D are ellipses whose area is independent of the magnitude of AB ; and that the angle 2B0 between their axes is cot"' — jr- . If in any position AB make an angle 6 with the axis of y, and a, P be the angles which the CONIC SECTIONS, ANALYTICAL, 143 tangents at C, D to the loci make -with the axes of y, x re- spectively, AB cot a. + cot 6 = cot /8 + tan 6 = -=-r, . 698. A circle being traced on a plane, the locus of the vertex of aU cones on that base whose principal elliptic sections have an eccentricity e, is the surface generated, by the revolution about its conjugate axis of an hyperbola of eccentricity e"'. 699. A straight line of given length slides between two fixed straight lines, and from its extremities two straight lines are drawn in given directions ; prove that the locus of their inter- section is au ellipse. 700. Two circles of radii r, r (r > r') touch each other, and a conic is described having real double contact with both ; prove that, when the points of contact are not on different branches of a hyperbola, the eccentricity >-^^/( 1 +-;) , and the latus rectum is greater than, equal to, or less than r — r', according as the conic is an ellipse, parabola, or hyperbola. If the contacts be on different branches of a hyperbola, e > ; , and the asymptotes always touch a fixed parabola. 701. If ABC be a triangle circumscribing an ellipse, S, S the foci, and if SA = SB = SC, then will S'A.S'B.S'O iSA.SB.SG = l-( and each angle of the triangle ABG wUl lie between the values _. 1 ± e 702. One angular point of a triangle, self conjugate to a given conic, is given j prove that the circles on the opposite sides as diameters will have a common radical axis, which is normal at ,144 BOOK OF MATHEMATICAL PROBLEMS. the given point to the similar, concentric, and similarly situated conic through that point. 703. If e be the eccentricity of the conic ax^ + 6/ + c + 2a y + 2h'x + 2c'xy = 0, 0,nd 01 the angle between the axes, e* _(fls-6)'sin=to+{(a + 6)cos) + y{k + h cos u) = hk cos to. 705. In the parabola (I)' -(!)'=■• a tangent is drawn meeting the axes in F, Q ; and perpendiculars are drawn from P, Q to the opposite axes respectively ; prove that the locus of the point of intersection of these perpendiculars is x + y cos to V + a; cos to ^ = + = = cos 0). k h 706. The condition that the straight line - + 1 = 1 should touch the conic \h aj\k p) hk CONIC SECTIONS, ANALYTICAL. 145 707. The asymptotes of the conic ■will touch the parabola (i)'*(0-- 708. The ec[uation of the director circle of the conic is (X'- 1) {a^ + y^ + 2xy cosu>) + h {x + y coso) + k(^y + x cos (o) — hk cos oi = 0. 709. A conic is drawn to touch four given straight lines, two of which are parallel ; prove that its asymptotes will touch a fixed hyperbola, and that this hyperbola touches the diagonals of the quadrilateral, formed by the four given straight lines, at their middle points. 710. If a rectangular hyperbola have double contact with a parabola, the centre of the hyperbola and the pole of the chord of contact will be equidistant from the directrix of the parabola, 711. The area of the ellipse of minimum eccentricity which can be drawn touching two given straight lines at distances h, k from their point of intersection is 7 7/72 7a\ • {¥ + ¥ -2hh cos a)^ ■nlik {h + a; ) sm u -^ '— ; Qf+h' + ^hkGOB and, if e be the minimum eccentricity, e* {h'-ky l-e" h'k'Bm'm' 712. Four points are such that ellipses can be described through them, and e is the least eccentricity of any such ellipse ; W. 10 146 BOOK OF MATHEMATICAL PROBLEMS. e' is the eccentricity of the hyperbola, which is the locus of the centres of all such conies ; prove that ■+-. — r=4. e'» - 1 e' - 1 Prove also that the equi-conjugates of the ellipse are parallel to the asymptotes of the hyperbola. 713. The equation of the conic of minimum eccentricity through four given points is x' ixy cos is the angle which the tangent at the point of contact makes with the axis. 724 The locus of the centre of a rectangular hyperbola x' if having contact of the third order with the ellipse —, + ?, = 1 is the curve .« + ,' = («' H-6')gH-^/ 725. The locus of a point, such that tangents drawn from it to a fixed conic intercept a constant length on a fixed tangent, is a conic having contact of the third order with the given conic at the extremity of the diameter through the fixed point of contact ; and that the locus is an ellipse, parabola, or hyperbola, according as the given length is less than, equal to, or greater than the diame- ter parallel to the fixed tangent. 726. On every straight line can be found two points, conju- gate to each other with respect to a given conic, and the distance between which subtends a right angle at a given point not on the straight line. 727. If a parabola meet an ellipse in four points such that the four normals to the ellipse at those points meet in one point, the axis of the parabola will be parallel to one of the equi-con- jugate diameters of the ellipse. 728. A conic is drawn through four given points lying on two parallel straight lines ; prove that the asymptotes touch the parabola which touches the other four joining lines. CONIC SECTIONS, ANALYTICAL. 149 VII. Envelopes. The equation of the tangent to a parabola in the form a y = mx+~ , m gives as the condition of equal roots in jw, y* = iax, and the equation of the tangent to an ellipse X a cos a + ?sina = ■I. written i in the form «'(l + D- -2«|+l-^ = a 0, ( gives as the condition of equal roots in «, («Htan|) So, in general, if the equation of a straight line, or curve, involve a parameter in the second degree, it follows, that through any proposed point can in general be drawn two straight lines, or curves, of the series represented by the equation. These two curves will be the tangents (rectilinear or curvilinear) from the proposed point to the envelope of the system. In order that they may be coincident, the point from which they are drawn must be a point on the envelope. Thus, "To find the envelope of a system of conies having a given focus, length and direction of major axis." The equation of any such conic may be taken — ^ '- = l+ecoa6, T where e is the parameter. The envelope is then the curve (^ cos 6)' = ia{r- a), or, r'sin'tf = (r-2a)V 150 BOOK OP MATHEMATICAL PEOBLEMS. 2a a pair of parabolas having the given point for focus, and their common latus rectum in the given direction and of twice the given length. This admits of very easy geometrical proof. In this case every one of the system of curves has real contact with the envelope; but it frequently happens that this is true for only a portion of the system. The same principle applies to systems whose equation involves higher powers of the parameter ; but in the following examples it will be found that the general equation of the system whose envelope is, required can always be expressed in the form U+2\V+X'W=0, where X is the parameter; and the cor- responding envelope is the curve UW= Y\ 729. The envelope of the circles ^'+(y-^)(y-y) = o, where )8, y are connected by the equation 1 11 P~ y-a' is the two circles a;' + y° ± 2ax = 0. 730. The envelope of the circles a''+(2/-i8)(2/-7) = 0, where )S, y are connected by the equation (^+a)(7-a) + A''=0, is the two circles j/' + (a; ± Kf = a'. 731. TheelHpse ^+C=1 P has its axes connected by the equation CONIC SECTIONS, ANALYTICAL. 151 prove that tte envelope is the two circles a;' + 3/' =fc 2ax = ; and if the relation between the axes be the envelope is px^ + qy^ + 2aq^x = 0, 732. Through each point of the eitraight line — i- -r^= 1 Gi x' v' is drawn a chord of the ellipse -5 + tj = 1 bisected in the point j prove that the envelope is the parabola (?-l)"*K^?-0-»- 733. The envelope of the ellipse 1 1 a / T ■ ^ /*! cos a v sin a\ X +y'-2{ax cos a+hy sm a) ( + ^— r — j / , 7, .\ /a;cosa wsina\' , . » i, , . + (a' + S'-c') ( "^i, — ) =a sin" a + 6' cos' a- c' is the two ellipses ^+1^-1 _^+_J^! 1 a' b'~ ' a'-c' b'-c' 734. A variable tangent to a parabola meets two fixed tangents in two points ; prove that the directrix of the parabola which touches the fixed tangents at these points envelopes an- other parabola. 735. A variable tangent to a parabola meets two fixed tan- gents, and a circle is described on the part intercepted as dia- meter; prove that the envelope of these circles is an ellipse which touches the fixed tangents at the points where the direc- trix meets them. 736. The director circle of a conic and one point of the conic being given, prove that the envelope is a fixed conic whose major axis is a diameter of the given circle. 152 BOOK OF MATHEMATICAL PROBLEMS. 737. If two points be taken on an ellipse such that the normals at these points intersect a given normal in the same point, the chord joining the two points envelopes a parabola whose directrix passes through the centre and whose focus is the foot of the perpendicular from the centre on a tangent per- pendicular to the given normal. 738. In an ellipse given the centre and directrix, prove that the envelope is two parabolas having their common focus at the given centre. 739. In an ellipse given one extremity of the minor axis and a directrix, prove that the envelope is a circle having its centre at the given point and touching the given straight line. 740. If three points be taken on an elUpse, such that their centre of gravity is a fixed point, the straight lines joining them two and two wUl touch a fixed conic. 741. If three points be taken on an ellipse, such that the centre of perpendiculars of the triangle formed by joining them is a fixed point, the joining lines will touch a fixed conic whose asymptotes are perpendicular respectively to the tangents drawn from the fixed point to the ellipse. VIII. Areal Coordinates. In this system, the position of a point P with respect to three fixed points A, B, G, not in one straight line, is determined by the values of the ratios of the triangles PBG, PGA, PAB to the triangle ABC, «d.j one of them PBG being positive or negative, as P and A lie on the same or opposite sides of BC. These are usually denoted by o, ^, y, which satisfy the equation o+^+y = 1. A point is therefore completely determined by equations of the form la = mP = ny. The general equation of a straight line is la + m^ + my = 0, and I, m, n are proportional to the perpendiculars from A, B, C on the, CONIC SECTIONS, ANALYTICAL. 153 straight line, perpendiculars drawn in opposite directions being of course affected witli opposite signs. The condition that the straight lines la + mP + ny = 0, Va + m'P + n'y = 0, shall be parallel is mn' — m'n + nL' — n'l + lm' — l'm = 0; and that they may be perpendicular is IV sin'^ + ... + ... = (mw' + m'w) sin ^ sin (7 cos .4 +... + ... li l = m = n, or V = m' = n', it will be noticed that both these conditions are satisfied. The straight line a + /8 + y = 0, + -irr= 0. The form (4) of the equation of a conic admits of our denoting any point on the conic by a single variable, analogous to the eccentric angle in the case of a conic referred to its axes, which is indeed a particular case of this form. Thus, the equation of the conic being la? + m/3' + ny* = 0, any point on it may be represented by the equations Jla^_Jmp_ ^^-sin/S-*^^ "^^' which we may call the point 0. The equation of the tangent at 6 is ^la cos + Jm^ sin 6 = J{— ri) y ; the equation of the chord through 6, <^ is ,, 6 + , „ . + d> ,, , — d> ^la cos -J- + Jm^ sin — ^ = J{- n) y cos —^ ; and the point of intersection of the tangents at 6, <^ is sll°- Jm,p J{- n) y cos -^ sin— 2" cos -y^ CONIC SECTIONS, ANALYTICAL. 155 The equations of any two given conies may be assumed in this form, since we have only to take the equations of their fout common points to be Za = ± m^S = * ny, or the equations of their four common tangents to be la. =fc m^ ± wy = 0. The triangle of reference will, however, be imaginary if the conies intersect in two real and two impossible points. Any point on the conic l^y + mya. + na,^ = may in like manner be taken to be a cos' ^ _ i8 sin' 6 _ y I m —n' and any point on the conic {laf + {ml3)^+{nyf=0 to be -—rh=^-irQ = '^y'> cos 6 sm* ' the tangents at these points being respectively tcos,*6 + ^ sin* 6 + ^=0; la mB . cos^ sin''^ ' but these equations are not often needed. The equation of a tangent to ^y = la' in the form tlie point of contact being . 13 y I /M is, however, frequently convenient. 742. The sides of the triangle of reference are bisected in points .4,, -SpC, ; the triangle A^Bfi^ is treated in the same way, and so on n times ; prove that the equation of Bfi^ is / 2"+' + (_ 1)- 156 BOOK OF MATHEMATICAL PEOBLEMS. 743. The equation of the straight line passing through the centres of the inscribed and circumscribed circles is -: — J (cos^-oosC) + -T^-T, (cosC-cosji) +-T^ (cos^-co35)=0. sin j4 '• ' sin i? ^ ' sin C ^ Prove that the point « iS y sin^ (to+w cos^) sin ^ (m + ra cos i?) sin C7 (m + m cos (7) lies on this straight line. 744. If o, p, y be perpendiculars from any point on three straight lines which meet in a point and make with each other angles A, B, C ; the equation la' + mj? + wf = will represent two straight lines, which will be real, coincident, or imaginary according as mn sin' A+nl sin' B + ltn sin' G is negative, zero, or positive. 745. If X, y, « be the perpendiculars from A, B, G on any straight line, and o, )3, y the areal co-ordinates of any point on that line, then a;a + j/yS + ay = Q ; and the perpendicular &om any point (a, )8, y) on the straight line is xa -h j/j3 + zy. 746. Within a triangle ABG are taken two points 0, C j AO, BO, GO meet the opposite sides in A', B, G', and the points of intersection of O'A, ffG'; O'B, G'A' ; O'G, A'B' are respect- ively B, E, F: prove that A'D, SE, G'F will meet in a point ; and this point remains the same if 0, 0' be interchanged in the construction. 747. If X, y, z be the perpendiculars from A, B, G on any tangent (1) to the inscribed circle xsraA + yshiB + z sin (7 = 22? sin A sinB sin (7; (2) to the circumscribed circle a; sin 2j1 + y sin 25 + z sin 2(7= 4iJ sin A sin J5 sin (7 ; (3) to the Nine Points' Circle X sin A cos (B — C) + ysai.B cos (G—A) + zsinC cob[A—B) = 2Ii shxA smB sin(7j CONIC SECTIONS, ANALYTICAL. 157 and (4) to the circle to vMch the triangle is self-conjugate se'tanA + y' tan5+ar'tanC=0. 748. If A', £', C be the feet of the perpendiculars let fall from the point la = TnP = My on the sides of the triangle of refer- ence, the straight lines drawn through A, B, G perpendicular to FC, G'A', A'B' will meet in the point la' mh' nc' ' 749. A triangle A'B'C has its angular points on the sides of the triangle ABG, and AA\ BB", GG' meet in a point, any straight line is drawn meeting the sides of the triangle A'B'G' in points which are joined respectively to the corresponding angles of the triangle ABG ; prove that the joining lines meet the sides of the triangle ABG in three points lying in one straight line. 750. The two points at which the escribed circles of the triangle of reference subtend equal angles lie on the straight line a(5-c) cot ^ + ^ (c - a) cot5 + y(a-5) cotC = 0. 751. Four straight lines form a quadrilateral, and from the middle points of the sides of a triangle formed by three of them perpendicidars are let fall on the line joining the middle points of the diagonals ; prove that these perpendiculars are inversely pro- portional to the perpendiculars from the angular points of the triangle on the fourth straight line. 752. A straight line meets the sides of a triangle in A!, B, G', the straight line joining A to the point {BE, GG') meets BG in D, and E, F are similarly determined. If be any point, the lines joining D, E, F to the respective intersections of DA, OB, OG with A'FG' will pass through a point 0', and Off will pass through a point whose position is independent of 0. 158 BOOK OF MATHEMATICAL PEOBLEMS. 753. The conic la' + m^' + wy' = will represent a circle if I tan A=m, tan B = n tan C. 754. The necessary and suflScient conditions that the equation Za" + mp,^ + nf + I'fiy + m'ya + n'a/S = may represent a circle are m + n — l' n + l — mf _l + 7n — n' a' " 6^ ~ ? ■ 755. The lengths of the tangents from A, B, C to a circle are p, q, r; prove that the equation of the circle is a'l3y + Vya + c^a^ = (a + j8 + -y) (pV + q'fi + r'y). 756. P, P,, Pj, Pg are the points of contact of the Nine Points' Circle with the inscribed and escribed circles ; prove thai (1) the equations of the tangents at these points are b — c 0— a a — o and the three equations obtained from this by changing the sign of a, b, ov e ; (2) PP^, PJP^ meet BG iu the same point* as the straight, linen bisecting the internal and external angles at A; (3) PP^, P^P^ intersect in the point -« ^ /3 y . 6>_c» c'-a' a'-V (4) the tangents at P, P,, P^, P^ all touch the ellipse which touches the sides of the triangle at their middle points. 757. The straight line la + m^ + wy = meets the sides of the triangle jIPC in A', B', G' ; prove that the circles on A A', BB' CG' liave the common radical axis and the circles will touch each other if {mn + Ttl+lm)' {a*+ -26V- ) = ^lmn{la* + -(ni + w)6V- }. CONIC SECTIONS, ANALYTICAL. 159 758. The equation of the straight line bisecting the diagonals of the quadrilateral whose four sides are la ± «ij8 ± ny = 0, is l'a + m''^ + n'y = 0, and the equation of the radical axis of the three circles on the diagonals is , {m!'-n'^{¥y + c'fi) + + =0, 759. One of four straight lines passes through the centre of one of the four circles ■which touch the diagonals of the quadrila- teral ; prove that the other three straight lines pass each through one of the other three centres. Prove that in this case the circles described on the three diagonals touch each other in a point lying on the circle circumscribing the triangle formed by the diagonals, and that their common tangent is a normal to this circle. 760. The equation of a circle passing through £ and C, and •whose segment on £G (on the same side sls A) contains an angle e, is Pysin' A + yasni'' B + a^ sm* 0=a{a + P +y) ^~^ -'. 761. The locus of the radical centre of three circular arcs on £0, GA, A£, respectively containing angles A + 9, £+0, C+d for different values of 6, is the straight line -A-rsm(£-G)+^^sm{G-A)+-J^sm{A-£) = 0. 8inA ^ ' am.£ ^ ' sinC ' If 5=90°, the radical centre is the centre of the circumscribed circle. 762. Prove that, if P be a point such that tan £PG- tan A = tan GPA - tan 5 = tan AP£ - tan G, there are two positions of P j and that the equation of the line joining them is acotii(tan5-tanC) + )8 cot 5 (tan (7- tan ^) + y cot G (tan A — tan B\ = 0. 160 BOOK OF MATHEMATICAL PROBLEMS. 763. A straight line drawn through the centre of the in- scribed circle meets the sides of the triangle ABG in A', E, C"; these points are joined to the centres of the correspcaiding escribed, circles; prove that the joining Jines meet two and two on the sides of the triangle ; and, if ^", B", C" be their points of inter- section, the circles on A' A", B'B", C'G" as diameters will touch each other in one point Ijdng on the circumscribed circle ; and their common tangent will be normal to the circumscribed circle. 764 A circle meets the sides of a triangle ABG in P, P' ; Q, Qf ; R, B' respectively, and AP, BQ, CE meet in the point la = mjS = ny; prove that AP, BQ, GR meet in the point I'a = m'P = ny ; where {m + n) (m + n') _ {n + T){n' +1") _ {I + m) (l' + TnT) a' ~ 6^ 7' • 7G5. The equation of the circle which passes through the centres of the escribed circles of the triangle of reference is 6ca^ + ca^ + ah-f + (a + 6 + c) (osySy + 6ya + ca;8) = 0. 766. The lines joining the feet of the perpendiculars meet the corresponding sides of the triangle ABG in ^', K, G'; prove that the circles on A A', BB\ GG' as diameters will touch each other if sec' A + sec' B + sec' C + 2 sec ^ sec 5 sec (7 = 7. 767. If the side BG subtend angles 6, ff at the foci of an inscribed conic, sin {e - A) sin {& - A) = sin 6 sin ff, 768. If P be any point on the miaimum ellipse circum- scribing a given triangle ABG, AP BP CP -I +. = sin BPG sin GPA sin APB ' cot BPG + cot GPA + cot APB = cot .4 + cot B + cotG; the angles BPG, GPA, APB being so taken that their sum is 360". CONIC SECTIONS, ANALYTICAL. 161 769. If a conic iPy'+ mya + na/S = be sucli that the nor- mals to it at the angular points of the triangle of reference meet, in a point, ^ (^'_n') + -^ {n' -1')+ -^{l'-m') = 0; .A^ ' sm .S ^ ' amU^ ' sm the point must lie on the curve a(/3'-/)(cos^-cos^cosC)+... + ...=0; and the centre must lie on the curre sm^ V * / trilinear co-ordinates being used. 770. The two conies circumscribing the triangle of reference, passing through the point (o, j8, y), and touching the Une xa + yP+zy=0 •will be real if xyz (xa + y^ + zy) Interpret this result geometrically. 771. A conic touches the sides of a triangle ABO in the points D, E, F, and AD meets the conic again in d; prove that the equation of the tangent at c? is 2 (myS + ny) = la ; AD, BE, GF meeting in the point la = mj3 = ny. 772. rind the two points in which the straight line j3 = hy meets the conic {la)K(mP)^+(ny)i = 0; and from the condition that one of these points may be at infinity, determine the directions of the asymptotes. Prove that the conic will be a rectangular hyperbola if aH' + b^m" + c^n' + Zbcmn cos A + 2coml cos B + 2ahlm cos (7=0. W. 11 162 BOOK OF MATHEMATICAL PEOBLEMS. 773; If ia + jMjS + My = be the equation of the axis of a paxabola ■which touchas the sides of the triangle of reference, la' mb' ng sin° G k sin A sin ^ ' 780. OA, OB touch a conic at A, B, and the tangent at P meets OA, OB in ff, A' respectively ; find the locus of the inter- section of AA', BF; and if AP, BP meet OB, OA in A', F' respectively, find the envelope of A"F'. 781. OA, OB touch a conic at A, B, and C, D are two other fixed points on the conic ; a tangent to the conic meets OA, OB in G', -D'j prove that the locus of the intersection of GG', BU is a conic passing through C, B and the intersections of OG, BB; and of OB, AG. 782. The locus of the foci of the conic ^y = ka' for different values of k is -Aj C-^ - ^^) + . y^ r (/3 cot-B-y cot G) = 0. sm^l \sin^j6 sm 0/ sin ^ sin C^ ' ^ 783. Prove that the equation 4^-y = a" represents a parabola; and that the tangential equation of the same parabola is yz — x\ 784. GA, GB are tangents to a conic, P any point oiji the conic, and AP, BP meet GB, GA in A', B' ; prove that the tri- angle A'B'P is self-conjugate to another conic touching the former at A, B. 785. The sides of a triangle ABG touch a conic, 0, 0^, 0^, 0^ are the centres of the circles which touch the sides ; a conic is described through B, G, 0, 0^ and one focus, and another through £, G, Oj, O3 and the same focus ; prove that the fourth point of intersection of these conies will be the second focus. 786. A conic passes through four given points j prove that the locus of the points of contact of tangents drawn to it from a given point is in general a curve of the third degree, which re- duces to 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 given points. 11—2 161 BOOK OF MATHEICATICAL PROBLEMS. 7S7. Given a point on a conic, and a triangle ABQ self- conjugate to the conic, AO, BO, GO meet the opposite sides in three points, and the lines joining these two and two meet the corresponding sides' va. A', B, G' ; prove that the intersections of BB, CG' ; GG', AA'; and AA', BB' also lie on the conic. -., 788. Any tangent to a conic meets the sides of a self-conju- gate triangle in J), E, F; the line joining A to the intersection oif BE, GF meets BG in A' ; E, G' are similarly determined ; prove that B'G', G'A', A'B' are also tangents to the conic. 789. /' Tf la. + m^ + wy = be the equation of an asymptote of a rectangular hyperbola self-conjugate to the triangle of reference, Id? mh' n(? ■■ . -, .+ + =0. m— n n — I l — m 790. A parabola is described touching the sides of a triangle ABC, Sis the focus, and the axis meets the circle circumscribing the triangle again in 0; prove that if with centre a rectangular hyperbola be described, to which the triangle is self-conjugate, one of its asymptotes will coincide with OS. 791. One directrix of the conic la' + m0' + ny' = passes through A; prove that 1 cot' 5 cot'C 7 = + ; and that the second focus lies on the line joining the feet of the perpendiculars from B, G on the opposite sides. 792. The equations determining the foci of the conic la' + m^' + ny'' = ^'A\ L m^ n)~am'B\ m n'^ l] are sin , CONIC: SECTIONS, ANALYTICAL. 165 793. If a triangle be self-conjugate to a parabola, the lines joining the middle points of the, sides are tangents to the parabola. Hence prove that the focus lies on the Nine Points' Circle, and rthat the directrix passes through the centre of the circumscribed circle. 794. A conic is drawn touching the four straight lines la ± m^ ± Tjy = ; prove that its equation is La!' '+ MIS' + 2Fy' = 0, Z, M, N satisfying the equation ' l'MN + mWL+n'LM = Q. • ,■ , Investigate the species of this conic with reference to the position of its centre on the straight line which is its locus. 795. Any conic through the four points la. = A, tnj3 = =fc My , > •will divide harmonically the straight line joining the points («!• ^1. 7i). K ^2' y,)> if 796. If a triangle be self-conjugate to a rectangular hyper- bola, and any conic be inscribed in the triangle, its foci will be conjugate points with respect to the hyperbola. 797. A given triangle is self-conjugate to a conic, and the centre of the conic lies on a given straight line parallel to one of the sides of the triangle ; prove that the asymptotes will envelope; a fixed conic which touches the other two sides of the triangle. 798. . The locus of the foci of all conies touching the four, straight lines la ± m^ d= wy = is (a + )S + y) Q'a' cot A + m'^ cot B + n'y' cot C) sin A sin B sin C = (Pa + m'^ + n'y) ()8y sin' ^ + ya sm' -S + a/? sin' C). 166 BOOK OP MATHEMATICAL PROBLEMS. 799. The straight lines ■will be conjugate with respect to all parabolas inscribed in the triangle of reference, if ya/ + y'z = zui + z'x = xy' + a!y : and with respect to all coqIcs touching the four straight lines la. ± m^ ± «y = 0, . » sa^ _yy^_ t^ I m n -, 800. A rectangular hyperbola is inscribed in the triangle ABC ; prove that the locus of the pole of the straight line which bisects the sides, ^5, AG is the circle a' (a' + 6» + c") + (/3= + 2ay3) {a' + 6» - c') + (y* + 27a) (a» -b' + c') = 0. 801. Four conies are described with respect to any one of which three of the four straight lines la^m^^ny — O form a self-conjugate triangle, and the fourth is the polar of a fixed point (a', ^', y') ; prove that the. four will have two common tangents meeting in (a, ^, y"), their equation being 802. A triangle circumscribes the conic W + m^' + ny' = 0, and two of its angular points lie on thecdnic prove that the locus of the third angular point is the conic -, ; TT-T+"'+ ..• =0; \»i n IJ CONIC SECTIONS, ANALYTICAL. 167 and that this will coincide ■with the second conic if Also prove that the three conies have four common tangents. 803. The angular points of a triangle lie on the conic . la' + m^ + wy' = 0, and two of its sides touch the conic «V + ni'j8' + Hy=0; prove that the envelope of the third side is the conic \m n 1/ and that this will coincide with the second conic if Also prove that the three conies have&ur common points. 804. A triangle is self-conjugate to the conic la' + m^ + n-Z^O, and two of its angular points lie on the conic ra' + m'P' + n'y'=0; prove that the locus of the third angular point is the conic , /m' n'\ , . n— + — )a'+... +...=0, \»» n/ and that this will coincide with the second if m n - -=+— + - = 0. I m n Prove also that the three conies have four common points, and that the envelope of the line joining the two angular points is the conic m n — +- m n = 0. 168 BOOK OF MATHEMATICAL PEOBLEMS. 805. A triangle is self-conjiigale to the conic ' 10? + m^ + n/ = 0, and two of its sides touch the conic prove that the envelope of the third side is the conic la." m « m, n + ... +... = 0; and that this will coincide with the second if Prove also that the three conies have four common tangents, and that the locus of the point of intersection of the two sides is the conic I {~ + —)a+ ... + ...=0. IX. Anha/rmonic Properties. The anharmonio ratio of four points A,£,G,D in. one straight line, denoted by \ABUJJ], means the ratio -5-^ : ^^ , or .^ „, ; the order of the letters marking the direction of measurement of any segment, and segments measured in opposite directions being affected with opposite signs. So, if A, B, G, D be any four points in a plane, and P any other point in the same plane, P {ABGD\ denotes -; — , „^ ' . — ^ttt: > ^^ same rules being oh- ' ' am APG .sm£P J) ' ° served as to direction of measurement and sign for the angles iu this expression as for the segments in the former. Either of these ratios is said to be harmonic when its value is — 1 ; in which case Al) is the harmonic mean between AB and AC, and J) A the harmonic- meaH_ between DB and BG. CONIC SKCTIONS, ANALYTICAl;. ' 16D The anliarmonic ratio of four points, or four straight lines, can never be equal to 1 j as that leads immediately to the condition AD , BC = 0, or sin APJ) . sin £FG = ; making two of the points, or two of the lines, coincident. If A, B, G, J) be four fixed points on a conic, and P ] any other point on the same conic, P {ABCD} is constant for all posir tions of P, and is harmonic when BG, AD are conjugates to the conic. Also, if the tangents at A, B, G, D meet the tangent at P in a, 6, c, d, the range {abed} is constant and equal to the former pencil. The range formed by four points in a straight line is equal to 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 = fi^v, w = /ij'y, u = [i^v, u = fj.^v, the anharmonic ratio of thp pencil formed by them, or of the range in which any straight line meets them, is 806. Two fixed straight lines meet in -4 ; B, G, D are three fixed points on another straight line through A ; any straight line through D meets the two former in B^, C; BS, GG' meet in P, BG', GS in Q : prove that the loci of P, Q are straight lines through A which form with the two former a pencil equal to {ABCD}. ■ 807. ABG is a triangle, two fixed straight lines intersect in a point on BG, any point is taken on A 0, and the straight lines joining it to B, G meet the two fixed lines in B^, B^, C,, G^, respectively; prove that Bfi^ and -BjC, jass each through a fixed point on BG. 808. Chords are draxvn tbrough a fixed point on a conic, making equal angles with a given direction ; prove that the straight Une joining their extremities . passes through a fixed point. 170 BOOK OP MATHEMATICAL PROBLEMS. 809. Through a given poiat are drawn chords PF, QQ to a given conic, so as to touch any the same confocal conic ; prove that the points of intersection of PQ^ P'Q', and of PQ', PQ are fixed. 810. If ^, 5 be t-wo fixed points, P, Q any two points on the same straight line, such that the range {APQB\ is harmonic, and a circle be described on PQ as diameter j all such circles will have a common radical axis, and will cut orthogonally any circle passing through A, B. 811. If two triangles be formed, each by two tangents to a conic and the chord of contact, the six angular points lie on a conic. 812. Four chords of a conic are drawn through a point, and two other conies are drawn through the point, and each through, four extremities of the chords, respectively opposite ; prove that these conies will have contact at the point. 813. Through a given point is drawn any straight line meeting a given conic in Q, Q', and P is taken on this line, so that the range \PQ(^P\ is constant j prove that the locus of P is an arc of a conic having double contact with the former. 814. Given two points A, B of a conic, the envelope of a chord PQ, such that the pencil {APQB} on the conic is equal to a given quantity, is a conic touching the former conic at A, B. 815. Through a fixed point is drawn any straight line meet- ing two fixed straight lines in Q, R respectively; E, F are two other fixed points, QE, RF meet in P; prove that the locus of P is a conic passing through E, F, and the point of intersec- tion of the two fixed straight lines. 816. If A, B, C be three points on a conic, and P, P'. any two other points, such that the pencils {PABC\, {PCBA} at any point on the conic are equal ', PP", CA, and the tangent at B, wUl meet in a point. CONIC SECTIONS, ANALYTICAL. 171 817. If A, B, C, A', B', C be six fixed points on a conic, sueli that {B'ABG} = {BA'FG'), and P, P be any two points of tlie conic, sucli that {PABG] = {P'A'FG'}, ^ PP" ■will pass through a fixed point on BB'. 818. A given straight line meets any conic which passes through four given points in two points ; prove that these points are conjugate with respect to the conic, which is the locus of the pole of the given straight line with respect to the series of conies through the four points. 819. The anharmonic ratio of the pencil subtended by the four points a,, a^ a^ a^ on an ellipse at any point on the ellipse is ct — a . a — a. sm ' „ ' sm -5-= — * sm ■ ' ° sm ° * 820. If tangents be drawn to a conic at A, B, C, D, and X,, Xj, Xj be the middle points of the diagonals joining the points of intersection, of the tangents at (1) A, B; G, D; (2) A, G ; B, D ; (3) A, I) ; B, G ; and be the centre ; the range {OXjXjXj} is equal to the pencil {ABCD\ at any point of the come. 821. A conic is drawn through four given points A, B,G, D) AB, GD meet in F,, AG, BB in Y^, AD, BG in Y^ and is the Centre oT the conic ; prove that the pencil {ABGD] on the conic is equal to the pencil {OY^YJT^ on the conic which is the locus of 0. 822. The anharmonic ratio of the points of intersection of the conies W + «i^' + wy' = P, IV + m'^* + n'y' = 0, , 172 BOOK OP MATHEMATICAL PEOBLEMS. ■with respect to the former is I' m' I in m' n' m n »' r n I n m n m V n" I n m' V ' m I or the reciprocal of on^e of these, according to the order in ■which the points are taken : and these are also the anharmonic ratios of ■the range formed by the four common tangents on any tangent to the latter. X. Reciprocal PoIoiTS, JPfojections. If there be a system of points, and straight lines, lying in the same plane, and ■we take the polars of the points and the poles of the straight lines -with respect to any conic in that plane, ■we obtain a system reciprocal to the former ; so that to a series of points lying on any cuitc in. the first system correspond a series of straight lines touching a certain other curve in the second system, and vice verad : 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. Thus, from any general theorem of position may be deduced a reciprocal theorem. It is in nearly all cases ad-visable to take a circle for the auxiliary conic, ■with respect to which the system is recipro- cated; the point {p) corresponding to any straight line being then found by drawing from 0, the centre of the circle, OP per- pendicular to the straight line, and taking on it a point p, such that OP . Op = k', k being the radius of the circle : and simDarly the straight line through ^ 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 0, or, more shortly, with respect to the point 0, draw Oa perpendicular to £G, and in it take any point a ; through a, 0,dra^w straight lines perpendicular to 00 ^ GA, meeting CONIC SECTIONS, ANALYTICAL. 173 in 6; and throngli 6, draw straight lines perpendicular to OA, AB, meeting in c ; then the points a, b, c ■will be the poles of the sides of the triangle, and the straight lines he, ca, ah the polars of the points A, B, C, with respect to some circle with centre 0. Now, suppose we want to find the point corresponding to the per- pendicular from A, it must lie on he, and on the line through at right angles to Oa, since Oa is parallel to the straight line whose reciprocal is required ; it is then determined. Hence, to the theorem that the three perpendiculars of a triangle meet in a point, corresponds the following : if through any point in the plane of a triangle aho be drawn straight lines at right angles to Oa, Oh, Oc, to meet the respectively opposite sides, the three points so determined will lie in one straight line. So, from the theorem, that the bisectors of the angles meet in a point, we get the following : the straight lines drawn through bisecting the external angles (or one external and two internal angles) between Ob, Oc; Oc, Oa; Oa, Ob, respectively, will meet the opposite sides in three points lying in a straight line. If a circle, with centre A and radius R, be reciprocated with respect to 0, the corresponding curve is a conic whose focus is 0, OA 2¥ major axis along OA, eccentricity -„- , and latus rectum -^ or 2 -= if we take the radius of the auxiliary circle unity. The cen- jn tre A is reciprocated into the directrix. Most of the focal pro- perties of conies may thus be deduced from well known properties of the circle. For instance, if be a point on the circle, and OP, OQ two chords at right angles, PQ passes through the centre. Reciprocating with respect to 0, to the circle corresponds a parabola, and to. the points P, Q two tangents to the parabola at right angles to each other, which therefore intersect on the directrix. Again, to find the condition that two conies which have a common focus should be such that triangles can be inscribed in one whose sides shall touch the other. 174 BOOK OF MATHEMATICAL PEQBLEMS. ,' Take two circles wHch have this property, and let B, r be their radii, 8 the distance between their centres; then Eeciprocate 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 major axes of the conies, and if 2Cj, 2c^ be the latera recta, Cj, e^ the eccentricities, whence I? 'i^ 2Rr = x^ + y' — 2xy cos a ; 1 2 e/ e' 2e,e, .-. — r± — =— 2 + -^ — cos a, or c,' ± 2ejC, = e/e,° + e^'c/ - 2e^e^c^e^ cos o j the relation required. 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 former focus being one of the limiting points of the system. The reciprocal of a conic with respect to any point in its plane is another conic which is an ellipse, parabola, or hyperbola, according as the point lies within, upon, or without the conic. Xo, the points of contact of the tangents from the point correspond the asymptotes, and to the polar of the point the centre of the reciprocal conic. So also to the asymptotes and centre of the original conic correspond the points, of contact and polar with respect to the reciprocal conic. As an example, we may investigate the elementary property that the tangent at any point of a conic makes equal angles with the focal distances. The reciprocal theorem is, that if we take any point in the plane of a conic, there exist two fixed straight lines (the reciprocals of the foci), such that if a tangent to the CONIC SECTIONS, ANALYTICAL. 175 conic at P meet them in Q, Qf, OP makes equal angles ■with OQ and OQ'. If, however, the point lie on the curve, the original curve "was a parabola ; and one of the straight lines being the reciprocal of the point at infinity on the parabola -will be the tangent at 0. Since the anharmonic ratio of the pencil formed by any four straight lines is equal to that of the range formed by 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 having the same anharmonic ratio. The Method of Projections enables us to make the proof of any general theorem of position depend upon that of a more simple particular case of the theorem. Given any figure in a plane, "we have five constants disposable to enable us to simplify "the pro- jected figure, three depending on the position of the vertex and two on the direction of the Plane of Projection. It is clear that relations of tangency, of pole and polar, and anharmonic ratio, are the same in the original and projected figures. As an example we will take the follo"wing : " To prove that if two triangles be self-conjugate to the same conic, their angular points lie on a conic" Let the t"wo triangles be ABC, DEF ; project the conic into a drcle "with its centre at D, then E, ¥ -will be at infinity, and DE, DF will be at right angles to one another. Draw a conic through ABODE, then since ABO is self-conjugate to a circle whose centre is D, D is the centre of perpendiculars of the triangle ABO, the conic is therefore a rectangular hyperbola, and E being one of its points at infinity, F will be the other. The theorem is therefore true. Again, retaining the centre at B, take any other conic instead of the circle; J)E, BF -will still be conjugate diameters, and therefore if any conic pass through A, B, 0, B, its asymptotes -will be parallel to a pair of conjugate diameters of the conic -whose centre is B, and to which ABO is self-conjugate. The same must therefore be the case -with respect to the four conies, each having 176; BOOK OF MATHEMATICAL PROBLEMS. its centre . at one of tte four' points, and the other three self- conjugate. These conies are therefore similar and similarly- situated. Moreover, if we draw the two parabolas . which can pass through the four points, their axes must be parallel respect- ively to coincident conjugate diameters of any one of the four ponies j i. e. to the asymptotes. But the axes of these para- bolas must be parallel to the asymptotes of the conic which is the locus of centres of all conies through the four points ; since when the centrfe is at infinity the conic becomes a parabola. Hence, finally, if we have four points in a plane, the four conies, each of which has its centre at one of the four points, and the other three self-conjugate, and the conic which is the locus of centres of conies through the four points, are all similar and similarly situated. Let A, B be any two fixed points on a' circle, oo, oo' the two impossible circular points at infinity, P any other point on the circle ;;then P {A oo co'P} is constant. Hence PA, PB and the circular points divide the line at infinity in a constant anhar- jnonic ratio. . Hence, two straight lineS including a constant angle, may be projected into two straight lines, which divide the straight line joining two given points (the projections of the circular points) in a constant anharmonic ratio. In particular, if APB be a right angle, AB passes through the centre (the pole of 00 oo'), and the ratio becomes harmonic. Thus, projecting the property of the director circle of a conic, we obtain the following theorem : " The locus of the intersection of tangents to a conic which divide harmonically the straight line joining two given points is a conic passing through the given points; and the straight line joining the two points has the same pole for both conies." If tangents be drawn to any conic through the circular points, their four points of intersection are the real and impossible foci of the conic. If the conic be a parabola, the line joining the cir- cular points is a t^ingent, and one of the real foci is at infinity, while the two impossible foci are the circular points. Many CONIC SECTIONS, ANALYTICAL. 177 properties of the foci, especially of the parabola, may thus be generalized by projections. Thus, remembering that the directrix is the polar of the focus, we see that if a conic be inscribed in a triangle ABC, and two, tangents be drawn dividing BQ harmoni- cally, their point of intersection lies on the polar oi A. So also, since the locus of the intersection of tangents to a parabola, includ- ing a constant angle, is a conic having the same focus and directrix, it follows that, if a conic be inscribed in a triangle ABG, and two tangents be drawn dividing BG in a constant range, the locus of their point of intersection is a conic having double contact with the former at the points where AB, AG touch it. The circular points at infinity have singular properties in relation to many other curves. All epicycloids and hypocycloids pass through them, the cardioid has cusps at them, and may be projected into a three-cusped hypocycloid. 823. If two conies have a common focus S, and two common tangents PQ, PQ^, the angles PSQ, P'SQ' wUl be equal or sup- plementary. 824. If two conies have a common focus and equal minor axes, their common tangents will be parallel. 825. If two conies have a common focus and equal latera recta, one of their common chords wUl pass through the common focus. 826. If S be the focus of a conic and A any point, the straight line drawn through S at right angles to SA will meet the polar of A on the directrix. 827. If be a fixed point on a conic. A, B, G any three other points on the conic, and the straight lines through at right angles to OA, OB, OG meet BG, GA, AB respectively in A', B', G', the straight line A'BC will meet the normal at in a fixed point. 828. Given a conic and a point ; prove that there are two straight lines, such that the distance between any two points on one of them, conjugate to the.conic, subtends a right angle at 0. w. 12 178 BOOK OP MATHEMATICAL PROBLEMS. 829. being a fixed point on a conic, OP, OQ any two chords, OR the chord normal at ; prove that there exists a straight line passing through the pole of OR, such that the tangents at P, Q intercept on it a length which subtends at an angle twice POQ. 830. On any straight line can be found two points, conjugate to a given conic, such that the distance between them subtends a right angle at a given point. 831. ABC is a triangle, any point, and straight lines through at right angles to OA, OB, OG meet the respectively opposite sides in A', S, C ; prove that any conic which touches the sides of the triangle and the straight line A'B'C will subtend a right angle at 0. 832. An ellipse is described about an acute-angled triangle ABC, and one focus is the centre of perpendiculars of the triangle; prove that its latus rectum is „ P cos 4 cos B cos G ^^ . A . B . a- sin 2 sin -sm^ 833. A parabola and hyperbola have a common focus and axis, and the parabola touches the directrix of the hyperbola; prove that any straight line through the focus is harmonically divided by any tangent to the parabola and the two parallel tangents to the hyperbola. 834. A series of conies are described having equal latera recta, the focus of a given parabola their common focus, and tangents to the parabola their directrices ; prove that the com- mon tangents of any two intersect on the directrix of the parabola, 835. With a given point as focus four conies can be drawn circumscribing a given triangle, and the l&,tus rectum of one of them will be equal to the sum of those of the other three. Also;, if any conic A be drawn touching the directrices of the four CONIC SECTIONS, ANALYTICAL. 179 conies, the polar of tlie given point with respect to it will be a tangent to the conic which has the given poiat for focus and touches the sides of the triangle, and the conic A will subtend a right angle at the given point. 836. Prove that with the centre of the circumscribed circle as focus, three hyperbolas can be described, passing through the angular points of the triangle ABC j that their eccentricities are cosec 5 cosec C, &c. ; their directrices the lines joining the middle points of the sides ; and that the fourth point of intersection of any two lies on the straight line joining one of the angles to the middle point of the opposite side, 837. Prom a point P on the circle circumscribing a triangle ABC, are drawn FA', PB, PC at right angles to PA, PB, PC to meet the corresponding sides ; prove that the straight line A'B'C passes through the centre of the circle. 838. A triangle is inscribed in an ellipse so that the centre of the inscribed circle coiacides with one of the foci ; prove that the radius of the inscribed circle is :; jjz, =; ; 2c being the 1 + ^(1 + * latus rectum and e the eccentricity, 839. A triangle is self-conjugate to a hyperbola, and one focus is equidistant from the sides of the triangle; prove that each '' distance is —rr^ r^, ^c being the latus rectum and e the ec- centricity. 840. Two conies have a common focus, and are such that triangles can be inscribed in one which are self-conjugate to the other ; prove that 2c^ + c/ = fij'c/ + e^c^ - 2e,e/jCj cos a ; c,, Cj being their latera recta j £,, e^ their eccentricities ; and a the angle between their axes. Prove also that in this case tri- angles can be circumscribed to the second, which are self-con- jugate to the first, 12—2 180 BOOK OF MATHEMATICAL FBOBLEMS. 841. Three tangents to a hyperbola are so drawn that the centre of perpendiculars of the triangle formed by them is at one of the foci; ^rove that the radius of the circle to which the triangle is self-conjugate is constant. 842. If four points lie on a circle, four parabolas can be described having a common focus, and each touching the side of a triangle formed by joining three of the points. 843. BB' is the minor axis of an ellipse, and -5 is the centre of curvature at B'; on the circle of curvature at 1^ is taken any poiat P, and tangents drawn from P to the ellipse meet the tangent at B in Q, Q" ; prove that a conic drawn to touch QB', Q'B', with its focus at B and directrix passing through B, wUl touch the circle at P, 844. If three tangents to a parabola form a triangle ABC, and perpendiculars jp^, 'p^, p^ be let fall on them from the focus S ; then will pj)^ sin BSG +pj)j sin CSA + pjp^ sin ASB = 0, the angles at S being measured in the same direction. 845. A triangle ABC circumscribes a parabola, and x, y, z are the perpendiculars from the focus on the sides ; prove that sin 2 J. sin 25 sin 2C 8 sin A sin 5 sin C a;' y z I I being the semi latus rectum. 846. A point S is taken within a triangle, such that the sides subtend at it equal angles, and four conies are described with S as focus passing through A, B, G ; prove that one of these conies will touch the other three, and that the tangent to this conic at A will meet BG in a point A', such that ASA' is a right angle. 847. With the centre of perpendiculars of a triangle as focus are described two conies, one of which touches the sides of the triangle, and the other passes through the feet of the perpen- diculars; prove that these conies will touch each other, and that CONIC SECTIONS, ANALYTICAL. 181 the point of contact will lie on the conic touching the sides of the triangle at the feet of the perpendiculars. 848. A conic is inscribed in a triangle, and one directrix passes throvtgh the centre of perpendiculars ; prove that the corresponding focus lies on the circle to which the triangle is self-conjugate. 849. With the centre of the circumscribing circle of a tri- angle as focus are described two ellipses, one touching the sides of the triangle, and the other passing through the middle points ; prove that these will touch each other. 850. Five points are taken, no three of which lie in one straight line, and with one of the points as focus are described four conies, each of which touches the sides of a triangle formed by joining three of the other points; prove that these conies will have a common tangent. 851. Through a fixed point are drawn any two straight lines Ineeting a given conic in P, P' ; Q, Qf ; and a given straight line in ^, R ; and RR' subtends a right angle at another fixed point. Prove that PQ, PQ, P'Q, P'Qf all touch a certain fixed conic. 852. Given a conic and a point in its plane ; prove that there exist two points L, such that if any straight line through L meet the polar of L in P, and P' be the pole of this straight line, PP" subtends a right angle at 0. 853. The envelope of a straight line which is divided har- monically by two given straight lines and a given conic, is a conic touching the two fixed straight lines at points on the polar of their point of intersection ; unless the given lines are conju- gate with respect to the conic, when only one such straight line can be drawn. 854. Two equal circles A, B, touch at S, a tangent to B meets A in P, Q, and is its pole with respect to A ; prove that the directrices of two of the conies described with focus S to circumscribe the triangle OPQ will touch the circle A. 182 BOOK OP MATHEMATICAL PROBLEMS. 855. A conic touches the sides of a triangle ABG in D, M, F, and AD, BE, OF meet in S ; three conies are described with *S' as focus, osculating the former at D, F, F ; prove that these three and the former will have one common tangent, which also touches the conic having *S^ as focus and touching the sides of the triangle ABC. 8.56. Given four straight lines, prove that two conies can be constructed so that an assigned straight line of the four is its directrix and the other three form a self-conjugate triangle : and that, whichever straight line be taken for directrix, the corre- sponding focus will be one of two £xed points. 857. Through a fixed point are drawn two chords PF', QQ' to a given conic, such that the two lines bisecting the angles at are also fixed ; prove that the straight lines PQ, P'Q, PQ', P'Q' all touch a fixed conic ; except when the two fixed straight lines are conjugates to the given conic. 858. OP, OQ; ffP, O'Q' are tangents to a conic, and the conic which touches the sides of the triangles OPQ, UPQ is drawn : prove that any tangent to the latter conic will be divided harmonically by the former conic and the lines PQ, P'Q. 859. Two conies circumscribe the triangle ABG, any straight line through A meets them again in P,Q; and the tangents at P, Q meet BC in P'Q' ; prove that the range {BFQfC} is con- stant. 860. The equation of the polar reciprocal of the evolute of the ellipse -^ + |j = 1 with respect to the centre is 861. If P, Q be two fixed points, and if on the side BG of a triangle ABG be taken a point A', such that the pencil A'{APQB] is harmonic ; and F, C' be similarly taken on the sides GA, AB; the straight lines AA', BE, GG' will meet in a point ; and the CONIC SECTIONS, ANALYTICAL. 183 four such points corresponding to the triangles formed by four given straight lines will lie on one straight line. 862. The locus of a point, such that the tangents drawn from it to a given conic are harmonically conjugate to the straight lines joining it to two given points is a conic passing through the two given points, and through the points of contact of the tangents drawn from the two points to the given conic. This locus reduces to a straight line, if the line joining the two points touches the given conic. 863. Two conies touch each other at 0, any straight line through meets them ir\. P, Q ; prove that the tangents at P, Q intersect in the straight line joining the two other points of intersection of the conic. 864. Four tangents A, B, G, D are drawn to a conic, the line joining the points of contact of A, B meets G, D respectively in -P, Q; prove that if a conic be described touching the four lines, and P be its point of contact with G, Q will be its point of contact with D. 865. Two conies S, S' intersect in A, B, G, D, and the pole of AB with respect to *S' is the pole of CB with respect to S' ; prove that the pole of CD with respect to S is the pole of AB with respect to S'. 866. A quadrilateral can be projected into a rhombus on any plane parallel to one of its diagonals, the vertex being any point on a circle in a certain parallel plane. 867. If ABG be a triangle circumscribing a conic. A' the point of contact of BG, D the point where AA' again meets the conic, and TP be any tangent meeting the tangent a,t D va. T ; the pencil T{ABGP} will be harmonic. 868. ABG is a triangle circumscribing a conic, TP, TQ two tangents, a conic is described about TPQBG, and is the pole of BG with respect to it ; prove that A {OBGT\ is harmonic. 184! BOOK OF MATHEMATICAL PROBLEMS. If T lie on the straight line joining A to the point of contact ■with BC, will coincide with A. 869. A. conic is described touching the sides of a triangle ABC, one of them BG in a fixed point A' ; B', C are two other fixed points on BG ; prove that the point of intersection of tangents drawn from B^, C" to the conic lies on a fixed straight line. 870. The sides of a triangle which is self-conjugate to a given rectangular hyperbola touch a parabola, and a diameter of the hyperbola is drawn through the focus of the parabola ; prove that the conjugate diameter is parallel to the axis of the parabola. 871. TP, TQ are two tangents to a parabola; a hyperbola described through T, P, Q, and having an asymptote parallel to the axis of the parabola meets the parabola again in R; prove that its other asymptote is parallel to the tangent to the parabola at R. 872. TP, TQ are two tangents to a hyperbola ; another hyperbola is described through T, P, Q with asymptotes parallel to those of the former ; prove that it will pass through the centre C of the former, and that GT will be a diameter, 873. A triangle is self-conjugate to a conic, and from any other two points conjugate to the conic tangents are drawn to a conic inscribed in the triangle ; prove that the other four points of intersection of these tangents will be two pairs of conjugate points to the first conic. 874. If a conic pass through four points, its asymptotes meet the conic which is the locus of centres in two points at the extremities of a diameter. 875. Four points and a straight line being given, four conies are described, such that with respect to any one of them three of the points are the angular points of a self-conjugate triangle,. CONIC SECTIONS, ANALYTICAL, 185 and tlie fourtli is the pole of the given straight line ; prove that these four conies will meet the given straight line in the same two points, and that these points are the points of contact of the two conies through the four points touching the line. 876. Four straight lines and a point being given, four conies are described, with respect to each of which three of the four lines form a self-conjugate triangle, and the fourth is the polar of the given point ; prove that all four will have two common tangents through the given point, and these tangents are tangents to the two conies through the point touching the four lines. If two tangents be drawn through the point to any conic touching the four lines, these wUl form a harmonic pencil with the two common tangents. 877. Four tangents are drawn to a conic, and from a point T on one of the diagonals of the quadrilateral formed by them two other tangents are drawn ; prove that the points of contact of these tangents lie on the conic passing through T, and through' the points of intersection of the four tangents which do not lie on the diagonal through T. 878. CA, CB are two tangents to a conic, any point P in AB is joined to the point in which its polar meets a fixed straight line ; prove that the envelope of the joining line is a conic touch- ing the sides of the triangle ABC and the fixed straight line. 879. ABC is a triangle whose sides are met by a straight line in A', B', C ; the straight line which joins A to the point {BB", CC) meets BC in a; and b, c are similarly determined. Four conies are drawn touching the sides of the triangle ABC, and meeting A!B'G' in the same two points ; prove that the other common chord of any two of these conies passes through either a, h, ov c; that these six common chords intersect by threes in four points; and that these four points are the poles of A'B'C with respect to the four conies which intersect A'B'C in the before-mentioned two points and touch the sides of the tri- angle ahc. 186 BOOK OP MATHEMATICAL PEOBLEMS. 880. The four conies which, can pass through three given points and touch two given straight lines are drawn, and their other common tangents drawn to every two ; prove that the six points of intersection will lie by threes on four straight lines ; and that the diagonals of the quadrilateral formed by these four straight lines pass each through one of the three given points. 881. Two conies intersect in A, B, C, D ; through D is drawn a straight line to meet the curves again in two points ," prove that the locus of the point of intersection of the tangents at these points is a curve of the fourth degree and third class, having cusps at A, B, 0, and touching both conies. 882. Prove that the envelope of the straight line joining the points of contact of parallel tangents to two given parabolas is a curve of the third degree and fourth class ; having three points of inflexion the tangents at which are the common tangents of the parabolas. ( 187 ) THEOEY OF EQUATIONS. 883. The product of two unequal roots of the equation aa^ + haf + cx + d = is unity ; prove that the third root is - — -z . 884. The roots of the equation x''—px + q = may be ex- pressed in the forms 2 i ^ , andp-£ i . p—p—p~... p—p—... Explain these results when p' < iq. 885. If the equation of —pa? + qx — r= have two equal roots, the third root must satisfy either of the equations x(x—py = 4:r, (x—p){3x+p) + 4:q = 0. 886. The roots of the equation a? —px' + qx — r = are the sines of the angles of a triangle ; prove that p* - ip'q + 8pr + 4r' = 0. 887. Determine the relation between q and r necessary in order that the equation x^ — qx + r=0 may be put into the form X* = (x' + mx + n)' ; and solve in this manner the equation 8a:' -36a! + 27 = 0. 888. Pind the condition necessary in order that the equation cols' + hx' + cx + d=Q may be put under the form X* = (a;' +px + g)' ; and solve by this method the equation 188 BOOK OF MATHEMATICAL PROBLEMS. 889. The roots of the equation x'—px' + qx — r = are in harmonical progression ; prove that those of the equation n(pq — n'r) x^ — {p^ — 2npq + 3 w V) x' + (pq — 3nr)x — r = are also in harmonical progression. 89G. Eeduce the equation x" —px' + qx- r = to the form 2/° =b 3y + m = by assuming x = ay + b; and solve this equation by assuming y = s =p — . Hence prove that the condition for equal roots is 4 (/ - Sqf = (2/ - 9pq + 21r)'. 891. In solving a cubic by Cardan's rule ; if a, yS, y be the roots of the complete cubic, the roots of the auxiliary quad- ratic are (2a-/3-y)(2;8-y-a)(2y-a-;8)d.3^(-3)(/3-y)(y-a)(a-;8) 64 892. Solve the equations (a;» + 4a! - 2)' + 3 = 4cB (Sa;' + 4), a;*_6a!'-8a;-6 = 0. 893. If a, P, y, S be the roots of the equation X* + qx' +rx + s=0, the roots of the equation 8'x* + qs{l-syx' + r{l-syx + {l-sy = ■will be j^ + y + 8 + -~, 894. In the equation X* —pa? + q3? — rx + s=(i] prove that the sum of two of the roots will be equal to the sum of the other two, if 8r - ipq + p' = Q; and that the . product of two will be equal to the product of the oth^r two, if ^"s = r'. THEORY OP EQUATIONS. 189 895. If a, p, y, S be the roots of a biquadratic, and the equation be solved by putting it in the form {x' + ax + by = c{x-dy, the values of 26 are ^y + aS, ya + /3S, ajS + yS; those of 4c are (^ + y-a-8)^ {y + a-13-By, (a + ^-y_S)=j and of d are /8y — aS ya — ^S a^ — yS P + y-a-6' y + a-^-B' a + /3-y-8' 896. In the method of solving a biquadratic in x by sub- stituting X = my + n, and making the resulting equation in y reciprocal ; prove that the three values of n are /8y — aS ya — ;88 ayS — yS ^, J and — y3 + y_a-8' y+a-j8-S' a + /3-y-8' and those of m' are (a-y)(a-8)(^-y)(ff-8) - (a + ^-y-Sr ''^°- a, )8, y, 8 being the roots of the biquadratic. 897. Prove that the eqiiation 3a!*+8a;*-6a!=-24a; + r=0 will have four real roots ifr< — 8> — 13j two real roots if r>— 8<19; and no real roots if r>19. 898. Prove that the equation SB" + ra;""'' + s = will have two equal roots if 899. If /(«) = (a; - a,) {x-a^...{x- aj ; and a„ «„ . . . o„ be all unequal; then will + -^7^-c+ + - /K) /'(«J /'(«J 190 BOOK OF MATHEMATICAL PROBLEMS. be equal to tlie sum of the homogeneous products of r dimensions of the n quantities a^, a^ ... a„. Prove also that />..) s^j + ... + 1 fi ly 900. The equation a. = will be an identical equation if 2(a) = 0, 2(a6) = 0, '^{aV)=0, S(«&°-')=0. 901, If w be a prime number, and a an impossible root of the equation a;°=l, then will 1 a a" a' a""' n + -^ + — -^ + ...+- m + I m + a m + a? m + a? '" m + a" ' 1 — (-«»)"' 902. If the system of equations of which the type is aj'jCj + a^x^ + . . . + «„'«!„ = c' be true for integral values of r from r=\ to r = n+ 1, they will be true for any value of r. 903. Having given cos na+p^ cos (w - 1) a +p^ cos (« - 2) a + ... +p_^ = 0, sinwa+Pj sin (n- l)a+^j, sin(n- 2) a + ... +^__= ; prove that 1 +p^ cosa+^j cos 2a + ... +p^ cosmo= 0, and 1 +p^ sin a + ^j sin 2a + ... +p^ sin na = 0. 904. Find the commensurable roots of the equations (1) «' + 5X* + 1 20a!" - 524a; - 24 = 0, (2) 6!B*-iB'-i2a!»- 27a! + 18 = 0. THEORY OF EQUATIONS. 191 905. One of the roots of the equation (l+a!)* + a!"+l = is the square of another root ; explain why, on attempting to solve the equation from the knowledge of this fact, the method fails. 906. If a, 13, y, B, ... be the roots of the equation then wUl 2(2a-^-y)(2^-y-a)(2y-a-/3) = (w - 1) (w - 2)p,' -3n{n- 2) p^p^ + Zv?p^ ; and determine what symmetrical function of the differences of the roots is equal to (»i-l)(9i-2)(w-3) . „ . -, . „, , = - -^ 2 " ^' + 2m (w - 2) {n - 3) p^p^ - dw" (w - 3)pji^ + ^r^Pv 907. The equation a" - 209a; + 56 = has two roots whose product is 1 j determine them. 908. Prove that all the roots of the equation (1 -a3)"-mwa3(l -as)" '+ — ^^j^ — - ^ ^ a' (1 -«)" ^ — ... to m + 1 terms = are real ; and that none lie beyond the limits and 1, n being integral and m, = 3?/ - 85.53 + 2O5,, Sp.Va - ^P^V! - 50^,p, + 5p^3 + 250;j. = Sy.Va - 32',5/ - 505,5^ + 5q^q^ + 25O5,. THEOEY OF EQUATIONS, 916. Prove that 1-n, 1, 1, 1 1, l-n, 1, 1 1, 1, l-n, 1 1, 1, l-n 193 = ; n being tie order of the determinant. 917. Prove that «!, 1, I, I 1, X, 1, 1 1, 1, X, 1 1, 1, 1, E(aj-l)""'(a! + w-lj; w be- ing the order of the deter- minant. 918. Prove that «„ 1, 1, 1 I, ajj, 1, I 1, 1, a^a' 1 1, 1, h «!„ + (_1)-(^_1)J X , x^ a;, being the roots of the equation x" -p,a^-' +p,:^-' -. + (- 1)>„ f= 0. 919. Prov6 that X, x", x", as', aj', X*, x', X x", X*, x", X, a? n[n-l) s(-l) ' a;" (a:" -I)"-' W, 13 194 BOOK OF MATHEMATICAL PKOBLEMS. 920. Prove that {cosg-cos(w+l) ef- {1 -cos n6]' 2(-l) ' (l-cosw^) COS 0, COS 26, cos 36, cosnd cos 2^, cos 3^, cosnO, cos6 cos3^, cos 6, cos 2^ cosn.^, cos 6, cos 26, ... cos (n — l)6 921, Tte product of the roots of the equation x^2, 1, 0, 0, 0, 1, x~2, 1, 0, 0, 0, 1, x-2, 1, 0, 0, 0, 0, 0, 1, x-2, 1 0, 1, x-2 = is TO + 1 ; and the sum of the roots is 2to, to being the order of the determinant, 922, ProTe that 1 - x^, x^ (1 - x^), x^x^ (l-xj, a;^a;. . . . a!„_, (1 - xj, x^x^ ... a;. -1, l~x^, x^{l-x^), x^...x^_^(l-xj, «,...«;, 0, -1, !-«« 0, 0, 0, 0, 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. ( 195 ) DIFFERENTIAL CALCULUS. 923. Having given . /tt \ . /2ir \ sma; sin ( — hajlsm { i-x] \m J \m J sin I (»i - 1) - + «! = 2~('"~') sin mx ; prove that cot a; + cot{ — I- a;) + cot ( ha;) + .... + cot j(»i — 1) — i-a3> = «i cot mx, 924. Prove that the limit of (cos a;)™ *, as x diminishes in- definitely, is e~*- 925. If y = cot-' X, -T^= (— 1 )"[«. — 1 sin ny sin"y. 926. Prove that, if w be a positive integer, the expression , . af-' 2a;"-' 3«"-" , ,, (x-viSf. +i =-+j n-+-j 5"+ ...+(w-l)a; + w ^ ' \n-\ p- 2 |w-3 ^ ' will he positive for all positive values of a;. 927. If , d y dy „ dx' dx " ' then •wUl 13—2 196 BOOK OF MATHEMATICAL PEOBLEMS. 928. Having given prove that 929. If sin (»» tan"' a;) = o,a! + «,«' + ... +ajc'+ ...', prove that (m + 1) (m + 2) a.„ + (2m* + m») a. + (m - 1) (n - 2) o„_, = 0. 930. K {log(l +a;)}' = a3a;° + a^aj*+ ... +ajd'+ ..., then will 6f-ir*' / 1 1 1\ («+2)«.„+(« + l)«.,i = -^^(l+2+g + ...+-j. 931. In the equation /{x + h)=/(x)+hf'{x + eh), tie limiting valne of 6, when A is indefinitely diminished, is | ; and, if 6 be constant, /(x) = A + Bx+ Cx', where A, B, G are in- dependent of X. 932. In the equation J{x + h) =/(cb) + hf'{x + 6h), prove that the first two terms of the expansion of B in ascending powers of h are 1, h f"(x)_ 2 2i/'{x)' snd, if y(a:) = sin a;, the first three terms of the expansion are 1 h cot X h^ •S + 2 24 48sin'«' provided that cot x he finite. DIFFERENTIAL CALCULUS. 197 933. In tbe equation. _ /(«■+ h) =/{a) + lf{a) + ...+ ^/'(fl + eh), the limiting value of 9, when h is indefiaitely diminished, is j; and, ii/(x) be a rational algebraical expression of w+ 1 dimensions in x, the value of 6 is always 5- . ' •' n + 1 934. In the equation F{x + h) -F{x) _ F'jx + Oh) f{x + h)-f{x) ~ f{x + 6hy prove that 6 will be constant if ^(a;) = sina; and y (a;) = cos a;. Prove also that, in all cases, the limiting value of 0, when h ia indefinitely diminished, is ^. 935. Prove that •' IT E -1 I- 1 1 ■ +rT^' + rTKi + •••*«>'»' 4 £" + 1 \ + V 1+3' ' 1 + 5' and that 936. Prove that 1 + g+ Q + ••••+ ->log(l+Ji) ^ 1 ; a a TT ' ■ \ i wa in the last being <^ . 937. If y = x cosxy, the general term of the expan^on of sin xy in terms of a; ' is (2w + 2)(2« + l) . .,,„ ^ 1x1 ^^ !-^ — , — ^(»-l)''°+ ... to»i+l terms L 198 BOOK OF MATHEMATICAL PEOBLEMS. 938. The limiting values of d' ( X Y" d* d' ( X )'*' dar\smx) ' ^°^'"*'°**^ ' dar{log(l+x)j ' are respectively 0, 0, 1, if w be odd ; and {1.3.5... (w-1)}', (- 1)^Lm, and 1, if M be even. 939. If/(a)=0 and <^(a) = 0, and if the limit, as x ap- f(x) proaches a, of ^rVr be finite; the limit of {/(a;)}*'*^ will be 1. 940. If /(a) = 1, ^ (a) = 00 , the limiting value of {/(«)}* '^> as X approaches a, is « , m being the limit of {x — a) <^(a;). 941. If z = €'/^{x + y) + ^'f,{x + y)+...+e"f,{x + yy, then will \dx dy J \dx dy )'" \dx dy J 942. If then will {p + q-l){p + q-2)...{jp + q-n)z = (i, where p'lfz denotes /y\ d'-^'-z «-y. a-vj|)-./-v. m; xj dx'd'if 943. If x, y be co-ordinates of a point referred to axes in- clined at an angle oi, and u any function of the position of the point ; 1 (d'u ^_2 .^) 1 (d'u d?u I d'u \'\ aia' \da? dy^ \dxdy} y will be independent of the particular axes. DIPPEEENTIAt CALCULUS. 199^ 944. Having given x + y = X, y = XY; prove that d'u d'u du _ yd'u _ d'u du ^ d^'^y d^dy~ di~ dX'~ dXdf~dX- 945. If 2a; = »•(£» + £"»), 2y = »• (c* - c'e) ; d^u d^ti _d^v, 1 d^-a, 1 d'U, do? ~ ly^^ dp ~T' ~de^ '^ 1- Ir' 946. If « = £»+* + €«-«, y = £«+tf - £»-* ; 947. If £' = 7-E™««, £* = »-e™«; ,■ sin 9 sin , x^ = r sin cos tfi, x^ = r cos $ sin ^, a;^ = r cos 6 cos i/r ; then will d^ d^ d^ d^_d^ I d^i 1 d'u, dx^' dx^ dx^ dxj dr" r' d6' r" sin^ 6 d<^' 1 d'u 3 du 2 , „ . du j+ - ~r + -5 cot 25 — ; . r' cos^ e d^' r dr v^ d9 ' 949. If x, y, z be three variables connected by one equation; prove the formulae for changing the dependent variable from » to a;; dx_\ dx _ q d'x _ r dz^ p^ dy~ p ' d^~ />° ' d^x _qr—p8 d^x _ pH—l'pqs+tfr dydz f ' dy' p' 200 BOOK OF MATHEMATICAL PEOBLEMS. 950. The distances of any point from two fixed points are r, r, and a maximum or minimum value oifir, r') for points lying on a given curve is c ; prove that the curve /(r, r') = c ■will touch the given curve. 951. In the Straight line bisecting the angle ^ of a tri- angle ABC is taken a point P ; prove that the difference of the angles APB,-APG ■will he a maximum ■when AP is a mean proportional between AB and AG. 952. Find the maximum and minimum values of a normal chord of a given ellipse : proving that, if one exist other than the axes, the eccentricity must be > — t, , and that the length of such q in 212 a chord ■will be —^ 5 , -whei-e 2a, 2J are the axes. ' (a? + 6^ 953. Kormals are drawn to an ellipse at the extremities of t^wo conjugate diameters : prove that a maximum or minimum distance of their point of intersection from the centre is -^5 — r— , , 2 provided the eccentricity is > — ^ . Examine which of the two it is. 954. If as + y + » = 3c, f{x) . f{y) . f{z) -will be a maximum or minimum •when x.= y = z=c, according as 955. The minimum area ■which can be included between two parabolas, ■whose axes are parallel and at a distance c, and /3 ' -which cut each other at right angles in two points, is c' -^ • 956. is the centre of curvature at a point P of a given ellipse, and 0P\ OQ are normals drawn from : prove that, if a" < 26', PQ' has its minimum value when the eccentdc angle „p. , _. f 2a'-/.« )i DIFFERENTIAL CALCULUS. 201,. 957. Prove that three parabolas of maximum latlis rectum can be drawn circumscribing a given triangle ; and tbat if a, j3, y be the angles •which the sides make with the axis of any one. of them, cot a + cot P + cot y =-0. 958. Find the plane sections of greatest and least area which can be drawn, through a given point on the surface of a paraboloid of revolution : proving that,, if 0^ , 6^ be the angles' which the planes of maximum and minimum section make with the axis, « 3 tan 6 tan p^ = tt • 959. If X, y, m be the distances of any point in a plane from three given points, andy(a;, y, z) be a maximum or minimum; 1 df_ 1 d/ _ 1 #. siu (y, z) dx sin (z, x) dy sin {x, y) dz ' (y, z) denoting the angle between the distances y, z. 960. If A, B, C, D be four points not in one plane,' and F a point the sum of whose distances from A, B, G, D is a mini- mum ; then will PA . PA' PB.PE _ PG.PC _ PD . PD ' . ~I3/ :BB' CC " DI)' ' PA, PB, PC, PD meeting the opposite faces of the tetrahedron in a; ff, C, B'. 961. If A, B, C, D be four paints not in one plane, aaid P a point at which l.PA' + m.PB' + n.PG^+r.PD' is a maximum or minimum, then will vol. PBGD _ vol. PCDA _ vol. PDAB ^ v&l. PABG I m n r 202 BOOK OP MATHEMATICAL PROBLEMS, 962. If u, X, y, z be the distances of any point from four given points not in one plane, «qA. f(yi,j x,y, «) be a maximum' or minimum; 1 (MS 1 (M^ 1 _ a» _ j« _ c^ + 2aie \du) ~ 1 - a" - &'" - c" + 'AoMc \ dx) l-a"-b'~ c" + 2a'bc' \dy) 1 - a'" - b" - c= + 2a'b'c \dz) ' o, h, c, a', h', c' denoting the cosines of the angles between the distances (y, z), (a, x), {x, y), {u, x), (u, y), (u, z) respectively. 963. P, P are contiguous points of a curve, PO, P'O are drawn at right angles to the radii vectores SP, SP" ; prove that the Umitiag value of PO, as P' moves up to P, is =*= -5^ . 964. *S', H are two fixed points, P is a point moving so that the rectangle SP, SP is constant : prove that straight lines drawn from S, H at right angles respectively to SP, HP will meet the tangent at P to the locus of P in points equidistant from P. 965. In the curve y' = Sasjc" — a;', the tangent at P meets the curve again in Q ; prove that tan QOx + 2 tan POx = 0, being the origin. Prove also that, if the tangent at P be a normal at Q, P lies on the curve iy (3a — x) — {2a — x) (16a — 5a;). 966. In the curve y' = a'x, the greatest acute angle between 3 two tangents which intersect on the curve is tan ' -j . 967. A tangent to the curve a;* + 2/" = a which makes an angle tan"' ^ with the axis of a; is also a normal to the DIFFERENTIAL CALCULUS. 203 curve. Also if two tangents be drawn to the curve from a point P lying on the curve (P not being the point of contact of either), the acute angle between these tangents cannot exceed 60°. 968. The tangent to the evolute of a parabola at a point where it meets the parabola is also a normal to the evolute. 969. If from a point on the evolute of the ellipse -5 + , j = 1, the two other normals to the ellipse be drawn ; the straight line joining the points where they meet the ellipse wiU be normal to the ellipse (aV+6y)(a'-6y = a'6*. 970. Prove that, for any curve of fhe third degree, there exists one pcdnt such that the points of contact of the tangents drawn from it to the curve lie on a circle. 971. A tangent to a given ellipse at P meets the axes in two points, through which are drawn straight lines at right angles to the axes meeting in p : prove that the normal at p to the locus of p and the line joining the centre of the ellipse to the centre of curvature at P make equal angles with the axes. 972. In a curve of the fourth degree, which has four real asymptotes, no two of which are parallel, the asymptotes will meet the curve again in eight points lying on a conic. Determine this conic in the case of the curve xy{«? -y") +a{x'' + 'it) = a'{{x + yy -a{x + y)+d?}: and prove that three of the asymptotes touch the curve in points not at infinity. 973. If the equation of a curve of the «'•> degree be and ^(«) = have two roots /ij and if also ^Jji,)=0; the equation of the corresponding rectilinear asymptotes will be 20* BOOK .OF MATHEMATICAL PEOBLEMS. 974. Two circles- of radii 6, a-h, respectively roll ■witMn s circle of radius a, their points of contact -witli the fixed cii'cle being originally coincident, and the circles rolling in opposite directions in such a manner that the velocities of points on the circles with respect to their respective centres are equal : prove that they will always intersect in the point which was originally the point of contact. 975. In a hypocycloid the radius of the rolling circle is ^ j- times the radius of the fixed circle (n integral) • prove that the locus of the point of intersection of perpendicular tan- gents is a circle ; and that the line joining the points of contact is also a tangent to a hypocycloid having the same fixed circle. 976. A circle is drawn to touch a cardioid and pass through the cusp : prove that the locus of its centre is a circle. If two such circles be drawn, and through their second point of intersection any straight line be drawn, the tangents to the circles at the ends of this straight line will intersect on the cardioid. 977. If S, II be foci of a lemniscate, FT the tangent at any point P, . „p_, SP~3ffP .inSPT= ^^^^^, ; and if 6, tft be the acute angles which the tangent makes with the focal distances, ^2 sin -~- = cos — -— . z ^ 978. Two circles touch the curve r" = a'' casmO in the points P, Q and touch each other in the pole iS : prove that the angle PSQ is equal to y— — ^ n being a positive or negative integer. 979. The locus of the centre of a circle touching the curve r" = a" cos m6 and passing through the pole is the curve (2r)" = o" cos n9i {w (1 - m) = m}. DIPFEEENTIAL CALCULUS. 205 980, In the curve r=aaec'6, 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 ^e, as n increases from to 00. If » be negative, there is no real point of inflexion. 981, ST is the perpendicular from the pole S on the tangent to a curve at P; 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 when there is a point of inflexion at Z in the locus of Y, the chord of curvature at P through S will be equal to iSP. 982, The general equation of a curve of the fourth degree having cusps at ^, £, C is ABC being the triangle of reference, 983, The equation will represent a catenary if 46c = a'. 984,- If as, y be rectangular co-ordinates of any point on a curve, p the radius of curvature at that point, ^ the angle which the tangent at the point makes with a fixed straight line, 985, 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 of P in Q ; prove that the radius of curvature of the evolute at is eq^ual to 30Q, 206 BOOK OF MATHEMATICAL PEOBLEMS. 986. The reciprocal polar of the evolute of a parabola Tvith respect to the focus is a curve whose equation is of the form r cos B = c sin' Q ; and that of the evolute of an ellipse is c _ ^ e sin 6 the focus being pole and the axis the initial line in each case. 987. A rectangular hyperbola, ■whose axes are parallel to the co-ordinate axes, has contact of the second order with a given curve at a given point (x, y) : prove that the co-ordinates {X, ¥) of the centre of the hyperbola are given by the equations X— «___ _\dx/ dx doif and that the central radius to the point («, y) is the tangent at {X, T) to the locus of the centre of the hyperbola. If the given curve be (1) the parabola ')f=4:ax, (2) the circle x' + y' = a', the locus of the centre of the hyperbola is (1) 4={x + 2a)' = 27a/, (2) x^ + y^={2af. 988. A series of rectangular hyperbolas have their axes parallel to the axes of the ellipse and have with it contact of the second order : prove that the locus of their centres is a curve similar to the evolute of the ellipse, and whose dimensions are to those of the evolute as a' + b' : a'-V. 989. A curve is such that any two corresponding points of its evolute and an involute are at a constant distance : prove that the liae joining the two points is also constant in direction. ' DIFFEEENTIAL CALCULUS. 207 990. Prove that in any epicycloid or liypocycloid, the radius of curvature is proportional to the perpendicular on the tangent from the centre of the fixed circle. 991. The curvature at any point of a lemniscate varies as the difference of the focal distances. 992. If the tangent and normal at a curve be taken as the axes of X, y, the co-ordinates of a neighbouring point are, ap- proximately, p being the radius of curvature at the origin, and s the arc measured from the origin. 993. A loop of a lemniscate rolls in contact with the axis of X, prove that the locus of the node is given by the equation and if p, p' be corresponding radii of curvature of this locus and of the lemniscate, 2pp' = a". 994. If the curve r" = a" cos m^ roll along a straight lin'e, the radius of curvature of the path of the pole is 'Hr 995. A rectangular hyperbola rolls on a straight line : prove that the radius of curvature of the path of the centre is half the distance from the centre to the point of contact; and that the length of any portion of the path of the centre is equal to the corresponding arc of the locus of the feet of the perpendiculars let fall from the centre on the tangent. 996. A plane curve rolls along a straight line ; prove that the radius of curvature of the path of any point fixed with respect 208 EOOK' OP MATHEMATICAL PEOBLEMS. to the curve is -. — - . ■where r is the distance from the fixed r— p sin ■ point to the point of contact, i/" being the mass of the lamina, 2a the major axis, o and e the eccentricity. 1017. If the areas of the curves ay{x-by={a'-x'){hx-ay, x' + y' = a', {b>a) be A, A'; prove that, as b decreases to a, the limiting value of A'-A is Grra. b — a 14—2 212 BOOK OF MATHEMATICAL PROBLEMS. 1018. Perpendiculars are let fall upon the tangents to an ellipse from a point within it, ■whose distance from the centre is c : prove that the area of the curve traced out by the feet of these perpendiculars is lia^+h' + c^. 1019. Find the whole length of the arc enveloped by the directrix of an ellipse rolling along a straight line during a com- plete revolution; and -prove that the curve will have two cusjds if the eccentricity of the ellipse exceed -^^^r — . 1020. Two catenaries touch each other at the vertex, and the linear dimensions of the outer are twice those of the inner; two common ordinates MPQ, mpq are drawn from the directrix of the outer: prove that the volume generated by the revolution of Fp about the directrix is Sir x area MQqm. 1021. Find the limiting values of (1) -^sm-sm — ^ sm — ... sinm — 1)- V , ^ ' \ n n , n ^ ' n) (2) rco roo /-co yi ri ri I j f j Vdx^dx^dx^dx^= I Ml V'u^^u'u^du^du^du^dutj where a::, + ajj + a;,. + x^ = u^, ,x, + x^ + Xt = u,u„ ^ >,_ ,, 1028. Prove that , dx^ dx^ dx^ dx^ dXi^ I j ... Vdx^i Jo -'0 214! BOOK OP MATHEMATICAL PEOBLEMS. where as, = r sin. 0^ cos 6^ x^ = r cos 0^ cos 6^, a, = r sin 0^ sin 0^ cos 6„ x^^r sin ^^ sin 0^ sin ^4, a!j = »• cos 6^ sin ^^ cos 6^, x^ = r cos ^, sin 6^ sin 6^. 1029. Prove that ijj...dx^dx^...dx^, is equal to n+1 \n{n + l)j ^G-0' the limits being given by the equation, — V. n*. * J- 2n < + a;/ + . . . + as„» - a3,a;^ - aj^x^ - . . . - a!._,a5„ - a. + \- = 0. " 1030. Prove that the limit of the sum of the series \ a? \ x'' when X is indefinitely increased is -^ . 1031. Prove that, if m be a positive integer, £(i-.r'iogQ«f.=i(i.i4......^), and that jJ{l-yr-lo,UxdyJ^(l.\.l.....'^. 1032. Prove that TT /■"* 1 (1) I sin 4a3 log cot atcZas = ^ , ,_, [a xcosx . IT (2) / .j r-K-dx =--a, ^ ' JjL 1 + sin' X 2 INTEGRAL CALCULUS. 215 (4) _[%in-(csiiia;)& = c + |-'+|! + ... to oo , (c J{n + 1) U (,. + 1)1 r (- + l) " 1051. Find the singular solution of the equation dy /<^2/V «(« — «)_ «-.^-..=..,|.(|) and trace the locus of the points whose coordinates satisfy the singular solution. 1052. Find the differential equation of a curve, such that the foot of the perpendicular from a fixed point on the tangent INTEGEAL CALCULUS. 219 lies on a fixed circle; and obtain the general integral and singular solution. 1053. Along the normal to a curve at P is measured a con- stant length. PQ; is a fixed point and the curve is such that the circle described about OPQ has a fixed tangent at ; find the diflferential equation of the curve, the general integral, and sin- gular solution. 1054. If PM, PG be the ordinate and normal from a point P of a curve to the axis of x, find a curve (1) in which PM' varies PM^ as PG; (2) in which the curvature varies as 3 : and prove that one species of curve satisfies both conditions. 1055. Find the orthogonal trajectory of the circles A. being the parameter. 1056. Integrate the equations : (1) x'^£=2y + x''-'y~<^yjy, (3) -^^ + 2x + 2y = -j^ + x + Zy = cosnt, ,,•. n d"^ c dz dz c\ dz a dz ^ ^ (5) m^,,-3m, + 4«/=0. 1057. The general solution of the equation **,(2aJ + l -«»+.) = «', is w^=a! + |a + 2Q} . 220 BOOK OF MATHEMATICAL PROBLEMS. 1058. Find the general solution of the equations /(« + y) =/(«) ^(y) + <^ (a')/(2/)> ^(« + y) =/(«') f{y) + <^ («) ^ (y) ; and hence prove that 2 cos x, 2 v — 1 sin x, can be expressed in the form F + h-", ¥-h~\ 1059. Prove that , e^' - 1 X a? B, x" B, x' 5 lOS = --) — ! —'-I .— *+ ^ x 2[2'2 |44[66 ^ '' j2w 231 ( 221 ) SOLID GEOMETEY. I. Straight Line and Plane. 1060. The co-ordinates of four points are a — h, a — c, a — d; b — c, b — d, b — a; c—d, a—a, c — b; d — a, d — b,d—c; respect- ively : prove that the straight line, joining the middle points of any two opposite edges of the tetrahedron of which they are the angular points, passes through the origin. 1061. Of the three acute angles which any straight line makes with three rectangular axes, any two are together greater than the third. 1062. The straight line joining the points (a, b, c), (a, V, c',) will pass through the origin if aa' + bV + 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 a, /3, y. 1063. From any point P are drawn PJf, FJV perpendicular to the planes of zx, zy, is the origin, and a, /?, y, 6 the angles which OP makes with the axes (rectangular) and with the plane OMJ^ : prove that 1111 siu^^ sin^a sin^;8 1064. The e.quations of a straight line being given in the form a + mz — np b + nx — lz c + ly-mx (1) I m 222 BOOK OF MATHEMATICAL PROBLEMS. ._: a + mz — ny _h + nx — lz _c+ly~mx ^^ X ^ V ' obtain them in the form L M N ' 1065. A straight line moves parallel to a fixed plane and intersects two fixed straight lines not in the same plane : prove that the locns of a point which divides the part intercepted in a constant ratio is a straight line. 1066. Determine what straight line is represented by the equations a + ma — ny_h + nx — lz_c + ly — mx •m — n n — l l — m Result. The line at oc in the plane 03 (m — w) + 2/ (m — Z) + » (Z — ot) = ; unless la + mh + wc = 0, when it is indeterminate. 1067. The equations Ix + my + OT« = 0, an? + 63/' + c^ = 0, represent two straight lines, the cosine of the angle between which is V(b + c) + m° (c + a) + n''{a + h) 'JF(b-cy + ... + .. . + 2m'n'{a-b){a-c)+... + ...' 1068. A straight line moves parallel to the plane y = z., and intersects the curves 2/ = 0, g' = mx ; » = 0, y' = — mx ; prove that the locus of its trace on the plane of yz is two straight lines at right angles to each other. SOLID GEOMETRY. 223 1069. A straight line always intersects at right angles the straight line x + y = s=0, and also intersects the curve 2/ = 0, oc/'^az : prove that the equation of its locus is a? — y^ = az. 1070. The equations ax + cy + Vz _ c'a3 + Jy + a'« h'x + ay + cz X y z represent in general three straight lines mutually at right angles ; but, if c - ca ab a r = b rr = C -, a b c they represent a plane and a straight line perpendicular to that plane. 1071. The two straight lines cos a sin a ' meet the axis of x in 0, 0' ; and P, P are points on the two such that (1) OP = hO'F; (2) OP.O'P' = c'; (3) 0P + 0'P' = '2,c; prove that the equation of the locus of PP' is (1) (x + a) [y sxaa + z cos a) = k{x — a) {y sin a — » cos a) ; /2N t y' , ^ -1- ^ ' a" c^cos^a c^'sin^o ' and xy az c , „ n ^ ' cos a sm a a the points being taken on the same side of the plane xy. 224 BOOK OF MATHEMATICAL PEOBLEMS. 1072. A triangle is projected orthogonally on eacli of three planes mutually at right angles: prove that the algebraical sum of the tetrahedrons which .have these projections for bases and a common vertex in the plane of the triangle is equal to the tetrahedron which has the triangle for base and the intersection of the plane for vertex. 1073. A plane is drawn, through the straight line X y « _ I m n' prove that the two other straight lines in which it meets the surface if) — c) yz (mz — ny) + (c — a) zx (nx — Iz) + (a — h) xy (ly — mx) = are at right angles to each other. 1074. If (?j, m,, Wj), (l^, m^, n^, (l^, m^, n.^ be the direction cosines of three straight lines which are, two and two, at right angles, and if ah cab c . then will T- -I H — = : and =-r^ = K '"K w„ Lli, m,m,jn.. n,n„n. 1075. The equations of the straight lines bisecting the angles between the two straight lines given by the equations Ix + my + nz = 0, ax' +hy^ ■¥ cz^ = Q, are Ix + my + nz=0, lyz (b — c) + mzx (c — a) + nxy (a — b) = 0. 1076. The straight lines bisecting the angles between the two lines given by the equations lx + my + nz = 0, ax^ + by' + cz' + 2a'yz + 2b' zx + 2c'xy = 0, lie on the cone x' {c'n - b'm) + ... + ... + ys{c'ni-b'n + (c -b)l} + ... + ... = 0. SOLID GEOMETEY. 2'2o 1077. If X, y be tlie lengths of two of the lines joining the middle points of opposite edges of a tetrahedron, oj the angle between these lines, and a, a those edges of the tetrahedron ■which are not met by either of the lines, a ~a cos 0) = — -. . 1078. The lengths of the three pairs of opposite edges of a tetrahedron are a, a ; h, b'; c, c: prove that, if 6 be the acute angle between the directions of a and a, cos 6 = ^ ^ y . Aaoi 1079. The line joining the centres of the two spheres -which touch the faces of the tetrahedron ABGD opposite to A, B respec- tively, and the other faces produced, will intersect the edge CD in a point P such that OF : PD :: AACB : AADB; and the edge AB (produced) in a point Q such that AQ iBQ -.-.ACAD : ACBD. 1080. On three straight lines, meeting in a point, are taken points A, a; B,h; G,c respectively: prove that the intersections of the planes ABC, abc; aBG, Abe; AbG, aBc; and ABc, abG all lie on one plane which divides each of the three straight lines har- monically. 1081. If through any point be drawn three straight lines each meeting two opposite edges of a tetrahedron ABGD ; and if a, a; b, /3; c, y be the points where these straight lines meet the edges BG, AD; GA, BD; AB,GD; then will £a.Gy.Dj3 = BI3.Ca.Dy, Gb.Aa.Dy=Gy.Ab.Da, Ac.B^.Da = Aa.Bc.D^, Ab.Bc.Ca^Ac.Ba.Ch. w. 15 226 BOOK OF MATHEMATICAL ^PROBLEMS. 1082. Any point is joined to the angular points of a tetra- hedron ABGD, and the joining lines meet the opposite faces in a, b, c, d: prove that Oa Ob Oc Od_ Aa'^Bb'^ Cc *J)d~ ' regard being had to the signs of the segments. Hence prove that the reciprocals of the radii of the eight spheres which can be drawn to touch the faces of the tetrahedron are the eight positive values of the expression 1111 =fc — ± — d= — ± — ; P, P^ Vz P* p ,p^, p^, p^ being the perpendiculars from the angular points on the opposite faces. 1083. Tf A, B, G, D be the areas of the faces of a tetra- hedron ; a, b, c, u, j8, y, the cosines of the dihedral angles {BC), (GA), {AB), {DA), (DB), (BG), respectively; then will A^ B^ C^ l-a'-b'-c'-2abo ~ l-a'~l3'-(^-2al3o ~ \-a^- V-f-2aby ^ ~l-a'-y-c'-'2,abc' 1084. With the same notation as in the last question, prove that for all real values of I, m, n, r, p + m' + n^ + r'> 2mna + Inl^ + 2lmy + 2lra + 2mrb + 2nrc ; except when I m n r A^B^G^D' 1085. Three straight lines are drawn, two and two at right angles, through a given point, and two of them lie respectively in two fixed planes : the locus of the third is a cone of the second degree, whose sections parallel to the fixed planes are circles. 1086. A point is taken within a tetrahedron ABGD so as to be the centre of gravity of the feet of the perpendiculars let SOLID GEOMETET. 227 fall from on the faces : prove that the distances of from the several faces are proportional respectively to the faces. 1087. The equation of the cone of revolution -which can be drawn touching a system of co-ordinate planes is {Ixf + {my)^ + {nzf = 0, the ratios I : m : n being given by the equations ■m' +n'~ 2mn cos a _n^ + P- 2nl cos ft _P + m'- 2lm cos y sinV sin^/3 ~ sin^y ' "where a, ^, y ai-e the angles between the axes. (See question 301). 1088. The inscribed sphere of a tetrahedron ABCD touches the faces in A', Jff, C, J)': prove that AJ!, BS, CG', DU will meet in a point, if a a. h & c v cos ^ cos ^ = cos jr cos ^ = cos jj cos ^ ; Z Z Z Z Z Z where a, o.; 5, j8j c, y are pairs of dihedral angles at opposite edges. II. linear Transformations. Geoieral Equation of the Second Degree. The following simple method of obtaiaing the conditions for a surface of revolution is worthy of notice. When the expression ax' + ly^ + cz^ + 2a'yz + 2h'zx + 2c'xy is transformed into AX^ + BY^ + CZ", we obtain the coefficients A, B, G from the equivalence of the conditions that h{!ii^ + y" + z') - ax'^ -by' - , and 7i{X'+Y'+Z')-AX'-BT'-C2;' 15—2 228 BOOK OP MATHEMATICAL PROBLEMS. may separate into (real or impossible) linear factors : which is the case when h=.A, B, or C. But if two of the three coincide as -S = C; then when h = B the two factors become coincident, or either expression is a complete square. The conditions that this may be simultaneously the case in the former expression give us {B-a)a' = -h'c', &c., 7 / / / / /If or B^a r = 6 — TT =0 r ii a, b', c be all finite. a b c If a' = 0, then 6'c' must also vanish; suppose then a', 6' = 0, therefore B = c, and we must have (c -a)x' + {c- b) y' - 2c'xy a perfect square, whence c" = (c-a)(c-b). In the case of oblique axes, inclined at angles a, ^, y, we must have h {3?+ y^+z^+ 2yzco3 a + 2zx cos /3 + 2xy cos yj-ax"— ... — 2dyz~ ... a complete square. It follows that the three equations (A - a) {h cos~a — a') = (/t cos ji - V) (h cos y - c'), {h - h) {h cos /? - 6') = {h cos y - c) {h cos a - a'), {h ~ c) {h cos y-o'} = (k cos a - a) {h cos p - b'), must be simultaneously true, and the two necessary conditions may be found by eliminating h. 1089. If there be two systems of rectangular co-ordinates, and ^i> ^a' ^-6 ^e the angles made by the axes of x', y', z with that of z, and <^j, <^2, ^3 the angles made by the planes oizx, zy', zz' with that of zx; then will tan^^, + ^ "°«('^.-y ^ = ' cos(<^3-,^Jcos(<^,-,^J "' witji two similar equations. SOLID GEOMETEY. 229 1090. By transformation of co-ordinates, prove that tiie equa- tion ^ + y^ + ^ + yz-¥ zx -^ xy = a" represents an oblate spheroid whose polar axis is to its equatoreal in the ratio 1:2; and the equations of whose polar axis are x = y = z. 1091. If a cone of the second degree touch one system of three planes, which are two and two at right angles, it will touch an infinite number of such systems : and if one system be co-ordi- nate planes, and {l^, m^, n^), (l„, m^, n^, {l^, m^, n^ be the direc- tion cosines of another; the equation of the cone will be 1092. Prove that the surface whose equation, referred to axes inclined each to each at an angle of 60°, is yz + zx + xy + 0^ =Q, is cut by the J)lane x+y + z = in a circle whose radius is a. 1093. In the expression ax' + by" + e^ -t- 2a ys + 2Vzx + ^c'xy + 2a"x + W^y + 2c"z + d; prove that {h + e) a'" +... + ...- a'h"c" - h'c"a" - c'a"b", and a"\bo - a") + ... + ... + 26"e" {b'c -aa')+ ... + ... are invariants for all systems of rectangular co-ordinates having the same origin. 1094. Prove also that the coefficients in the following equa- tion in h h + a,, h cos 7 + c', h cos ^ + b', a" = 0, h cos y + c', h + b, h cos a + a', h" hcosP + h', h eos a + a, h + e, c" a", b", c", d a, 13, y being the angles between the axes, are invariants for all systems of Qo^rdinates having the same origin. 230 BOOK OF MATHEMATICAL PROBLEMS. 1095. Assuming the formulae for transforming from a system of co-ordinate axes inclined at angles o, /8, y to another inclined , at angles a, /?', -y' to be prove that 1 = ;^= + Z/ + Z/ + 2^3 cos a + 21^1, cos 13 + 2\l^ cos y ; ■with similar equations in m and n; and that cos a = m^n^ + m^n^ + in./h^ + {m^n^ + m^n^ cos a + {m^n^ + m^n^ cos ^ + {m^n^ + m^n^ cos y ; with similar equations mn, I; and I, m. 1096. If ax^ + hy' + c:^ become AX' + BY'+CZ^ by any transformation of co-ordinates, the positive and negative coefficients will be in like number in the two expressions. 1097. The equation ax'+ ... +2a'yz+ ... + 2a"oa+ .,. +d = will in general represent a paraboloid of revolution, if a b' c' b c a c a ¥ ^ a c b b a c c b a and a cylinder of revolution if, in addition to these conditions, V J^'f V a b c ^ -7 + T7 + - = 0. a b c 1098. The surface whose equation, referred to axes inclined at angles o, /?, y, is ax^ + by^ + ce'' = 1, will be one of revolution if a cos "■ _ b cos /? c cos y cos a - COS j8 COS y cos fi — cos y cos a cos y — COS a COS P ' 1099. The surface whose equation, referred to axes inclined at angles o, /8, y, is ayz + bzx + cxy = 1, will be one of revolu- tion if a b c ^ 1 =t cos a 1 ± cos /8 1 * cos y ' one, or three, of the ambiguities being taken negative, SOLID GEOMETRY. 231 1100. Prove that the equation of the surface can be obtained in the form a;^ + ?/^ - »^ = d'' in an infinite number of ways, provided that either a' or ¥ is greater than c" j and that the new axes of x, y lie on the cone "■' (h' - ,?) + t («^ - 0==) - C (a^ + ¥) = 0. g^y "I • y c 1101. The equation of a given hyperboloid may be obtained in the form ayz + hzx + cxy = 1 in an infinite number of ways; and, if a, /3, y be the angles be- tween the axes in any such case, the expression cibc 1— cosV — cos^/S — coa^y + 2 cos a. cos ^ cos y will be constant. ■ 1102. Prove that the only conoid of the second degree is a hyperbolic paraboloid; and that it will be a right conoid, if the two principal sections be equal parabolas. 1103. The equation a^ + ... +... + 2a'ys-l- ... + ... =0 will represent a cone of revolution, if -7- + -T =17- + ^ =0. a 0—0 c—a 1104. The radius r of the central circular sections of the sur- face ayz + hex + cxy = 1 is given by the equation ahcr'^ + {of + 6' -I- c°) r" = 4 ; and the direction-cosines of the sections by the equations ?(m'-^w') _ m(^^^+^') _ w(^-^m°) _ ^^^^ a ' h 232 BOOK OP MATHEMATICAL PEOBLEMS. 1105. If a cone be described having a plane section of a given sphere for base, and vertex at a point F on the sphere, the subcontrary sections ■will be parallel to the tangent plane at V. 1106. If a cone whose vertex is the origin and base a plane section of the surface ax' + hi/' + c^=l be a cone of revolution, the plane must touch one of the cylinders (b-a)y' + {c-a)z'=l, {c-V}!^ + {a-h)x' = \, {a-c)a?+(b-c)y'=l. 1107. A cone is described whose base is a given conic and one of whose axes passes through a fixed point in the plane of the conic : prove that the locus of the vertex is a circle. 1108. If the locus of the feet of the perpendiculars let fall from a fixed point on the tangent planes to the cone ax' + hy' + c^ = Q be a plane curve, it will be a circle; and in order that this may be the case, the poiat must lie on one of the systems of straight lines (of which only one is possible) a, = 0, -^ + J?!- = 0; &c. b — a c — a 1109. Prove also that, if the point lie on one of these straight lines, the plane of the circle will be perpendicular to thj other : and that a plane section of the cone perpendicular to one of the straight lines will have one of its foci on that straight line , • -i , , /((i-a)(o-a)) and its eccentricity equal to / i '-^ > . 1110. If a plane cut the cone ayz + hzx + cxy = in two straight lines at right angles to each other, the normal to the plane through the origin will also lie on the cone. 1111. Prove that, when 66'= c'a', and cc' = a'b', the equation ax'+ ... + 2a'i/s+ ... +2a"x + ... +/= represents in general a paraboloid whose axis is parallel to th$ straight line ai = 0, c'y + b's=0. SOLID GEOMETRY. 233 1112. Prove that the locus of tangent lines, drawn from the origin to the surface u^ax'+ ... + 2a'yz+ ... + 2a"x+ ... +/=0, is fu - {a"x + h"y + c"a +/)^ = ; and investigate the condition that the surface may be a cone from the consideration that tliis locus will then become two planes. 1113. The section of the surface yz + zx + xy = o? by the plane Ix + my +nz = p will be a parabola if and that of the surface x' + y' + z'' — 2yz - 2zx — 2xy = a' will be a parabola if jnn + nl + lm=0. 1114 The semiaxes of a central section of the surface ayz + hzx + cxy + aba = made by a plane whose direction-cosines are I, m, n, are given by the equation r* (2bcmn + ...- aH^ -■■■)- iaicr' (amn +...) + ia^iV = 0. 1115. Prove that the section of the surface aa;^+ ... + 2ayz+ ... + 2a"x + ... +d=0 by the plane Ix + my + nz = will be a rectangular hyperbola, if P{b + c) + m'{c + a) + 7v'{a + 6) = 2a'mn + 2h'nl + 2chn; and a parabola, if r(6c-a'^)+ ... + ... +2mm(JV-aa') + ••• + ... =0. Explain why this last equation becomes identical if h'c — aa, c'a — W, and ah' = cc. 234 BOOK OF MATHEMATICAL PROBLEMS. 1116. The generators drawa through the point (x, y, z) of the surface ayz + hzx + cxy + aha = will be at right angles, if x' + y' + z'=a!'+¥ + c\ 1117. Normals are drawn to a conicoid at points lying along a generator: prove that they will lie on a hyperbolic paraboloid whose principal sections are equal parabolas. 1118. The two conicoids {A + b'+c')x'+{B + c' + aI')y''+{C + a' + ¥)^ — 2bcyz — 2cazx — 2abxy = e*, /■x' y°' z\/a? ¥ c' ,\ fax hy czV , have their axes coincident in direction. 1119. The two conicoids ax" + ... +2a'ye+ ... = 1, Aof + By'+C^ = l, have one, and in general only one, system of conjugate diameters coincident in direction ; but, if 1 / h'c'\ 1 /, c'a'\ 1 / a'b'\ there will be an infinite number of such systems, the direction of one of the diameters -being the same in all. 1120. Prove that eight conicoids can in general be drawn, containing a given conic and touching four given planes. 1121. A, B is the shortest distance between two generators, of the same system, of a conicoid; and any opposite generator meets them in P, Q respectively : prove that the lengths x, y of AP, BQ are connected by a constant relation of the form axy + 'bx+cy + d = Q. SOLID GEOMETEY. 235 1122. A, B; P, Q are the points where two fixed generators are met by two of the opposite system; if A, B be fixed, the lengths X, y oi AP, BQ will be connected by a constant relation of the form axy + 6a; + cy = 0. 1123. A hyperboloid of revolution is drawn containing two given straight lines which do not intersect : prove that the locus of its axis is a hyperbolic paraboloid, and that its centre lies on one of the generating lines through the vertex of this paraboloid. III. Conicoids referred to their axes. 1124. The curve traced ovX on the surface ^ ^ — = « by the he ■' extremities of the latera recta of sections made by planes through the axis of x lies on the cone f+z''=ix\ 1125. The locus of the middle poiilts of all straight lines passing through a fixed point and terminated by two fixed planes is a hyperbolic cylinder. 1126. An ellipsoid and hyperboloid are concentric and con- focal : prove that a tangent plane to the asymptotic cone of the hyperboloid will cut the ellipsoid in a plane of constant area, 1127. The 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 which the given point and the centre of the given surface are extremities of a diameter. 1128. An ellipse and a circle have a common diameter, and on any chord of the ellipse parallel to this diameter is described a circle in a plane parallel to that of the given circle ; prove that the locus of these circles is ah ellipsoid. 236 BOOK OF MATHEMATICAL PEOBLEMS. 1129. An ellipsoid is generated by the motion of a point fixed in a certain straight line, whicli moves so that three other fixed points on it lie in the co-ordinate planes : prove that there are four snch systems of points; and that if the corresponding straight lines be dra'wn through any point on the ellipsoid, the angle between any two is equal to that between the othei' two. 1130. 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 two elliptic cylin- ders. 1131. At each point of a generating line of a conicoid is drawn a straight line touching the conicoid and at right angles to the generating line : prove that the locus of such straight lines is a hyperbolic paraboloid whose principal sections are equal parabolas. 1132. The locus of the axes of sections of the surface which contain the line X 2/ _ « is the cone (& — c) yz (rnz — ny) + (c — a) zx {nx — Iz) + (a — 6) xy {ly - mx) = 0. 1133. The three acute angles made by any system of equal conjugate diameters of an ellipsoid will be together equal to two right angles, if 2 (2a' -V- c') (25^ -e'- a') (2c' -a'- ¥) = 27a'6=c= ; 2a, 26, 2c being the axes. 1134. From different points of the straight line X y z SOLID GEOMKTEY. 237 asymptotic straight lines are drawn to the hyperboloid prove that they will lie on the two planes 2«^ 1135. The asymptotes of sections of the conicoid ax' + h't^ + c^ = \, made by planes parallel to Ix + my + nz — lie on the two planes {phc + Tufca + n^ab) {ax' + hy^ + cz') = abc {Ix + my + nzy. 1136. The locus of points, from which rectilinear asyinptotes can he drawn to the conicoid ax' + by' + cs/' = 1 at right angles to each othei', is the cone a' {b + c)x' +¥■ (c + a) y' + c' (a + b) ^ =0. 1137. A sphere is described, having for a great circle a plane section of a given conicoid : prove that the plane of the circle in which it again meets the conicoid intersects the plane of the former circle in a straight line which lies in one of two fixed planes. 2 2 2 1138. In the hyperboloid — + ^-j^ — =1, (a > b), the spheres, a of which one series of circular sections of the hyperboloid are great circles, will have a common radical plane. 1139. The plane containing two parallel generators of a conicoid will pass through the centre. Two generators of the paraboloid 2 2 X y , V = 4« a 238 BOOK OF MATHEMATICAjl, .KOBLEjIS. are drawn through the point (X, 0, Z) : prove that the angle between them is COS' \a + h + z)' 1140. The perpendiculars let fall from the vertex of a hyper- bolic paraboloid on the generators will lie on two cones of the second degree, whose circular sections are parallel to the princi- pal parabolic sections of the paraboloid. 1141. HA, A' be one of the real axes of a hyperboloid of one sheet, and P, P' the points where any generator meets' the genera^ tors of the opposite system through A, A' respectively ; the rect- angle AP, A'P will be constant. 1142. In a hyperboloid of revolution of one sheet, the shortest distance between two generators of the same system is not greater than the diameter of the principal circular section. 1143. The equation of the cone generated by straight lines, drawn through the origin parallel to normals to the ellipsoid ^ y' «' . at points where it is met by the confocal surface x' _i _2 ^2"*" ■•■ ~ ' ^ a'-cP'V-d' 1144. The points on a conicoid, the normals at which inter- sect the normal at a given point, all lie on a cone of the second degree having its vertex at the given point. 1145. Straight lines are drawn in a given direction, and the tangent planes drawn through each straight line to a given coni- coid are at right angles to each other : prove that the locus of such straight lines is a cylinder of revolution, or a plane. SOLID GEOMETRY. 239 1146. A cone is described having for base the section of the conicoid ax' + hi/' + ca^ = 1 made by the plane lx + my + m« = 0, < and intersects the conicoid in a second plane perpendicular to the given plane : prove that the vertex must lie on the surface (P + rf + 'd) {ax' + ly' + cs^ - 1) = 2 l}x+my + nz) (alx + hmy + cnz). 1147. The six normals drawn to the ellipsoid x^ y"- ^ , — I- — + — = 1 ft= 6^ a» from the point («„, y^ «J all lie on the cone ' Of _ en— ^5_ + (c= _ a^) J!h- + («2 _ jn _f!!_ = 0. 'x-x^ 'y-y^ «-»„ 1148. The six normals drawn to the conicoid ax' + 'by'' + c!?=\ from any point on one of the lines a (6 — c) a; = ± S (c — a) 2/ = ± c (a — 6) « will lie on a cone of revolution. 1149. A section of the conicoid aoi? + hy^ + cz'^l is made by a plane parallel to the axis of «, and the trace of the plane on xy ia normal to the ellipse ax' hf c^ {a-cf {h-cf (c'-aby prove that the normals to the ellipsoid at points in this plane will all intersect the same straight Hne. 1150. If the normals to the ellipsoid x' f z' , I- — -I =1 a' V c' at points on the plane I- +m%+n-= 1 a c 240 BOOK OP MATHEMATICAL PEOBLEMS. all intersect the same straight line, the normals at points on the plane 4 + f +^ + l = b al bm en will also intersect that line. Prove also that the condition for this is (mW - V) {V - iff + {rev - m') (c" - ay + (I'm' - n') (a" - bj = 0. If l = m=n=l, the straight line which the normals inter- sect is ax {V - c") = by {c' - a") = cz {a" - V). 1151. The normals to the paraboloid ^ + — = 2x b c at points on the plane px+ qy +tz=\ will all meet one straight line if ^(q^b + i^c) p'{b-c)+2p{q'b-r'c) = b-c 1152. PQ is a chord of a conicoid, normal at F ; any plane conjugate to FQ meets the conicoid in a curve A : prove that one axis of the cone whose vertex is F, and base the curve A, is the normal at F, and that the other axes are parallel to the axes of any section parallel to the tangent plane at F. 1153. Straight lines are drawn through the point (a;„, y„, «J, such that their conjugates with respect to the paraboloid I4 = 2x a are perpendicular to them respectively : prove that these straight lines must lie on the cone y„ z„ .+ ^Z^=0: y-y^ «-»o '«-«'o and that their conjugates will envelope the parabola a ..„; (-ff.(fy.(«- 5)^=0. SOLID GEOMETET. 241 1154. If a straight line be perpendicular to its conjugate with respect to any conicoid, it will be perpendicular to its conjugate with respect to any conicoid confocal with the former. 1155. Any generator of the surface y" + «* — a;' = m will be perpendicular to its conjugate with respect to the surface ax'+ ... + 2a'yz+ ... = 1, if be — a'" = ca — b'^ = ah — c' ; and aa' = b'c. 1156. In the two conicoids aar' + 6y + c«==l, Ax' + By' + 0^? = !, eight generators of the first will be perpendicular respectively to their conjugates with respect to the second. 1157. is a fixed point, F a point such that the polar planes of 0, P with respect to a given conicoid are perpendicular to each other ; prove that the locus of P is the plane bisecting chords perpendicular to the polar plane of 0. 1158. A hyperbolic paraboloid, whose principal sections are equal, is drawn through two given non-intersecting straight lines ; prove that the locus of its vertex is a straight line. 1159. If two conicoids have two common generators of the same system, they will also have two common generators of the opposite system. 1160. If two given straight lines be generators of the same system of a conicoid, the polar plane of any given point, with respect to the conicoid, will pass through a fixed point. 1161. If two conicoids touch each other in three points, they will touch each other in an infinite number ; or will have four common generates. w. 16 242 BOOK OP MATHEMATICAL PROBLEMS. IV. Tetrahedral Coordinates. 1162. When the opposite edges of the tetrahedron ABC B we, two and two, at right angles, the three shortest distances between the opposite edges meet ia the point a {AB' + AC + AB^- k) = p.{BC' + BB' + BA^ -k)- ... = ..., : k bang the sum of the squares on any pair of opposite edges. 1163. Determine the condition that the straight line may touch the conicoid l/Sy + mya + na^ + I'aS + m'fiS + n'yS = ; find thence prove that the equation of the tangent plane, at the point a — P = y=<), is I'a + m'P + ny = 0. 1164. Any conicoid which touches seven of the planes ± ?a ± m,^ d= ray ± rS = will touch the eighth ; and its centre will lie on the jilaue Prove that this plane bisects the part of each. edge of the fan-i damental tetrahedron which is intercepted by the given planes. 1165. If a hyperbolic paraboloid be drawn containing the sides AB, BG, CB, BA of a quadrilateral which is not plajQe^ and /-" be any point on this surface, vol. PBGB X vol . PBAB = vol. PBA C x vol. PA BC. Also, if any tangent plane meet AB, CB. in P, Q respectively, AP : BP :: BQ : CQ. SOLID GEOMETET. 243 1166. The locus of the centres of all conicoids which have four common generators, two of each system, is a straight line, 1167. Perpendiculars are let fall from the point (a, (3, v, S) on the faces of the fundamental tetrahedron, and the feet .of these perpendiculars lie iu one plane; prove that J_ J_ J_ _J__„ Pi°- P,^ Pb7 K^ Pi' Pi' Pi' Pi I'eing the perpendiculars of the tetrahedron. 1168. If a tetrahedron be self-conjugate to a given sphere, any two opposite edges will be at right angles to each other, and all the plane angles at one of the solid angles will be obtuse. 1169. If the opposite edges of a tetrahedron be, two and two, at right angles to each other; the circumscribed sphere, the sphere bisecting the edges, and the sphere to which the tetra- hedron is self-conjugate will have a common radical plane. 1170. A tetrahedron is such that a sphere can be described touching its six edges; prove that any two of the four tangent cones drawn to this sphere from the angular points will have a common tangent plane and a common plane section; and the planes of these common sections will all six meet in a point. 1171. A tetrahedron is such that the straight lines joining its angular points to the points of contact of the inscribed sphere with the respectively opposite faces meet in a point; prove that, at any point of contact, the sides of the face on which the point lies subtend equal angles. 1172. If a conicoid circumscribe a tetrahedron ABCD, and the tangent planes at A, B, G, D form a tetrahedron A'BG'jy ; then \i AA!, BE intersect, GO', DD' will also intersect. 1173. Four points are taken, on a conicoid; prove that, if the straight line. joining one of the points to the pole of the plane 16—2 244! BOOK OF MATHEMATICAL PROBLEMS. through the other three pass through the centre, the tangent plane at that point is parallel to the plane through the three points. 1174. The equation of a conicoid is mn^y + vh/a + Ima.^ + ?ra8 + ... + . . . = ; prove that it can never be a ruled surface, and that it will be a paraboloid if 1175. The surface ?/3y + mya. + wayS + Z'aS + m'ySS + w'yS = win be a cylinder, if U (to + w - Z) + mm! (w + ?- m) + nn{l + m - n) = llrrm, and m {ml + «.'-?)+ mm' (w + Z - m) + nn {l+m'-n')= 21m n. 1176. If Z, m, n, r^be respectively the rectangles of segments of chords drawn from four points A, B, G, D (not in one plane) to meet a certain sphere, and p be the radius of the sphere ; then wiU = 0. 0, 1, 1, 1, 1, 1 1, 0, AB\ AQ\ AB\ Z + p" 1, BA\ 0, BG\ BD\ m+fi' 1, CA\ GB\ 0, GB\ n + p" 1, DA\ BB\ BG\ 0, r + p' 1, l + p^, m + p', n+p', r + p", 1177. The perpendiculars u, x, y, z let fall from the angular points of a given finite tetrahedron on a plane are connected by the equation pu' + qa? + ry^ + sa* + 'Mxy + Imyu + Inux + iTIuz + ludxz + ^ny% = j SOLID GEOMETET, 245 prove that the envelope is a conicoid, which degenerates into a plane curve if p, n, m, I' =0. n, q, I, m' m, I, r, n' I, m, n, s V. Focal Curves, Reciprocal Polars. 1178. The equations of the foqal lines of the cone ayz + hzx + cxy = 0, a^j.g (cy + M' ^ {az+cxY ^ {bx + ayf y'+a' ^ + v? x' + y'^ 1179. A parallelogram of minimum area is circumscribed about the focal ellipse of a given ellipsoid, and from its angular points tsiken in .order are let fall perpendiculars p^, p^, p^ p^ on any tangent plane to the ellipsoid ; prove that 2c being the length of that axis ■which is perpendicular to the plane of the focal ellipse. 1180. If •53-,, OTj) 'STj, w^ be the perpendiculars on any tan- gent plane from the extremities of two conjugate diameters of the focal ellipse, and p the perpendicular from the centre • 1181. With any two points of the focal ellipse as foci, can be described a prolate spheroid touching an ellipsoid along a plane curve ; provided the tangents to the focal at the two points intersect without the ellipse. 1182. The four straight lines drawn in a given direction, and intersecting both focal curves of an ellipsoid, lie upon a cylin- 246 BOOK OP MATHEMATICAL PEOBLEMS. der of revolution -whose radius is ij{(x'—p'); a being tlie semi major axis, and p the perpendicular from the centre on the tan,- gent plane normal to the given direction. 1183. If with a given point as vertex, a cone of revolution be described whose base is a plane section of a given conicoid ; this base will touch a fixed cone whose vertex lies on one of the axes of the enveloping cone drawn from the given point. 1184. The straight line joining the points of contact of a common tangent plane to the two conicoids ax^ ■¥ly'' + cz"" ^\, {a-k)x' + (h-k)y' + (c-k)z''=l fcubtends a right angle at the centre. 1185. Through a given point can in general be drawn two straight lines,' either of which is a focal line of any cone, en- veloping a given conicoid, and having its vertex on- the straight line. If two enveloping cones be drawn with their vertices one on each of these straight lines, a prolate conicoid of revolution can be inscribed in them, having its focus at the given point. 1186. If any point be taken on the umbilical focal conic of a conicoid, there exist two fixed points L, such that if any plane A be drawn through Z and a be its pole, Oa is at right angles to the plane through and the line of intersection of A with the polar of L. 118'?. With a given point as vertex there can in general be drawn one tetrahedron sdf-conjugate to a given conicoid, and such that the edges meeting in the point are two and two a,t right angles; but if the given point lie on a focal curve, an infinite number. SOLID GEOMETKY. 247 VI. General Functional and Differential Equations. 1188. A surface is generated by a straight line which always intersects two fixed straight lines y = mx, s = c ; y = — mx, z = — c: prove that the equation of the surface generated is of the form max — yz . fmzx — cyy : . fmzx — cy\ 1189. The general functional equation of surfaces generated by a straight line, intersecting the axis of z, and meeting the plane of xy in the circle x^ +y' = a', is ,f^{.^^f($j^; x' + and the general differential equation is (flj^ + y^ (px + qy- z)' = a" {px + qy)'. 1190. The general functional equation of all surfaces, gene- rated by a straight line which always intersects the' axis of z, is ■ "/(I)-*(9^ and the differential equation is / rx' + 2sxy + ty' = 0. 1191. The differential equation of a famUy of surfaces, such that tSie perpendicular from the origin on the normal always lies in the plane of ay, is z{p' + q') + (px + qy) = 0. 1192. The differential equation of a family of surfaces, gene- rated by a straigl^t line which is always parallel to the plane of xy and whose intercept between the planes of sx, yz is constant and equal to c, is 0»a;+5'y)'(p'+2')=c>V- 24-8 BOOK OF MATHEMATICAL PROBLEMS. 1193. The general diiferential equations of surfaces, generated by a straight line (1) always parallel to the plane Ix + my + xz = 0; (2) always intersecting the straight line X _y z I m n' are respectively, (1) (m + nqfr-^{m + nq)(l + np)s + {l + npft = ; (2) {ly-mxf{q'r-2pqs+pH) + 2(ly — mx) {nx — Is) [qr — ps) + 2{ly — mx) (ny — inz) (as —pt) + {nx — hyr + 2 (nx — Iz) («y — mz) s + (ny — m£ft= 0. VII. Envelopes. 1194. The envelope of the plane Ix + my + nz = a ; I, m, «, being parameters connected by the relations l + m + n = Q, P +7n? + n^=\ is the cylinder (y-z)'+{z-xY + {x-yy=Za\ I 1195. Find the envelope of the plsmes (1) -cos(e + ^) + fcos(^-^) + *sin(^+^) = sin(d-<^), (2) - cos (fl - ^) + T (cos ^ + COS ^) + - (sin^ +sin^) = 1, both when 6, are parameters, and when only is a parameter. I 1196. The envelope of the plane X y z + . „ . . + ^ = « sin 6 cos (^ sin d sin ^ co&B is the surface X +y +z =a . SOLID GEOMETRY. 249 • 1197. The envelope of all paraboloids, to wliicli a given tetrahedron is self-conjugate, is the planes each of which bisects three edges of the tetrahedron. 1198. A prolate spheroid can be described having two oppo- site umbilici of an ellipsoid as foci and touching the ellipsoid along a plane curve : and this spheroid will be the envelope of a series of spheres, having one system of circular sections of the ellipsoid as great circles. 1199. Spheres are described on a series of parallel chords of an ellipsoid as diameters : prove that they will have double con- tact with another ellipsoid ; and that the focal ellipse of the latter will be the diametral section of the former conjugate to the chords. Also, if a, b, c be the axes of the former, and a, P, y of the latter, a' + b' + (^ = a.'+P'-y% y being that axis which is perpendicular to the plane bisecting the chords. 1200. The envelope of a sphere, intersecting a given conicoid in two planes and passing through the centre, is a surface of the fourth degree, touching the conicoid along a spherical conic. VIII. Curvature. 1201. If from any point of a curve equal small lengths S* be measured in the same direction along the curve, and along the circle of absolute curvature, respectively; the distance between the extremities of these lengths is ultimately 8a» /ri 1/M'\ p, a- being the radii of curvature and torsion respectively at the point. . 250 BOOK OP MATHEMATICAL PBOBLEMS. 1202. Two surfaces Lave complete contact of the w"* order at a point : prove that there are n + 1 directions of normal sections for which the curves of section will have contact of the n + l* order : and hence prove that two conicoids which have double contact with each other will intersect in plane curves. 1203. Prove that it is in general possible to determine a paraboloid whose principal sections shall be equal parabolas, and which shall have a complete contact of the second order with a given surface at a given point. 1204. Prove that a paraboloid can in general be drawn 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 contact of the third order. 1205. A skew surface is capable of generation in two ways by- straight lines ; at any point of it the absolute magnitudes of the principal radii of curvature are a, b : prove that the angle be- tween the generators which intersect in the point is wo — , a + b 1206. The points on the surface a:2/» = a (y« + zx + xy\ at which the principal radii of curvature are equal and opposite, lie on the cone «' (y + ») + 2/* (« + a;) + «* (a; + y) = : and on the surface a;y« = «'(»;+ 2/ + «) all such points lie on the cone a;'(y + «) + /(« + a!) +»'(a; + «/) = 0. SOLID GEOMETRY. 251 1207. A surface is generated by a straight line intersecting the two straight lines 2/ = a; tan a,") ?/ = — a; tan a,"| » = c, j « = -c, j and X, /A are the distances of the points where the generator meets these straight lines from the points where the axis of z meets them : prove that the principal radii of curvature at a point on the first straight line are given by the equation cy sin' 2of- 2cp sin 2a ^ (A. - /a cos 2a) (4c'' + / sin' 2a) . ' ^ 'ic" + /x'sin''2a)^ (|)<- 1208. 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. If, at a point on the axis of x, r be the radius of the circle and 6 the angle which the diameter through the point makes with the axis of », the principal radii, of curvature at the point are given by the equation pv.gy(p-.)=o. 1209. A surface is generated by a straight line which always intersects a given circle and the straight line through the centre perpendicular to the plane of the circle ; B is the angle which the generator makes with the fixed straight line, 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 cur- vature at the point where the generator meets the fixed straight line are dB "04, sia e (cos e ± 1) ' and that at the point where it meets the circle they are given by the equation P [~ri) +apcos() = a, a being the radius of the circle. 252 BOOK OP MATHEMATICAL PROBLEMS. 1210. If I, m, n be the direction cosines of the normal at a^ 1/ s^ any point of the conicoid — •" x "" — ~^> ^^^ "^ *^® angle between the geodesic lines through that point and through the umbilici ; then will ,^_ {Pa(o-b) + m'b(o + a-2b) + n''c{a-h)Y *'°* ''~ZV(6-c)= + ... + ...-2OTW6c(a-6)(a-c)-. ...... ' the axis of y being parallel tc the circular sections. ( 253 ) STATICS. I. Composition and JSesolution of Forces. 1211. If be the centre of the circle circumscribing a tri- angle ABG, and B, E, F the middle points of the sides, the system of forces OA, OB, 00 wUl be equivalent to the system OD, OF, OF. 1212. Forces P, Q, R act along the sides of a triangle ABG and their resultant passes through the centres of the inscribed and circumscribed circles ; prove that P _ Q ^ R cos B — cos cos — cos A cos A — cos B ' 1213. ABCD is a quadrilateral inscribed in a circle, and forces inversely proportional to AB, BG, AD, DO act along the sides in the directions indicated by the order of the letters ; prove that their resultant acts along the line joining the intersection of AC, BD to the intersection of the tangents at B, D. 1214. In a triangular lamina ABG, AD, BE, GF are the per- pendiculars, and forces BD, CD, GE, AE, AF, BF are applied to the lamina ; prove that their resultant passes through the centre of the circumscribed circle, 1215. Three eq'ual forces P act at the angles of a triangle ABG perpendicular respectively to the opposite sides ; prove that their resultant is equal to i>y(l-8sin^sin|sin|). 1216. If be the centre of the circumscribed circle, and L the centre of perpendiculars of a triangle ABG, three forces repre- 254 BOOK OF MAT^HEMATICAL PROBLEMS. sented by LA, LB, LG will have a resultant along LO and equal to twice LO. 1217. Three parallel forces act at the angular points of a triangle ABC, and are to' each other as b + c : c + a : a+b; prove that their resultant passes through the centre of the in- scribed circle of the triangle whose angular points- bisect the sides of the former. 1218. The position of a point P such that forces acting along PA, PB, PC, and equal to IP A, mPB, nPG respectively, are in equilibrium, is given by the equations I m n' areal coordinates being used. 1219. Eorces act along the sides of a triangle ABC and are proportional to the sides ; AA', BB', CG' are the bisectors of the angles ; prove that if the forces be turned in the same direction about A', B', G' respectively, through an angle , _i/ .B-G n-A ,A-B\ tan ( — cot — = — cot — = — cot — - — 1 , there will be equilibrium. 1220. A system of forces whose components are (X, J"), (^2' ^2) ••■ a.ct at the points {x^, y^, {x^, y^ .. and are equiva- lent to a single couple ; prove that if each force be turned about its point of application through an angle 6 there will be equili- brium, if X [Xx + Yy) 1221. If Z, iSf, iV be the sums of the moments of a given system of forces about three rectangular axes, and X, Y, Z the sums of the components along these axes, then will LX+MY+NZ be independent of the particular system of axes. STATICS. 255 1222. The necessary and sufficient conditions of equilibrium of a system of forces acting on a rigid body are- that the sum of the moments of all the forces about each of the edges of any finite tetrahedron shall be severally equal to zero. 1223. Forces acting on a rigid body are represented by the edges of a tetrahedron, three acting from one angular point and the other three along the sides taken in order of the opposite face ; prove that, whichever angular point be taken, the product of the resultant force and minimum couple ■will be the same. 1224. Three forces act along three non-intersecting straight lines, any other straight line is drawn meeting the three ; prove that the shortest distance of this straight line from the central axis of the forces is proportional to the cotangent of the angle its direction makes with the central axis.. 1225. A portion of a curve surface of continuous curvature is cut off by a plane, and at a point in each element of the portion a force proportional 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. 1223. If a system of forces acting on a rigid body be reducible to a couple, it is always 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 produce equilibrium. 1227. A system of forces are reduced to a force acting through an assumed point and a couple; prove that, if the assumed point be taken on a fixed straight line, and through it the axis of the couple be drawn, the extremity of the axis will lie on another fixed straight line. 1228. Prove that the central axis of two forces P, Q in- tersects the shortest distance c between their lines of action, and divides it in the ratio Q{Q + P cos 6) : F (P'+ Q cos &), 256 BOOK OP MATHEMATICAL PEOBLEMS. 6 being the angle between their directions. Prove that the moment of the principal couple is cPQ sin e ^{F' + Q' + 2FQcoaey 1229. If a system of forces be reduced, to two, one of which F acts along a fixed straight line ; then will 1 cos 6 c sin fl ■ + • F~ R " G ' 6 being the angle which the given straight line makes with the central axis, c the shortest distance between them, £ the result- ant force, and G the principal couple. 1230. If a system of forces be reduced to two at right angles to each other, the shortest distance between their lines of action 2G cannot be less than -j- . 1231. If a system of forces be reduced to two P, Q, and the shortest distances of their lines of action from the central axis be X, y respectively ; then wUl G'{g'- P") = m (PW - QY), 1232. Two forces act along the straight lines 33 = a, y = «tana; x = —a, y = — «tana; prove that their central axis lies on the surface ' sin 2a the coordinates being rectangular, 1233. Two forces given in magnitude act along two given non-intersecting straight lines, 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. STATICS. 257 II. Centre of Gravity/. 1234. 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, 2W, ZW ; prove that the weight must be at one of the angular points of a hexagon whose opposite sides are equal and parallel, and "whose area is to that of the board as 3 : 25. 1235. If particles be placed at the angular points of a tetra- hedron, proportional respectively to the areas of the opposite faces, their centre of gravity will be the centre of the sphere inscribed in the tetrahedron. 1236. A uniform wire is bent into the form of three sides of a polygon AB, BG, CD, and AB=CD = a, BC=l; prove that, if the centre of gravity of the wire be at the intersection of AG, BD, each of the angles B, C is equal to V 2aV • 1237. A thin uniform wire is bent into the form of a triangle ABC, and particles of weights P, Q, R are placed at the angular points ; prove that, if the centre of gravity of the particles coin- cide with that of the wire, F : Q : B :: h + c : c + a : a + b. 1238. A polygon is such that a,, a^,, a^... being- the angles made by its sides with any fixed straight line, S (cos 2a) = 0, S (sin 2a) = ; prove that there exists a point which is the centre of gravity of n equal particles placed at the feet of the perpendiculars from on the sides; and that the centre of gravity of w equal particles, at ' the feet of the perpendiculars from any other point P, bisects OP, w. 17 258 BOOK OP MATHEMATICAL PEOBLEMS. 1239. The limiting position of the centre of gravity of the area included between the area of a quadrant of an ellipse bounded by the axes and the corresponding quadrant of the auxiliary circle, as the ellipse approaches the circle as its limit, will be a point whose distance from the major axis is twice its distance from the minor axis. 1240. A curve is divided symmetrically by the axis of a;, and is such that the centre of gravity of the area included between the ordinates aj = 0, x = h is at a distance mh from the origin ; prove that the equation of the curve is Sm-l 1241. The circle is the only curve in which the centre of gravity of the area included between any two radii vectores and the curve lies on the straight line bisecting the angle between the radii. 1242. Determine 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 of the extremities. Prove that the curve is a lemniscate, the node being pole. 1243. Two rods AB, BC rigidly united at B and suspend- ed freely from A, rest inclined at angles a, /3 to the vertical; prove that AB_ //, sin^\ 1244. AB, BC Bxe two uniform rods freely jointed at B and moveable about A which is fixed ; find at what point in BC a prop must be placed so that the rods may be at rest in a hori- zontal straight line. 1245. Three equal uniform rods, jointed together at their extremities, rest in one horizontal line on three pegs, each rod in contact with one peg; find the positions of equilibrium. STATICS. 259 The length of each, rod being a, and the pegs at equal dist- ances 6 ; prove that there wUl be three positions of equilibrium if 306 < 36a > 5 (4 + ^7) a. 1246. A rectangular board is supported -with its plane verti- cal by two smooth pegs, and rests with one diagonal parallel to the line joining the pegs ; prove that the other diagonal will be vertical. 1247. A rectangular 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 0—a sin 6'. 1248. A portion of a parabola, cut off by a focal chord in- clined 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 j prove that the latus rectum of the parabola is c^5 sin" a. 1249. A uniform rod AB of length 2a is freely moveable about A; a smooth ring of weight. P slides on the rod and has attached to it a fine string, which, passing over a pulley at a height b vertically above A, supports a weight Q hanging freely ; find the position of equilibrium of the system ; and prove that, if in this position the rod and string be equally inclined to the vertical, 2Q{Qb-Way = F'Wab. 1250. A uniform rod, of length c, rests 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 made by the rod ■with the horizon, tan 6 = -^ tan (j>, where a cos <^ + A = c cos 9. Explain the result when a = 25 = c, A= 0. 17—2 260 BOOK OF MATHEMATICAL PEOBLEMS. III. Smooth Bodies under forces in one plane. 1251. A small smooth lieavy ring is capable of sliding on a fine elliptic -wire wliose major axis is vertical] two strings at- tached to the ring pass through small smooth rings at the foci and sustain given weights j prove that, if there be equilibrium in any position in which the whole string is not vertical, there will be equilibrium in every position. Prove also that the pressure on the curve will be a maximum or minimum when the sliding ring is at either extremity of the major axis, and when its focal distances have between them the same ratio as the two sustained weights. 1252. Two spheres of densities p, o- and radii a, h, rest in a paraboloid whose axis is vertical, and touch each other at the focus; prove that pW = a-%"'. Also if W, F' be their weights, and \R, R the pressures on the paraboloid at the points of contact, ^_^_l/^_^\ W~W'~2\W' WJ' 1253. Four uniform rods freely jointed at their extremities form a parallelogram, and at the middle points of the rods are small smooth rings joined by rigid rods without weight. The parallelogram is suspended freely from one of its angular points; find the tensions of the rods and the reactions of the rings, and prove that (1) if the parallelogram be a rectangle the tensions are equal, (2) if a rhombus the reactions are equal. 1254. An elliptic lamina of axes 2ai 2h, rests with its plane vertical on two smooth pegs in the same horizontal line at a distance c; prove that, if c < h^2 or >' a J 2, the only positions STATICS. 261 of equilibrium are ■when one axis is vertical ; and that, if c > bJ2 < a^2, the positions in which an axis is vertical are stable, and there are unstable positions of equilibrium in which the pegs- are at the extremities of conjugate diameters. IV. Friction. 1255. Find the least coefficient of friction between a given elliptic cylinder and a particle, 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. 1256. 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 weight. 1257. A weight w rests on a rough inclined plane (;«, < 1) supported by a string, which, passing over a smooth pulley at the highest point of the plane, sustains a weight > iiw < w hanging vertically; prove that the angle between the two positions of the plane in which w is in a state bordering on motion is 2 tan-' fi. 1258. Two weights of similar material connected by a fine string rest on a rough vertical circular arc on which the string lies j prove that the angle subtended at the centre by the distance between the limiting positions of either weight is 2 tan"' /*. 1259. A uniform rod rests with one extremity against a rough vertical wall, the other being supported by a string of equal- 262 BOOK or MATHEMATICAL PROBLEMS. length fastened to a point in the wall j prove that the least angle' which the string can make with the wall is tan"' ( -) • 1260. A uniform rod of weight TT rests with one end against a rough vertical plane and with the other end attached to a string, which passes over a smooth pulley vertically above the former end and supports a weight P. Pind the limiting positions of equili- brium, and prove that equilibrium will be impossible unless P be greater than TFcos e, tan e being the coefficient of friction. 1261. A heavy uniform rod of weight W rests inclined at an angle B to the vertical in contact with a rough cylinder of revo- lution, whose axis is horizontal and whose diameter is equal in length to the rod. The rod is maintained in its position by a fine string in a state of tension, which passes from one end of the rod to the other round the cylinder] prove that the tension of the string must be not less than TT/cos^ ; A _(^___sinej. 1262. Two weights support each other on a rough double inclined plane, by means of a fine string passing over the vertex, and both weights are on the point of motion; prove that if the plane be tilted till both weights are again on the point of motion, the angle through which the planes must be turned is 2 tan"' /x. 1263. A square lamina has a string of length equal to that of a side attached at one of the angular points; the string is also attached to a point in a rough vertical wall, and the lamina rests with its plane vertical and perpendicular to the wall; prove that, if the coefficient of friction be 1, the angle which the string makes IT 1 1 with the wall lies between -r and -^ tan"' -=, . 4 2 2 1264. Two weights P, Q of similar material resting on a rough double inclined plane are connected by a fine string passing. STATICS. 263 over the common vertex, and Q is on the point of motion down the plane J prove that the weight which may be added to F without producing motion is P sin 2e sin (a + /3) sin (a — e) sin (;8 — e) ' o, j3 being the angles of inclination to the horizon, and tan e the coefficient of friction. 1265. A uniform rod rests with one extremity against a rough vertical wall (/^ = o). the other extremity being supported by a string three times the length of the rod attached to a point in the wall ; prove that the angle which the string makes with the wall in the limiting position of equilibrium is tan ' 27 OJ^ *^ 3" 1266. A weight IT is supported on a rough inclined plane of inclination a by a force P, whose direction makes an angle t with the plane, and whose component in the plane makes an angle d with the line of greatest inclination in the plane ; prove that, for equilibrium to be possible, , 2 sin' a sin' 6 cos' i 1 + cos 2a cos 2t — sin 2a sin 2l cos 6 ' 1267. A given weight, resting upon a rough inclined plane, is connected with a weight P by means of a string passing over a rough peg, P hanging freely. The coefficients of friction for the peg and plane are tan \, tan X', (\ > X'). Prove that the inclina- tion of the string to the plane in limiting equilibrium, when P is a maximum or minimum, is X — X', 264 BOOK OF MATHEMATICAL PROBLEMS. V. Ulastio Strings. 1268. A string whose elasticity varies as the distance from one extremity is stretched by any force; prove that its extensioi^ is equal to that of a string of the same length, of imiform elas^ ticity equal to that at the middle point of the former, stretched by the same force. I 1269. An elastic string rests on a rough inclined plane, -with its upper extremity fis:edj prove that its extension will lie between the limits P sin (a ± e) _ 2\ cos € ' u, being the inclination of the plane, tan c the coefficient of friction, and I, A. natural lengths of the string and of a portion of it whose weight is equal to the coefficient^of elasticity. 1270. Two weights P, Q are connected by an elastic string without weight, which passes over two small rough pegs A, B in the same horizontal line at a distance a, Q is just sustained by P, and AP = 6, BQ = o ; P, Q are then interchanged, and AQ'^ V, BP = c : obtain equations for determining the natural length of the string, its elasticity, and the coefficients of friction at A and B. 1271. A weight P just supports another weight Q by means of a fine elastic string passing over a rough circular cylinder whose axis is horizontal, W is the coefficient of elasticity, and a the radius of the cylinder; prove that the extension of the part of the string in contact with the cylinder is a. (Q+W\ 1272. An elastic string is laid on a cyoloidal arc whose plane is vertical and vertex upwards, and when stretched by its own STATICS. 265 weight is in contact with the whole of the cycloid, the. natural length of the string being equal to the circumference of the gener- ating circle ; prove that the coefficient of elasticity is the weight of a portion of the string whose natilral length is twice the dia- meter of the generating circle. 1273. A heavy elastic string, whose natural length is 21, is placed symmetrically on the arc of a smooth vertical cycloid, and when in equilibrium a portion of string, whose natural length is X, hangs vertically at each cusp ; prove that l — x 2 J{aX) = {\ + x) tan -j^^ l 2a being the length of the axis of the cycloid, and \ the natural length of a portion of the string whose weight is the coefficient of elasticity. 1274. A heavy elastic string hangs symmetrically over a smooth circular arc, whose plane is vertical, a portion whose natu- ral length is 2a, and stretched length 3a, hanging vertically on each sidej prove that the natural length of the part in contact is 4a V2 log (1 + ^2), 4a being the radius. 1275. A heavy cone resting symmetrically on a rough sphere may be displaced through an angle of j without upsetting, if the height of the cone be not greater than half a great circle of the sphere. 266 BOOK OF MATHEMATICAL PEOBLEMS. VI, Catefmries, 1276. An endless heavy chain, of length 21, is passed over a smooth circular cylinder, 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^ the angle between the radii drawn to the points where the chain leaves the cylinder; prove that tan dt + —. — ^ log tan ( t + o ) ■ 1277. ABGD are four smooth pegs forming a square, AB, CD being horizontal, and an endless uniform inextensible string passes round the four, hanging in two festoons ; prove that sin a log cot jr sin ^ log cot ^ Z it cot a cot )S Z - ^— — ^— + ij — A = 2, log cot 2 log cot 2 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. 1278. A heavy uniform chain rests on a rough circular arc whose plane is vertical, the length of the chain being equal to a quadrant of the circle, and one extremity being at the highest point when the chain is on the point of motion; prove that cf = . 2/^ 1279. A heavy uniform chain rests in limiting equilibrium on a rough cycloidal arc, whose axis is vertical and vertex up- STATICS. 267 ■wards, one extremity being at the vertex and the other at the cusp; prove that 1280. A heavy uniform chain fastened at two points rests in the form of a parabola under the action of two forces, one {A) pai-allel to the axis and constant, and the other {F) tending from the focus ; prove that F=ZA + m cos being the angle which the tangent at any point makes with the tangent at the vertex, and m a constant. 1281. Find the law of repulsive force tending from a focus under 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. 1282. A uniform chain rests in the form of a cycloid whose axis is vertical under the action of gravity and a certain normal force, and the tension at the vertex vanishes; prove that the tension at any point is proportional to the vertical height above the vertex, and that the normal force at any point is ^-^(3cos=^-l), 2 cos ^ ^ ' where 6 is the angle which the normal makes with the vertical. 1283. 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 wUl vary inversely as cos' /' (<(>). 1284. A string 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 ; 268 BOOK OF MATHEMATICAL PROBLEMS. prove that the repulsive force at any point is inversely propor- tional to the chord of curvature through the centre of force. 1285. The coordiaates' of a point P of a rigid body, referred to a system of fixed rectangular axes, are x, y, z; prove that, if the body receive any small displacement, the displacements of P parallel to the axes may be represented by a+e^s-e^, p+e^x-e^z, y+e^y-e^x; a, P, y, 6^, 6^, 6^ being independent of (x, y, z). Hence prove that the sum of the virtual moments of any forces acting on a rigid, body is of the form ax+p7+yz+ e^L+eju+ej^, X, T, Z being the sums of the resolved parts of the forces along the axes, and L, M, N the sums of the moments about the axes. ( 269 ) DYNAMICS, ELEMENTAEY. I. Bectilinear motion, Impulses. 1286. A ball A impinges obliquely on another ball B, and after impact tbe directions of motion of A and B make equal angles -with ,4 's previous direction, find 6; and prove tliat wben A = B, d = i2iQ.~^ Je, e being the mutual elasticity. 1287. A smooth inelastic ball, mass m,, is lying on a horizon- tal table in contact with a vertical wall, and is struck by another ball, mass 7n', moving in a direction perpendicular to the wall and incliaed at an angle «. 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 tiirned, is given by the equation cot 6 cot a = 1 H . m 1288. Equal particles A^, A^, ... A^ are fastened at equal intervals a on a fine string, of length (n — 1) a, and are then laid on a horizontal table at n consecutive angular points of a regular polygon of^ sides (p>w) each equal to a; a blow P is applied to A^in direction A,Aj,; prove that the impulsive ten- sion of the string A^A^^^ is , (l+sina)"-'--(l-sina)°-- ^°°^ " (l+siria)»-(l-sina)» ' a denoting — . 1289. AF, PB are chords of a circle whose diameter AB is vertical; particles falling down AP, PB respectively, start from 270 BOOK OP MATHEMATICAL PROBLEMS. A, P sitnultaneously ; prove that the least distance between them during the motion is equal to the distance of P from AB. 1290. A number of heavy particles start at once from the vertex of an oblique circular cone whose base is horizontal, and fall down generating lines of the cone ; prove that at any sub- sequent moment they will lie in a subcontrary section. 1291. The locus of a point P, such that the times of fall- ing down PA, PE' to two given points A, B td&j be equal, is a rectangular hyperbola. 1292. The locus of a point P, such that the time of falling down PA to a given point A is equal to the time of falling verti- cally from P to a given straight line, is one branch of a hyperbola of which one asymptote is vertical, and the other perpendicular to the given straight line. 1293. A parabola is placed with its axis vertical and vertex downwards ; 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. 1294. 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. 1295. Two weights W, nW 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 gravity of the weights describes a straight line with uniform acceleration nsinj8-sino ,, . „ , „. * ^ (w + 1)' ^{^+^^ cos (a + ^)+ 1}; a, P being the angles of inclination of the planes. DYNAMICS, ELEMENTAET. 271 1296. In the system of pullies in ■which each hangs by a separate string, P just supports W ; prove that, if P be replaced by another 'weight Q, the centre of gravity of Q and W will descend ■with uniform acceleration W{Q-PY ^ l,P'+QW){Q+Wy the ■weights of the pullies being neglected. 1297. The radii of t-wo circles -whose centres are in the same horizontal line are a, h, and the distance between their centres c{>a + h); prove that the shortest time of descent from one to the other down a straight line is //2 c°-('^. 1300. A uniform string of length 2a is in equilibrium, passing over a small smooth pulley ; if it be just displaced, the velocity of the string ■when the -whole has just passed the pulley is ^Jag- II. Paratolia Motion. 1301, A heavy particle is projected in a given direction, determine its velocity in order that it may pass through a fixed point. 272 BOOK .OP MATHEMATICAL PEOBLEMS. 1302. If a particle moving under tlie action of gravity pass through two given points, the locus of the focus of its parabolic path -will be a hyperbola. 1303. A heavy imperfectly elastic paxticle is projected from a point in a horizontal plane in such a manner that at its highest point it impinges directly on a vertical plane, from which it re- bounds, and, after another rebound from the horizontal plane, returns to the point of projection; prove that the coefficient of elasticity is ^. 1304. If rj, r^j, r^ be three distances of a projectile from the point of projection, and a,, a^, a^ the angular elevations of these points above the point of projection, r, cos" Oj sin (a^ — a^) + r^ cos° a^ sin {a.^ — a^ + r^ cos" a^ sin (a^ — a^) = 0. 1305. The axis of a parabola is vertical and the vertex downwards, and a circle has its centre at a point P on the para- bola, and passes through the focus ;S^; a perfectly elastic particle sliding down PS is reflected at the circle and then moves freely under the action of gravity ; find where it next meets the circle, and, if it be at the lowest point, prove that SP is equal to two thirds of the latus rectum. 1306. A particle is projected up an inclined plane of given inclination so as after leaving the plane to describe a parabola . prove that, for difierent lengths of the plane, the loci of the focus and vertex of the parabolic path are both straight lines. 1307. A perfectlyelastic particle is projected from the middle point 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 is to a side of the square :: 45 : 32. DYNAMICS, ELKMENTAEY. 273 1308. A perfectly elastic particle 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 straight line ; deter- mine the angle of projection in order that the particle may after reflection return to the point of projection and be again reflected in the same path. Prove that the inclination of each plane must be 45°. 1309. An imperfectly elastic particle let fall on a fixed inclined plane bounds on to another fixed inclined plane, the line of inter- section being horizontal, and the time between the two planes is given ; prove that the locus of the point from which the particle is let fall is in general a parabolic cylinder ; but that it is a plane if tan a tan (a -I- /3) = e, a, p being the inclinations of the planes, and e the elasticity. 1310. A particle is projected from a given point, and its resolved velocity parallel to a given straight line is given ; prove that the locus of the focus of the parabolic path is a parabola of which the given point is the focus, and whose axis makes with the vertical an angle double that made by the given straight line. 1311. A particle, projected from a point in an inclined plane, after the r"* impact begins to move at right angles to the plane, and at the v}^ iiapact is at the point of projection ; prove that e" - 2e' -H 1 = 0, e being the elasticity. 1312. A particle is projected from a given point in a hori- zontal plane at an angle a to the horizon, and, after one rebound at a vertical plane, returns to the point of projection ; prove that the point of impact must lie on the straight line 2/(1 -t- e) = a; tan a, X, y being measured horizontally and vertically from the point of projection. If the velocity of projection and not the direction be given, the locus of the point of impact is an ellipse, W. 18 274 BOOK OF MATHEMATICAL PKOBLEMS. 1313. An imperfectly elastic particle is projected from a given point with given velocity, so as, after one rebound at an inclined plane passing through the point, to return to the point of projection ; prove that the locus of the point where the particle strikes the inclined plane is the ellipse a!^+(l+e)'2^'=4e%, X, y being horizontal and vertical through the given point, e the elasticity, and h the space due to the given velocity. 1314. 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 para- bolic path described, before meeting the horizontal plane through the point of projection, is greatest, the range is -=- , and the height of the vertex is -^ . o ] 315. A heavy particle is projected from a point in a plane whose inclination to the horizon is 30° with given velocity, in a vertical plane perpendicular to the inclined plane j prove that, if all directions of projection in that vertical plane are equally pro- bable, the chance of the range on the inclined plane being at least one third of the greatest possible range is \, III. Motion on a Smooth Curve vmder tlie action of Gramty. 1316. A heavy particle is projected from the vertex of a smooth parabolic 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 (A — Ic), DYNAMICS, ELEMENTARY. 275 1317. A particle is projected so as to move on a parabolic arc whose axis is vertical and vertex upwards; prove that the pressure on the curve in any position is proportional to the cur- vature. 1318. 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 plane. 1319. A particle P is attached by two strings to fixed points A, B in the same horizontal line, and is projected so as just tp describe a vertical circle ; PB is cut when the particle is in its lowest position, and P proceeds to describe a horizontal circle ; prove that cos 2PAB = \, o and that, if the tension of PA be unaltered, the angle APB is a right angle. 1320. Two given weights are attached 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 ; determine the inclinations of the two parts of the string to the vertical. 1321. A parabola is placed with its axis horizontal and plane vertical, and a heavy smooth particle is projected upwards from the vertex so as to move on the concave side of the curve ; prove that the vertical space described before leaving the curve is two thirds of the greatest height attained. If 2^ be the angle described about the focus before leaving the curve h = a (tan' ^ + 3 tan 6), h being the space due to the velocity of projection and ia the latus rectum ; also the latus rectum of the subsequent path is ia tan' 6. 18—2 276. BOOK .OP MATHEMATICAL ]?EOBLEMS. 1322. A heavy particle is projected so as to move on a circu- lar arc whose plane is vertical, and afterwards describe a purabola freely ; prove that the loctis of the focus of the parabolic path is an epicycloid formed by a circle of radius j rolling on a circle of radius ^ ; a being the radius of the given circle. 1323. A cycloidal arc is placed with its axis vertical and vertex upwards, and a heavy particle is projected from the cusp up the concave side of the curve with velocity due to a height A ; prove that the latus rectum of the parabola described after leaving the curve is ^, where a is the length of the axis of the cycloid. 1324. If a cycloidal arc be placed with its axis vertical and vertex upwards, and a heavy particle be projected from the cusp up the COD cave side of the curve ; the focus of the parabola described by the particle after leaving the curve will lie on a fixed cycloid of half the dimensions. 1325. In a certain curve the vertical ordinate of any point bears to the vertical chord of curvature at that point the constant ratio 1 : m, and a particle is projected, from the point where the tangent is vertical, along the curve with any velocity; prove that the height ascended before leaving the curve : height due to velocity of projection :; 4 : i + m, 1326. A smooth heavy particle is projected from the lowest point of a vertical circular arc with a velocity due to a space eqiial in length to the diameter 2a : the length of the arc is such that the range of the particle on the horizontal plane through the point of projection is the greatest possible ; prove that this range is equal to a J(9 + 6 ^3), ( 277 ) NEWTON. 1327. Two triangles CAB, cAh, have a common angle A and the sum of the sides containing that angle the same in each; BG^ be intersect in D ; prove that, ultimately, when 6 moves up to B, CD : DB :: AB : AC. 1328. 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 revolution of the arc PQ about the axis bears to the area MpqN a constant ratio. 1329. 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 intercepted area of the outer. 1330. AB is a diameter of a circle, P a point on the circle near A, and the tangent at P meets BA produced in T ; prove that ultimately the difference of BA, BP bears to AT the ratio 1 : 2. 1331. If PQ be an arc of continued curvature, and R the point between P and Q at which the tangent is parallel to PQ ; then the ultimate ratio PB, : BQ, when PQ is indefinitely dimin- ished, is one of equality. 1332. If ^ be a point on a curve, and P, Q two neighbour- ing points, the ultimate ratio of the triangle formed by the tan- gents at these points to that formed by the normals is ^ ©". 278 BOOK OF MATHEMATICAL PEOBLEMS. when P, Q move up to coincidence with A j p being the radius of curvature at A, and s the arc to A measured from a fixed point. 1333. A parabola is described about a force in the focus, and along the focal distance SP is measured SQ equal to a constant length ; QR is drawn perpendicular to the tangent at P to meet the axis in R; prove that QR : 2SQ : : velocity at P : velocity at the vertex. 1334. Prove that the equation 2V' = F.PVi3 true when a body moves in a resisting medium, F being the extraneous force, and P V the chord of curvature in direction of P. 1335. Two points P, Q move in the following manner; P describes an ellipse under acceleration tending to the centre, and Q describes relatively to P an ellipse of which P is the centre under acceleration tending to P, and the periodic times of these ellipses are the same; prove that the absolute path of Q is an ellipse concentric with the path of P. 1336. A particle describes a hyperbola under a force tending to one focus ; prove that the rate at which areas are described by the central radius vector is inversely proportional to the distance of the particle from the centre of force. 1337. A rectangular hyperbola is described by a point under acceleration parallel to one of the asymptotes ; prove that at a point P the acceleration is ^^ GW' PM being drawn in direction of the acceleration to meet the other asymptote, C the centre, and U the component velocity perpendi- cular to the acceleration. 1338. A point describes a cycloid under acceleration tending to the centre of the generating circle; prove that the velocity at any point varies as the radius of curvature. 1339. A particle constrained to move on an equiangular spiral is attracted to the pole by a force proportional to the NEWTON. 27d distance ; prove tliat, in whatever position the particle be placed at starting, the time of describing a given angle about the centre of force ■will be the same. 1340. An endless string, on -which runs a small smooth bead, incloses 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 a» the rectangle under the focal distances. 1341. A parabola 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. 1343. A particle is describing an ellipse about a centre of force /Ar"^, at a certain point /a receives a small increment S/a, and the eccentricity is unaltered ; prove that the point is one extremity of the minor axis, and that the major axis 2a is dimin- ished by . 1343. A particle is describing an ellipse about a centre of force fir~^, and at a certain point /* is suddenly increased by S/a ; prove the following equations for determining the corresponding alterations in the major axis 2a, the eccentricity e, and the longi- tude of the apse ■or, rSa eBe r eS'sr 8/x a {2a -r)~ I —e^ a — r sin (0 - ■ar) /x ' r, 6 being polar co-ordinates of the point at which the change takes place. 1344. In an elliptic orbit about the focus, when the particle is at a distance r from the centre of force, the direction of motion is suddenly turned through a small angle S/S ; prove that the consequent alteration in the longitude of the apse is 2a being the length of the major axis, and e the eccentricity. 280 BOOK OF MATHEMATICAL PEOBLEMS. 1345; At any point in an elliptic orbit about the focus, the velocity v receives a small increment hv; prove that the alterations in the eccentricity e, and the longitude of the apse 'S'', will be given by the equations V (a - r) " ab ^{csV - (r - af] " /xae (2a - r) ' 1346. An ellipse is described by a particle under the action of two forces tending to the foci, each varying inversely as the square of the distance ; prove that a, h being the axes, and a>, u> the angular velocities at any point about the foci. 1347. Two fixed points of a lamina slide along two straight lines fixed in space, so that the angular velocity of the line joining the points is constant ; prove that (1) every fixed point of the lamina describes an ellipse under acceleration tending to the intersection of the two fixed straight lines and proportional to the distance : (2) every fixed straight line of the lamina envelopes during its motion an involute of a four-cusped hypooycloid : (3) the motion of the lamina may be completely represented by supposing a circle fixed in the lamina to roll uniformly with internal contact on a circle of twice its radius fixed in space : (4) for a series of points of the lamina lying in a straight line, the foci of the ellipses described lie on a rectangular hyperbola. 1348. If a lamina move in its plane so that two fixed points in it describe straight lines with acceleration _/,_/' ; the accelera- tion of the centre of instantaneous rotation is 6 being the angle between the lines. NEWTON. 281 1349. A lamina moves in its own plane, so that two points, fixed in the lamina, describe straight lines with equal accelera- tions; 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. 1350. Two points fixed in a lamina move along two straight lines fixed in space, and the velocity of one of the points is uni- form; prove that every point in the lamina moves so that its acceleration is constant in direction and varies inversely as the cube of the distance of the point from a fixed straight line. 1351. Two ellipses are described about a common attractive force in their centre ; the axes of the two are coincident in direc- tion, and the sum of the axes of one is equal to the difference of the axes of the other ; prove that if the desci-ibing particles be at corresponding extremities of the major axes at the same moment and be moving in opposite directions, the line joining them will be of constant length during the motion and will revolve with uni- form angular velocity. 1352. A lamina moves in such a manner that two straight lines fixed in the lamina pass through two points fixed in space ; prove that the motion of the lamina may be completely repre- sented by supposing a circle fixed in the lamina to roll with internal contact on a circle of half its radius fixed in space. 1353. A triangular lamina ABG moves so that the point A lies on a straight line be fixed in space, and the side BG passes through a point a fixed in space, and the triangles ABC, abc are equal and similar ; prove that the motion of the lamina may be completely represented by supposing a parabola fixed in the lamina to roll on an equal parabola fixed in space. 1354. Two particles describe curves under the action of central attractive forces, 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. ( 282 ) DYNAMICS OF A POINT. I, Eectilinear Motion, Kinematics. 1355. A heavy particle is attached by an elastic string to a fixed point, from which the particle is allowed to fall freely; when the particle is in its lowest position the length of the string is twice its natural length ; prove that the coefficient of elasticity is four times the weight of the particle, and find the time during which the string is extended beyond its natural length. 1356. 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 G in BA produced, BG being equal to ^BA, and the particle just reaches the centre of force; prove that the velocity of the particle will be greatest at a point which divides GA in the ratio 8 : 7. 1357. A particle is attracted to a fixed point by a force = /x (dist.)"'', and repelled from the point by a constant force f; the particle is placed at a distance a from the centre, at which point the attractive force is four times as great as the repulsive, and projected directly from the centre with velocity v ; prove that, (1) the particle will move to infinity or not according as «>\ _ d<^ ds df^dt'" -rif), andi-?f.^^V97- \dtj' r dt\ dt) dt dt 1361. A point describes a curve of double curvature, and its polar co-ordinates at time t are (r, 0, ) • prove that its acceler- ations, (1) along the radius vector, (3) perpendicular to the radiiis vector in the plane of 0, and (3) perpendicular to the plane of 0, are respectively 284 BOOK OF MATHEMATICAL PEOBLEMS. 1362. A point describes a parabola in. such a manner that its velocity, at a distance r from the focus, is y{¥c-«->}. f, c being constant ; prove that its acceleration is compounded of fc' y parallel to the axis, and -5- from the focus. 1363. A point describes a semi ellipse bounded by the minor axis, and its velocity at a distance r from the focus is f{a-r)\l "V(2a-r)j ' where 2a is the length of the major axis, and/a constant accelera- tion ; prove that the acceleration of the point is compounded of two, each varying universely as the square of the distance, one tending to the nearer focus and the other from the farther focus. 1364. A point is describing a circle, and its velocity at an angular distance 6 from a fixed point on the circle varies as V(l+cns'e) sin^ e ' prove that its acceleration is compounded of two tending to fixed points at the extremities ofa diameter, each varying inversely as the fifth power of the distance and equal at equal distances. 1365. A point describes a circle, under acceleration constant and not tending to the centre ; prove that the point oscillates through a quadrant, and that the direction of the acceleration always touches a certain hypocycloid. 1366. A parabola is described with accelerations F, A, tending to the focus and parallel to the axis respectively ; prove that r bein" the focal distance. DYNAMICS OF A POINT, 285 1367. A point describes an ellipse ■with accelerations /(r ), <^ (r^) tending to the foci ; prove that P ~&r ^''''■^(''')^ = 7"^ ^ ^''^ "^ W^ ' r^, r^ being the focal distances. 1368. The parabola i/' = iax is described by a point under accelerations X, F parallel to the axes; prove that _ dY dX ^.^ . 1369. A point describes a parabola under acceleration which makes a constant angle a with the normal, 6 is the angle described from the vertex about the focus in a time t ; prove that ~j cc cos' - £ ' . dt 3 Find also the law of acceleration. 1370. A point describes a circle of radius 4a with uniform angular velocity w about the centre, and another point Q describes a circle of radius a with angular velocity 2a) about P ; prove that the acceleration of Q varies as the distance of P from a certain fixed point. 1371. An equiangular spiral is described ,by a point with constant acceleration in a direction making an angle i^ with the normal ; prove that ddi _ . sm j^ = 2 Sm d> + cot a cos A ; du a being the constant angle of the spiral, and 6 the angle through which the tangent has turned from a given position. 1372. The only curve which can be described under a con- stant acceleration in a direction making a constant angle with the normal is an equiangular spiral, 286 BOOK OP MATHEMATICAL PROBLEMS. 1373. A parabola, if = 1ex, is described by a point under acceleration making a constant angle a with, the axis, and the velocity when the acceleration is normal is i> ; prove that the acceleration at any point (x, y) is (c cos a — y sin of ' If the point, when the acceleration is normal, be moving towards the vertex, the time in which the direction of motion will turn through a right angle is 2c V (sin 3a + sin o) " 1374. The intrinsic equation of a curve being s=f{)^^{<\>-e)=f"{'i>)- 1376. A catenary is described by a point under acceleration, whose vertical component is constant [f) ; prove that the hori- zontal component at a point where the tangent makes an angle ^ with the horizon is . cos <^ ,, , . / ■ a , (1 + w cos A + cos , that of acceleration has turned through 2^ in the same sense; prove that the acceleration cccosrf>— , ds the velocity oc cos - 26) ° { \2 4/ J ' ortiat d>-2e = '^. 1384. P describes a circle under acceleration tending to S and varying as the distance, 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, ^^J^Y'^'J^^' and the distance of S from the centre is j-- ; a being the m cos ti 6 radius, and m, a constant. 1385. A point describes an arc of a circle, so that its acce- leration is always proportional to the «*■" power of its velocity; prove that the direction of the acceleration touches a certain epicycloid, generated by a circle of radius „ ^^ rolling on one 6 —2m ° 2 — re of radius a -z — ; a being the radius of the circle described. DYNAMICS OP A POINT. 289 II. Central Forces. 1386. Prove that tlie parabola y^=4Mx can be described under a constant force parallel to the axis of y, and a force proportional to y parallel to the axis of x. Also under two forces ifn. (c + x), jty, parallel to the axes of x and y. 1387. A cardioid is described with constant angular velocity about the cusp under a constant force to the cusp and another constant force; prove that the magnitude of the latter is twice that of the former, and that its direction always touches an epi- cycloid generated by a circle of radius a rolling on one of radius 2a; 8a being the length of the axis. 1388. The force to the pole, under which the hyperbola r cos 20 = a cos can be described, will vary as t-^ . ' ■' cos' 6 1389. ST is the perpendicular from the pole jS' of a lem- niscate upon the tangent at F, and the locus of Y is described by a particle under a force to *S' : prove that this force £= /SP~". 1390. The force to the pole under which the pedal of a given curve r=/(p) can be described will vary as 2r'' r dr p^ p* dp' If the given curve be » » 3 a^ = r^ sin ^ 6, the force will be constant. 1391. A parabola is described about a centre of force in C, the centre of curvature at the vertex ; prove that the force at any w. 19 290 BOOK OF MATHEMATICAL PROBLEMS. GP point P of the parabola varies as „ — cp\3' -^ being the vertex and S the focus. > 1392. The force tending to the pole under -which the evolute pf the curve r=f{p) can be described will vary as {r'-p'f fdrV d'r (; [dpj dp' 1393. An orbit described under a constant force tending to a fixed point will be the pedal of one of the curves represented hj the equation r' = Ap^ + Bp\ 1394. A particle is projected from a point A at right angles to a straight line SA, and attracted to aS by a force varying as cosec PSA ; prove that the rate of describing areas about A will be uniformly accelerated. 1395. A particle is projected at a distance a with velocity equal to that in a circle at the same distance and at an angle 45° with the distance, and attracted to a fixed point by a force which -^ + — j : determine the orbit de- scribed, and prove that the time to the centre of force is h-{^~^^' 1396. A particle is attracted to a fixed point by a force which at a distance r is equal to /otr-'(3a»+3aV*-?-"), and is projected from a point at a distance a from the centre with velocity equal to that in a circle at the same distance and in DYNAMICS OF A POINT. 291 a direction making an angle cot"' 2 with the distance ; deter- mine the orbit, and prove that the time to the centre of force ^ log 2. 4V/X 1397. 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. 1398. 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 : ^n ; prove that peer", p being the perpendicular from the centre of force on the tangent at a point whose distance is r. Prove also that the force varies inversely 1399. A particle acted on by a central force ~j{ j is projected at a distance a, at an angle 45°, and with velocity — —: determine the orbit and prove that the time, from projec- a tiontoanapseis 3-^(^y-^-lj. 1400. In an orbit described under a central force, the velo- city of T, the foot of the perpendicular from the centre of force on the tangent, is constant; prove that the chord of curvature of the orbit through the centre of force is constant. 1401. A particle describing a parabola about a force in the focus comes to an apse, at which point the law of force changes, and the force varies inversely as the distance till the particle next comes to an apse, when the former law is restored. No instanta- neous chauges being supposad, prove that the major axis of the 19—2 292 BOOK OP MATHEMATICAL PROBLEMS. new orbit will be r, 4a being the latus rectum of the m— 1 ma m parabola, and m tbe root of the equation -, = l-loga;, ■which lies between 1 and e. 1402. In an orbit described under a central) force a straight line is drawn from a fixed point perpendicular to the tangent and proportional to the force, and this straight line describes equal areas in equal times ; prove that the equation of the orbit is of the form 1 1 /r\| ps cTT \a J Prove that the rectangular hyperbola is a particular case. 1403. A chain of uniform material rests under normal and tangential forces —n,t\ prove that the curve in which it rests could be described by a particle, whose mass is equal to that of a unit of length of the chain, under the action of normal and tangential forces Sti, t. 1404. A centre offeree varying inversely as the w"' 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 : velocity of the particle : : 5 — w : \~n. If ^ = 3, the time of describing a semicircle is 4a° /-. V /^ 1405. A particle P is repelled from a fixed point (S* by a force varying as (dist.)"^, and attracts another particle Q with a force varying as (dist.)~° ; initially P and § are equidistant from S in opposite directions, P is at rest, and the accelerations of the two forces are equal ; prove that if Q be projected at right angles to SQ with proper velocity, it wUl describe a parabola of which S is the focus. DYNAMICS OP A POINT. 293 III. Constrained Motion on Curves or Surfaces: Particles joined hy Strings. 1406. A particle is constrained to move on a curve under tte action of forces, such that if projected from a certain point of the curve with velocity v, it would describe the curve freely. Prove that, when projected from that point with velocity F, the pressure on the curve is always m , p being the radius of curvature at any point, and m the mass of the particle, 1407. A particle is acted on by two forces, one parallel to a fixed straight line and constant, and the other tending from a fixed point and varying as (dist.)"'', and the particle is initially at rest at a point where the.se forces are equal ; prove that it will proceed to describe a parabola whose focus is the fixed point and axis parallel to the fixed line. 1408. Two particles A, B are together in a smooth circular tube ; A attracts B with a force whose acceleration is ft. . distance, and moves along the tube with uniform angular velocity 2Jfi; B is initially at rest ; prove that the angle ^ subtended at the centre hj AB after a time t is given by the equation Jga; prove tliat when it again moves hori- zontally, its vertical depth below the highest point of the sphere is a mean proportional between the radius and its initial distance from the lowest point. 1415. A particle in motion on a smooth surface z = ^(x, y) under the action of gravity describes a curve in a horizontal plane : if u be its velocity, prove that u'(d'z^ds\' d'z dzdz d^(dz\'-\ ( fdzV (dzV)' g{da?\dy) dxdy dxdy di/^\dxj} \\dx' \dy) j ~ ' the axis of z being vertical. 1416. In a smooth surface of revolution whose axis is verti- cal, a heavy particle is projected so as to move on the surface and describe nearly a horizontal circle ; prove that the time of a vertical oscillation is ye hr sin a "1 \g (k + 3r cos a sin'' a)J ' k being the distance from the axis, r the radius of curvature of the generating curve, and u, the inclination of the tangent to the vertical, in the mean position of the particle. 1417. A particle slides in a vertical plane down a rough cycloidal arc whose axis is vertical, starting frorn the cusp and coming to rest at the vertex; prove that the coefficient of friction is given by the equation /xV=l. 1418. Three equal and similar particles repelling each other with forces varying as the distance are connected by equal inex- tensible strings and are at rest : if one of the strings be cut, the subsequent angular velocity of either of the other strings will vary as /( ^ — ^^^ I, 6 being the angle between them. •' ^ \ 2 + cosO J 296 BOOK OP MATHEMATICAL PEOBLEMS. 1419. A heavy particle is attached to a fixed point by a fine string of length a, and when the string is horizontal and at itS| full length, the particle is projected horizontally at right angles to the string with a velocity due to a height 2a cot a; prove that the greatest depth to which it will fall is a tan ^ . 1420. Two heavy particles are placed on a smooth cycloidal arc whose axis is vertical, and are connected by a fine string passing along the arc : c is the distance of either particle mea- sured along the arc from its position of equilibrium ; prove that the time of arriving at a distance s from the position of equi- librium is J- -l,g. a being the radius of the generating circle. 1421. An elliptic wire is placed with its minor axis vertical, and on it slides a smooth ring to which are attached strings, passing through fixed rings at the foci and sustaining each a particle of weight equal to that of the ring : determine the velo- city which the particle must have at the highest point that the velocity at the lowest point may be equal to that at the extremity of the major axis. 1422. Two particles of masses /», q are connected by a fine string passing through a small fixed ring ; p hangs vertically, and q is held so that the adjacent part of the string is horizontal : if q be let go, the initial tension of the string is — — , and the initial radius of curvature of g's path is ^ {p'+(^-Hg)f . a being the initial distance of q from the ring. 1423. Two particles of masses m,, m lying on a smooth hori- zontal table are connected by an inextensible string at its full m' 1111 1 J - + -=- + -; P ip p a a DYNAMICS OF A POINT. 297 length, and passing througli a small fixed ring in the table ; the particles are at distances a, 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, if mv'd = m'v"a, either particle will describe a circle uniformly; and that, if mv'a' = m'v"a', their second apsidal distances will be a, a respectively. 1424. Two particles m, m connected by a string which passes through a small fixed ring are held so that the string is horizontal, their distances from the ring being a, a, and are simultaneously let go ; prove that m P p, p being the initial radii of curvature of their paths. 1425. 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 on the point of motion, B is projected horizontally with a velocity u in the plane perpendicidar to the edge of the table ; prove that A will begin to move with acceleration =■ — , and that the initial radius of . p.-\-\ a curvature of 5's path will be (/^ + 1) a, /<. being the coefficient of friction. 1426. A smooth wire in the form of a circle is made to re- volve uniformly in a horizontal plane about a point A in its circumference, with angular velocity u. A small ring P slides on the wire and is initially at rest at its greatest distance c from A ; prove that its distance from ^ at a time t is —^^ —. , and that the tangent to P's path in space bisects the angle between FA and the radius to F. 298 BOOK OF MATHEMATICAL PEOBLEMS. 1427. Two equal particles are connected by an inextensible string; one lies. on a smooth horizontal table, and the string passes through a small fixed ring in the edge of the table to the other ■which is vertically below the ring : the former particle is pro- jected on the table in a direction at right angles to the string with a velocity . / , ^ ,, , c being its distance from the ring: •' V n{n + l) ° prove that its next apsidal distance will be - , and that its ve- locity will then be to its initial velocity a,s n : 1. Prove also that the radius of curvature of the projected parti- 4c cle's initial path is — ; rr — s . ^ n (n + 1) + 2 1428. Two particles, whose masses are p, q, are connected by a fine string passing through a small fixed ring; p hangs verti- cally, and q is projected so as to describe a path which is nearly a horizontal circle ; prove that the distance of p at any time from its mean position will be A sin {mt + B) +A' sin (nt + B') ; m, n being the positive roots of the equation L' - ^ (1 - cos a)| L? - -^— (1 + 3 cosV)| = ^ cos a (1 - cos a); where c is the mean distance of q from the ring, and p =fq cos a. 1429. A particle is placed in a rough tube f ;«,= -A which revolves uniformly in one plane about one extremity, and no forces but the pressures of the tube are in operation ; prove that the equation of the particle's path is 5r = a (4£'2 + £"=*). 1430. A rectilinear tube inclined at an angle a to the verti- cal revolves with uniform angular velocity tu about a vertical y: DYNAMICS OF A. POINT. 299 axis which, intersects it, and a particle is projected from the O cos tt stationary point of the tube with a velocity '^„ .' — - ; find its CO ;y (sin a) position at any time before it attains relative equilibrium; and prove that the equilibrium is unstable. 1431. A smooth parabolic tube of latus rectum I is made to revolve about its axis, which is vertical, with angular velocity J , and a particle is projected from the vertex 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 twice that due to the velocity of projection. 1432. A smooth parabolic tube revolves with uniform angular velocity about its axis, which is vertical, and a particle is placed within the tube very near to the lowest point : find the least angular velocity which the tube can have in order that the . particle may ascend ; and, if it ascend, prove that its velocity will be proportional to its distance from the axis. Prove also, that if any position of the particle in the tube be one of relative equili- brium, every position will be such. 1433. A curved tube is revolving uniformly about a vertical axis in its plane, and is symmetrical with respect to that axis ; the angular velocity is /-, a being 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 at the vertex is an ellipse or hyperbola. 1434. A circular tube of radius a revolves uniformly about a vertical diameter with angular velocity n I -, and a particle is projected from its lowest point with such velocity as just to reach the highest point; prove that the time of describing the first quadrant is 7; {n + 1)5- 300 BOOK OP MATHEMATICAL PKOBLEMS. 1435. A circular tube containing a smooth particle revolves about a vertical diameter with uniform angular velocity a>, find the position of relative equilibrium of the particle; and prove that it will oscillate about this position in a time — : , a being ■^ tii Bin a the angle which the normal at the point makes with the vertical. 1436. A heavy particle is placed in a tube in the form of a plane curve, which revolves with uniform angular velocity to, about a vertical axis in its plane, and the particle oscillates about its position of relative equilibrium; prove that the time of oscil- lation is 2x // p sin a \ to 'y \k — p sin a cos" aj ' h being the distance from the axis of revolution, a the angle made by the normal with the vertical, and p the radius of curvature of the curve, at the point of equilibrium. 1437. A straight tube, inclined to the vertical at an angle a, revolves with uniform angular velocity ta about a vertical axis whose shortest distance from the tube is a, and contains a smooth particle placed initially at its shortest distance from the axis ; prove that - __ ff COSg f af stna . -lit sin a _ oi , ^ t coJsma_ ^-ut sinni . ui' sin'' a ^ ' ' s being the space described along the tube in a time t. 1438. A heavy particle is attached to two points in the same horizontal line, at a distance a, by two elastic strings, each of natural length a, and is set free when the strings are a,t their natural length ; prove that the initial radius of curvature of its path is -^ — ; the coefficients of elasticity being respectively m and n times the weight of the particle. 1439. A uniform heavy chain is placed on the arc of a smooth vertical circle, its length being equal to a quadrant, and one ex- DYNAMICS OF A POINT. 301 tremifcy being at the highest point of the circle ; prove that, in the beginning of the motion, the resultant vertical pressure on the circle : the resultant horizontal pressure : : ir" — 4 : 4. TV. Motion of uniform strings. 1440. A heavy uniform string PQ of which P is the lower extremity, 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 point R of the string is _^ V ("sin y , .. sin a , .. ) W -!- X '- cos (y + 6) cos (a + 6) > , where B, 2a, 2y are the angles AOP, POQ, POP respectively, and W is the weight of the string. 1441. A portion of a heavy uniform string is placed on the arc of a four-cusped hypocycloid, occu^jying the space between two adjacent cusps, and runs off the curve at the lower cusp, the tangent at which is vertical ; prove that the velocity which the string will have when the whole of it has just left the curve, will he the velocity due to nine-tenths of the length of the string. 1442. A uniform string is placed on the arc of a smooth curve in a vertical plane, and moves under the action of gravity ; prove that 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. 1443. A uniform heavy string is placed on the arc of a smooth cycloid, whose axis is vertical and vertex upwards; deter- mine the motion, and prove that, so long as the string is "wholly in contact with the cycloid, the tension at any given point of the string ii constant throughout the motion, and is a maximum at the middle point. 302 BOOK OF MATHEMATICAL PEOBLEMS. 1444. A uniform heavy chain of length I 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 c^ with the vertical j prove that the diiference between the depths of the extremities is I cos <^. 1445. A uniform inextensible string is at rest in a smooth groove which it just fits, and a tangential impulse F is applied at one extremity; prove that the normal impulse, referred to a unit of Ps length of the string, at a distance s from the other extremity is — , a being the whole length of the string, and p the radius of curvature at the point under consideration. 1446. A straight tube of uniform bore is revolving uniformly in a horizontal plane about a point 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 along the tube, and that the tension at any point of the chain is where x is the distance of the point from the middle point, m the mass of the chain, and m the angular velocity .^ 1447. A circular tube, of radius a, revolving with uniform angular velocity oi about a vertical diameter contains a heavy uniform chain, which subtends an angle 2a at the centre ; prove that the, chain will remain in relative equilibrium if the radius through its middle point makes with the vertical an angle cos \aui' cos aj ' DYNAMICS OF A POINT. 303 and that the tension will be a maximum either at the lowest poiut of the tube, or at a point whose projection on the axis bisects the distance between the projections of the extremities, Y. BesisUng Medium, Eodograph. 1448. A heavy particle is projected vertically upwards, 3.nd / 1 \2 resistance of the air is m — ; x, y are the respective heights of two points in the ascent and descent at which the particle has the same velocity; prove that f ' +e ' =2. 1449. A heavy particle moves in a medium in which the re- sistance varies as the square of the velocity ; v, v are its velocities at the two points where its direction of motion makes an angle a with the horizon, and u the velocity at the highest point; prove that 1 1 2 cos' g v' v"'~ u" 1450. A particle is moving under the action of gravity in a medium whose resistance varies as the square of the velocity ; p, p are the radii of curvature of its path at two points at each of which the direction of motion makes an angle a. with the horizon, and r the radius of curvature at the highest point; prove that 1 1 2cos'a - + — = ■ . P P »• 1451. A small smooth bead slides on a fine wire whose plane 2s . is vertical, and the height of any point of which is a sin — , s being the arc measured from the lowest point, in a medium whose 304 BOOK OF MATHEMATICAL PROBLEMS. resistance is m- — —, and starts from the point where s=-^ ; c o prove that the velocity acquired in falling to the lowest point is ^ag. 1452. A heavy particle slides on a smooth curve, whose plane is vertical, iu a medium whose resistance varies as the square of the velocity, and in any time describes a space which is to the space described in the same time by a particle falling freely in vacuo as 1 :2m; prove that the curve is a cycloid, the vertex being the highest point, and that the starting point of the particle divides the arc between two cusps in the ratio 2n—l : 2n+l. 1453. A point describes a straight line under acceleration tending to a fixed point and varying as the distance ; prove that the corresponding point of the hodograph will move under the same law of acceleration. 1454. The curves r" = a" cos m6, r" = a" cos nO will be each similar to the hodograph of the other when described about a centre of force in the pole, provided that i.Ul=0. m n Prove this property geometrically for both curves when m=l. 1455. A point describes a curve in such a manner that its hodograph is described' as if under the action of a central force, and T, iV^are the tangential and normal accelerations of the point; prove that dp \' p p being the radius of curvature, s the arc m3asured from a fixed point, and c a constant. DYNAMICS OP A. POINT. 305 1456. A point describes half the arc of a cardioid, oscillating symmetrically about the vertex, in such a way that the hodograph is a circle, the origin being in the circumference; prove that the acceleration of the point describing the cardioid will vary as 2r — 3a; r being the distance from the cusp, and 2a the length of the axis. 1457. 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 P sweeps out equal areas in equal times ; prove that the direction of P's acceleration makes with the normal at P an angle tan' ■^(|tan<^), , that of acceleration has turned through an angle 2(p in the same sense; prove that the hodograph of the path is a circle described about a point in the circumference. 1462. 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 hodo-, graph is a circle. w. 20 (306 ) DYNAMICS OF A EIGID BODY. I. Moments of Inertia, Principal Axes. 1463. If m be the mass of an ellipsoid, of wHch. the density at any point is proportional to the product of the distances of the point from the principal planes, the moment of inertia about one of the axes is — ^-j j 2b, 2c being the axes of the correspond- ing principal section. 1464. If a straight line be at every point of its course a principal axis of a given rigid body, it must pass through the centre of gravity, 1465. If ^, £, G be the principal moments of inertia at the centre of gravity of a rigid body, a' + h' + c^+r^ a principal moment at a point whose co-ordinates referred to the principal axes through the centre of gravity are a, h, c; the equation determining r is g' ¥ e' the mass being unity. 1466. The^locus of the points at which two of the principal moments of inertia of a given rigid body are equal is the focal curves of the ellipsoid of gyration for the centre of gravity. 1467. The locus of the points at which one of the principal axes of a given rigid body passes through a fixed point, in one of the principal planes through the centre of gravity, is a circle. 1468. The locus of the points at which one of the principal axes of a given rigid body is parallel to a given straight line is a rectangular hyperbola. DYNAMICS OF A EIGID BODY. 307 1469. Tn a triangular lamina any one of the sides ia a principal axis at the point bisecting the distance between its middle point and the foot of the perpendicular on it from the opposite angle, 1470. If straight lines be drawn in the plane of a given lamina through a given point, the locus of the pointy at which they are respectively principal axes is a curve of the third degree. 1471. 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)yz + b{G-A)zx + c{A-B)xy = 0, A, B, G being the principal moments of inertia at the given point, a, b, c the co-ordinates of the centre of gravity ; and the principal axes at the given point the axes of reference. Prove that the locus of the points at which these straight- lines are principal axes is the curve cy — bz az — cx' 1472. If the principal axes at any point be parallel to the principal axes through the centre of gravity, the point must lie on one of the principal axes at the centre of gravity. II. Motion about a Fixed Axis. 1473. A circular disc rolls in one plane on a fixed plane, its centre describing a straight line with uniform acceleration y": find the magnitude and position of the resultant of the impressed forces. 1474. A piece of wire of given length is bent into the form of an isosceles triangle, and revolves' about an axis through its vertex perpendicular to its plane ; prove that the centre of oscilla- tion will be at the least possible distance from the axis of revolu- tion when the triangle is right-angled, 1475. A heavy sphere of radius a, and a heavy rod of radius 2as swing, the one about a horizontal tangent, and the other about 20—2 808 BOOK OF MATHEMATICAL PROBLEMS. a horizontal axis through, one extremity, through 90° to their lowest positions, and the pressures on the axes in those positions are equal; prove that their weights are as 35 : 34. 1476. The centre of percussion of a triangular lamina, one of whose sides is a fixed axis, bisects the straight line joining the opposite angle with the middle point of the side. 1477. A lamina ABGB is moveable about AB which is parallel to CD ; prove that its centre of percussion will be at the intersection of AG, BD, if AB'' = ZCD\ 1478. In the motion of a rigid body about a horizontal axis under the action of gravity, prove that the pressure of the axis can only be reduced to a single force, throughout the motion, when the axis of revolution is a principal axis at the point M which is nearest to the centre of gravity. If the axis be a princi- pal axis, but at another point N, and the pressures on the axis be reduced to two, at M and N respectively, the former will be equal and opposite to the weight of the body. 1479. A rough uniform rod, length 2a, is placed with a length c(> a) projecting over the edge of a horizontal table, the rod being perpendicular to the edge; prove that the rod will begin to slide over the edge when it has turned through an angle , tan-' f""' o'+9(c-a/' 1480. A uniform beam capable of motion about one extre- mity is in equilibrium; find at what point a blow must be applied, p'erpendicular to the rod, in order that the impulse on the fixed extremity may be equal to one-eighth of the blow. , 1481. A uniform beam, moveable about its middle point, is in equilibrium in a horizontal position ; a perfectly elastic particle whose mass is one fourth that of the beam is dropt upon one extremity, and is afterwards grazed by the other extremity; prove that the height from which, the particle falls is to the length of the beam :: 49 (2w + l),r : 48, n being integral. DYNAMICS OF A EIGID BOOT. S09 1482. A uniform rod is revolving about its middle point, ■which is fixed, on a smooth horizontal table, when it strikes a smooth inelastic particle at rest, whose mass is to its own as 1 : 6, and its angular velocity is immediately diminished one ninth j find the point of impact, and prove that when the particle leaves the rod the direction of motion of the ball will mate with the rod ati angle of 45°. 1483. A smooth uniform rod is moving uniformly on a hori- zontal table about one extremity, and impinges upon a particle of mass equal to its own, the distance of the point of impact from the fixed extremity being to the length of the rod :: 1 : m ; prove that the final velocity of the particle will be to its initial ve- locity as ^{{5m'-l){m' + 3)} : im. 1484. A uniform rod is moving on a horizontal table about one extremity, and driving before it a particle of mass equal to its own, which starts from rest close to the fixed extremity; prove that when the particle has described a distance r along the rod, its direction of motion will make with the rod an angle h tan Jir' + k")' k being the radius of gyration of the rod about the axis of re- volution. 1485. A uniform circular disc of mass m is capable of motion about its centre in a vertical plane, 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 m sin 5 + iiup = ix.{m + Qp) cos 6. 1486. A uniform rod, capable of motion in a vertical plane about its middle point, has attached to its extremities by strings two equal particles which hang freely ; when the beam is in equl- 310 BOOK OP MATHEMATICAL PROBLEMS. librium inclined at an angle a to the vertical, one of the strings is cut] prove that the initial tension of the other string is mpg m + 3p sin^ a ' and that the radius of curvature of the initial path of the particle is Ql -- ; m, p being the masses of the rod and of either m cos a particle, and I the length of the string. 1487. A uniform rod, moveable about one extremity, is held in a horizontal 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 let go is mpga (4cb — 3c) ima" + 3pc^ ' m, p being the masses of the rod and particle, 2a the length of the rod, and c the distance of the string from the fixed extremity. Prove that the initial path of the particle is the curve whose equation, referred to horizontal and vertical axes, is ma (4a — 3c) y' + 90c^l (ma +pc) x = 0; I being the length of the string. 1488. A uniform rod, moveable about one extremity, has attached to the other extremity a heavy particle by means of a string, the rod and string are initially in one horizontal straight line and without motion; prove that the radius of curvature of the initial path of the particle is ■ —r , a, h being the lengths of a -r oO the rod and string. 1489. A uniform rod, of length 2a and mass m, capable of motion about one extremity, is held in a horizontal position, and on it slides a small smooth ring of mass p : if the rod be let go, the radius of curvature of the path of the ring is initially 4:a — 3c \ m aj ' c being the initial distance of the ring from the fixed extremity. DYNAMICS OP A EIGID BODY. 311 1490. A tmiform rod, capable of motion about one extre- mity, has attached to it at the other extremity a particle by means of a string, and the system is abandoned freely to the action of gravity, when the rod makes an angle a with the ver- tical ; prove that the radius of curvature of the particle's initial path is _,m + 2/) 'sin^a m cos a (2 — 3 sin" a) ' m, p being the masses of the rod and particle, and I the length of the string. 1491. A uniform rod is moveable about one extremity on a smooth horizontal table, and to the other extremity is fastened a particle by means of a string. Initially, the rod and string are in one straight line, the particle is at rest, and the rod has an angu- lar velocity m ; prove that when the rod and string are next in a straight line, the angular velocity of the rod is to that of the string as b : a, or as b{3p{a-by-ma''] : a {3p (a - b)' + ma {a - 2b)} ; m, p being the masses of the rod and particle, and a, b the lengths of the rod and string. III. Motion in Two Dimensions. 1492. Two equal uniform rods AB, BG, freely jointed at B and moveable about A, start from rest in a horizontal position, BC passing over a smooth peg whose distance from A is 4a Sin a f > -^ 1 ; prove that when BG leaves the peg, the angular velocity of ^5 is / /Sgr cos a \ y \ii 1 + cos'' 2a) ' 2a being the length of either rod. Sl^ BOOiC 0:P MATHEMATICAL PEOBEEMS. / 1493. A lamina, having its centre of gravity fixed, is at rest and is struck by a blow at the point (a, b) perpendicular to its plane; prove that the equation of the instantaneous axis is Aax + Bb7/=0, the axes of co-ordinates being principal axes at the centre of gravity, and A, £ the principal moments of inertia. If (a, h) lie on a certain straight line, there will be no impulse on the fixed point. 1494. A uniform beam revolves uniformly about one ex- tremity in such a manner as to describe uniformly a cone of revolution about a vertical axis ; determine the pressure on the fixed extremity and the relation between the angle, of the cone and the time of revolution. If 6, be the angles which the vertical makes with the rod and with the direction of pressure respectively, then will 4 tan (j) = 3 tan 6. 1495. A fine string, of length 26, is attached to two points in the same horizontal line, at a distance 2a, and carries a particle^ at its middle point ; a uniform rod, of length 2c and mass m, has a ring at each end through which the string passes, and is let fall from a symmetrical position in the same straight line as the two points ; prove that in order that the rod may reach the particle {a + b- 2c) {m' + 2mp) > 2 (2c - a) p'. 1496. A circular disc rolls on a rough cycloidal arc whose axis is vertical and verfex downwards, the length of the arc being such that the curvature at either extremity is equal to that of the circle; prove that, if the contact be initially at one extremity, the point on the auxiliary circle of the cycloid correspondi^g to the point of contact will move -with uniform velocity; and that this velocity will be independent of the radius of the disc. 1497. A sphere rolls from rest down a given length i! of a rough inclined plane, and then traverses a smooth portion of the plane of length ml; iind the impulse which the sphere sustains when perfect rolling again commences, and prove that the subse- quent velocity is less than if the whole plane had been rough. DYNAMICS OF A RIGID BODY. 313 If m=120, the subsequent initial Telocity is less than if tlie whole plane had been rough, in the ratio 67 : 77. 1498. A straight tube AB, of small bore, containing a smooth rod of the same length, is closed at the end B, and is in motion about the fixed end A with angular velocity u; B being opened, prove that the initial tension of the rod at any point P is equal AP . PB to Mil? — ' — ; M. being the mass. ZA.JS 1499. The ends of a uniform heavy rod are fixed by smooth rings to the arc of a circle, which is made to revolve uniformly about a fixed vertical diameter ; find the positions of relative equi- librium, and prove that any such position in which the rod is not horizontal is stable. 1500. A smooth semicircular disc rests with its plane vertical and vertex upwards on a smooth horizontal table, and on it rest two equal uniform rods, each of which passes through two fixed rings in a vertical line. The disc is slightly displaced, and in the ensuing motion one rod leaves the disc when the other is at th^ vertex ; prove that m 2 (2 sin a — 1 — sin /3) — sin ^ cos'' /3 p ~ sin' ^ m, p being the masses of the disc and of either rod, a the angle which the radius to either point of contact initially makes with the horizon, and ^ = cos"' (2 cos a). 1501. A uniform rod moves with one extremity on a smooth horizontal plane, and the other 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 when the string is vertical its angular velocity is ^ <, + 1 -, ,. g a+l-h , ,. and the angular acceleration of the rod is j— -—— , I, a being the lengths of the string and rod, and h the height of the fixed ^oint above the plaoe. 314 BOOK OF MATHEMATICAL PEOBLEMS. 1502. A sphere is resting on a rough horizontar plane, half its weight being supported by an elastic string attached to the highest point, the natural length of the string being the radios a, and the stretched length the diameter of the spherej prove that the time of small oscillations of the sphere paraUel to a vertical plane is ir / —rr- . 1503. A hea-vy uniform rod, resting in stable equilibrium ■witTiin a smooth prolate spheroid whose axis is vertical, is slightly displaced in a vertical plane ; prove that the length of the simple isochronous pendulum is afe + ^-jJ 2a being the length of the rod, and e the eccentricity of the generating ellipse. 1504 A uniform beam rests with one end on a smooth hori- zontal 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 oscUlatiou in a vertical plane is t . / , a being the mean inclination of the rod to the vertical. 1505. Two equal uniform rods AB, BC, freely jointed at B, are placed on a smooth horizontal table, at right angles to each other, and a blow is applied at A at right angles to AB; prove that the initial velocities of A, C are in the ratio 8:1. 1506. Two equal uniform rods AB, BC, freely jointed at B, are laid on a smooth horizontal table so as to include an angle o, and a blow is applied at A at right angles to AB; determine the initial velocity of C. 1507. Five equal uniform rods, freely jointed at their ex- tremities, are laid in one straight line on a horizontal table, and a blow applied at the middle point at right angles to the line; prove that V o) — O V being the initial velocity of the central rod, u, 12 the initial an- DYNAMICS OF A EIGID BODY. 315 gular velocities of the otlier pairs of rods, and 2a the length of each rod, 1508. Four equal uniform rods AB, B€, GB, DE, freely jointed at B, G, 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. 1509. Two equal uniform rods AB, BC, freely jointed at B and moveable about A, are lying on a smooth horizontal table inclined to each other at an angle a; a blow is applied at G at right angles to BG in a direction tending to decrease the angle ABC ; prove that the initial angular velocities of AB, BG are in the ratio cos a : 8 — 3 cos° a ; that 6, the least value of the angle ABG during the motion, is given by the equation 8 (5 - 3 cos &) (2 - cos' a) = (1 - cos a)' (16-9 cos'' a) ; and, when being respectively the angles which the rod makes with the vertical, and which its projection on a horizontal plane makes with the initial position. If the least value of ^ be ^ , the resolved o vertical pressure on the fixed point when = - is to the weight of o the rod as 31": 16. 1513. A centre of force = /i . distance is at a point 0, and from another point A at a, distance a, are projected simultaneously an infinite number of equal particles in a direction at right angles to OJ., with velocities in arithmetical progression from — ~^ to s '} prove that, if after any lapse of time they become sud- denly rigidly connected, the system will revolve with angular velocity ^^ . DYNAMICS OP A EIGID BODY. 317 1514. A uniform rod, moveable about one extremity, moves in such, a manner as to make always nearly the same angle a ■with, the vertical ; prove that the time of its small oscillations is //2a cos a \ V V% 1 + 3 cos' a) ' a being the length of the rod, 1515. A uniform rod is suspended by two strings of equal lengths, attached to its extremities and to two fixed points in the same horizontal line whose distance is equal to the length of the rod ; an angular velocity is communicated to the rod, about a vertical axis through its centre, such that it just rises to the hori- zontal plane in 'which are the fixed points; find the impulsive couple, and prove that the tension of either string is instanta- neously increased in the ratio 7 : 1. 1516. If two equal uniform rods AB, BG, freely jointed at B, rotate uniformly about a vertical axis through A, which is fixed, with angular velocity u, the equations to determine the angles a, p, which the rods make with the vertical are (8 sin a + 3 sin yS) cos a _ (3 sin a + 2 sin /8) cos fi _ Zg 3 sin a sin /3 ~ 2au' ' 2a being the length of either rod. 1517. A perfectly rough horizontal plane is made to rotate with constant angular velocity about a vertical axis which meets the plane in 0; a sphere is projected on it at a point P, so that its centre is initially in the same state of motion as if the sphere had been placed freely on the plane at a point Q ; prove that the centre will describe uniformly a circle of radius OQ, and whose centre B is such that OR is parallel and equal to QP. 1518. A rough plane, inclined at an angle a to the horizon, is made to revolve with uniform angular velocity to about a nor- mal, and a sphere without motion is placed upon it; prove that the path of the centre will be a prolate, a common, or a curtate 318 BOOK OF MATHEMATICAL PROBLEMS. cycloid, according as the initial point of contact is without, upon or within the circle 2(0^, {x' + y") = Z5gx sin a, the axis of y being horizontal, and the axis of x the line of greatest slope. If the initial point of contact be the centre of this circle, the path will be a horizontal straight line. 1519. A rough hollow circular cylinder, whose axis is ver- tical, is made to rotate with uniform angular velocity with their initial positions, {k' + ab cos {.!> ~ 6)} j^ +{b' + ab cos {- 6)}^^= (a + b)v, where Ji?=a' -^ , and ■;; is the initial velocity of the particle. 1534. A circular disc capable of motion about a vertical axis through its centre, normal to its plane, is set in motion with angular velocity O, and at a given point is placed freely a rough uniform sphere : prove the equations of motion 'w-'-{dt)-'r''dt='^' r, 6 being the polar co-ordinates of the point of contact measured from the centre of the disc, w the angular velocity of the disc, 7pc' b the initial value of r, and k" = -j— , where m, p are the masses of the sphere and disc, and c the radius of the disc. 1535. 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 disc : prove that DYNAMICS OP A EIGID BODY, 323 ■where 6, are the angles through which the string and the disc have turned at a time t, a is the radius, and k'= ^+ — ,\m, p A p being the masses of the disc and particle. 1536. Two equal circular discs lying flat on a smooth hori- zontal table, are connected" by a fine string coiled round each which is wound up till the discs are in contact with each other, and are on the same side of the tangent string. One of the discs has its centre fixed and can move fi:eely about it, the other disc is projected with a velocity u at right angles to the tangent string; prove that the angle through which either disc will have turned 1 + ■=— ^ — 1, and that the angle through which the string will have turned is -^tan"' — r^ j a being the radius of either disc. 1537. A smooth tube, mass m, lying on a horizontal table, contains a particle, mass^, which just fits itj the system is set in motion by a blow at right angles to the tube : prove that (^^(<^^^)=i''-''W+^h r being the distance of the particle from the centre of the tube, when the tube has turned through an angle 6, o the initial value of r, J. = -=- — — , where m, p, are the masses of the tube and particle, and 2a the length of the tube. 1538. A circular disc of mass m and diameter d, can move on a smooth horizontal plane about a fixed point A in its circumfe- rence, and a fine string is wound round it carrying a particle of mass p, which is initially projected from the disc, at the other end of the diameter through A, with a velocity u normal to the disc, the disc being then at rest. Prove that the angular velocity of 21—2 324 BOOK OF MATHEMATICAL PROBLEMS. the string will vanish -when the length of the string unwound is that which initially subtended at A an angle 6, such that ip{etxa.6 +\)cos'9 + Zm = Q; and that the angular velocity of the disc is then 1539. A rough sphere, radius a, moves on the concave surface of a vertical circular cylinder, radius a + h, and the centre of the sphere initially moves horizontally with a velocity v : prove that the depth of the centre below its initial position aifter a time t is (l-ooswi!)j (n'=^^. Prove that, in order that perfect rolling ma,y be maintained, the coefficient of friction must be not less than t.- „ - . 1540. A right circular cylinder is fixed with its axis hori- zoiltal and a rough sphere is projected so as to move in contact with the cylinder, Taeing initially at the lowest point and its centre moving in a direction making an angle a with the axis of the cylinder : prove that, in order that the sphere may reach the highest point, its initial velocity must be not less than ./ \-fj -^4- ); a,a + b being the radii of the sphere and cylinder respectively. Prove the equations of motion fdtjA' M^sinV lOff,, — < =^ — j5 yj-(l -cos ft>), m- dz di = McOsaCos(<^ /~j, = «,y|cosasin(,^yj): 5YNAMICS OF. A EIGID BODY. 325 s being the distance- described by the centre parallel to. the axis, <^ the angle through which the common normal has turned, and o> the angular velocity of the sphere about that normal, at the end of a time t. 1541. A sphere, radius qi, is in njotion on the surface of a right circular cylinder, 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 axis with such a velocity that the sphere just makes complete revolutions : prove the equations of motion dm dz dcji "'It ^didi' dzy 2 , , 10 (1) + ^ a u> =-^gis cos a; s being the distance described by the centre parallel to the axis, (^ the angle through which the common normal has turned, and with the vertical ; prove that X 3 /I i 2 I 2 cos' a - 1 tan" 6 + tan" pc'. 1570. An elliptic tube half full of heavy incompressible fluid revolves about a fixed vertical axis in its plane with angular velo- city (0 : prove that the angle which the straight line joining the free surfiices of the fluid mates with the vertical will be p being the distance of the axis from the centre of the ellipse. tan" 332 BOOK OF MATHEMATICAL PROBLEMS. 1571. A hollow cone very nearly filled witli fluid rotates Tiniformly about a generating line which is. vertical : prove that the pressure on the base is 3 IF" tan a, (a<>>' ,, _ , > „> . Y ■ I — (1 + 5 cos a) + 8 sin a > ; W being the weight of the fluid, 2a (<^ the vertical angle, a the radius of the base, and J HTDEOSTATICS. 333 1575. A hollow paraboloid, Laving a base perpendicular to the axis and at a distance from the vertex equal to the latus rectum, is placed with its axis vertical and vertex upwards, and contains seven-eighths of its volume of heavy uniform fluid. Find the angular velocity with which it must revolve about the axis in order that the surface of the fluid may be confocal with the paraboloid; and prove that in this case the pressure on the base will be greater than it was when the fluid was at rest in the ratio 2J2 : 2^2-1. 1576. If a uniform fluid be acted on by two central forces, each varying as the distance from a fixed point and equal at equal distances from those points, one attractive and the 'other repulsive, the surfaces of equal pressure will be planes. 1577. In a uniform fluid under the action of two forces tending to fixed points , and varying inversely as the square of the distance, one attractive and one repulsive, prove that one surface of equal pressure is a sphere. 1578. A mass of elastic fluid is confined within a hollow sphere and repelled from the centre of the sphere by a force - ; prove that the whole pressure on the sphere : the whole pressure which would be exerted if no force acted : : 3k + fi. : 3k, the pres- sure being k . density. 1579. A quantity of uniform incompressible fluid, not acted on by gravity, just fills a hollow sphere, and is repelled from a point on the surface of the sphere by a force equal to /^ . distance; the fluid revolves about the diameter through the centre of force with uuiform angular velocity w; find the whole pressure on the sphere, and prove that, if when the angular velocity is diminished one half the whole pressure is also diminished one half, w' = 6/t. 1580. All space being supposed filled with an elastic fluid, the whole mass of which is known, which is attracted to a given 334 BOOK OP MATHEMATICAL PEOBLEMS. point by a force varying as the distance; find the pressure at any point. 1581. Water is contained in a vessel having a horizontal base, and a right cone is supported, partly by the water and partly by the base on which the vertex rests : prove that, for stable equi- librium, the depth of the fluid must be greater than h ^{m cos^a), m being the number measuring the specific gravity of the cone, h the length of the axis, and 2a the vertical angle. 1582. A solid paraboloid is divided into two parts by a plane through the axis, and the parts are united by a hinge at the vertex; the system is placed in a heavy uniform fluid with its axis vertical and vertex downwards, and floats without separation of the parts : prove that the ratio of the density of the solid to that of the fluid must be greater than as*, x being given by the equation where h, ia are the lengths of the axis and the latus rectum re- spectively. 1583. A right cone is floating with its axis vertical and vertex downwards in a fluid whose density varies as the depth : prove that, for stable equilibrium, cos a' cos l/f' ' where ' are tl^e angles of incidence and refraction at the first surface, and \j/, ij/ are the angles of incidence and refraction at emergence, 1608. "When a ray of white light is refracted through a prism in a principal plane, so that the dispersion of two given colours is a minimum, ' sin(3£-2i)_ 2. sm <;!> /«. ^' being the angle of refraction at the first surface and t the refracting angle. GEOMETRICAL OPTICS. 341 1609. rind the focal length of a lens equivalent to a sys- tem of three convex lenses on a common axis, of focal lengths 36 inches, 4 in., and 9 in. respectively, placed at intervals of 24 in. and 13 in., fpr a pencil proceeding from a point 18 in. in front of the first lens. 1610. Two thin lenses of focal lengths f^, f^ are on a com- mon axis and separated by an interval a; the axis of an eccentric pencU. before incidence cuts the axis of the lenses at a distance d from the first lens ; prove that 11 1 a/1 1\ F being the focal length of the equivalent single lens, 1611. The focal length ^ of a single lens, equivalent to a system of three lenses of focal lengths f^,f^, f^ separated by intervals a, b, for an eccentrical pencil parallel to the axis, is given by the equation . -^ /i A fz f\fi fj /a Vi fj /i/s/s" 1612. Prove that the magnifying power of a combination of three convex lenses of focal lengths f^,/^,/^ on a common axis at intervals a, b, will be independent of the 'position of the object, if ( 342 ) SPHERICAL TEIGONOMETRY AND PLANE ASTEONOMY. 1613. In a spterical triangle ABC, a=h = -=, c=s; prove that the spherical excess is cos ' - . y 1614. In an equilateral spherical triangle ABC, A', B, G' are the middle points of the sides : prove that „ . B-C , BG 2 sm = tan -^ . 1615. In an equilateral spherical triangle, whose sides are each a and angles A, 2 cos = sin 5- = 1. 1616. ABG is a spherical triangle, each of whose sides ia a quadi'ant, P any point within the triangle : prove that cos' AP + cos' BP + cos' GP=\; cos AP cos BP cos GP + cot BPG cot CPA cot APB = ; and that tan BGP tan GAP tan ABP = 1. 1617. A point P is taken within a spherical triangle ABO whose sides axe all quadrants, and another triangle is described whose sides are equal to 2AP, 2BP, 2CP respectively : prove that the area of the latter triangle is twice that of the former. 1618. A spherical triangle ABG is equal and similar to its polar triangle : prove that sec^A + sec'5 + sec'C + 2 sec ^ sec B aecC= 1. 1619. If the sum of the sides of a spherical triangle be given, its area is greatest when the triangle is equilateral. SPHERICAL TRIGONOMETRY AND PLANE ASTRONOMT. 343 1620. I n a spLerical triangle ABC, a + b + o = '7r: prove that cos a = tan -^ tan -^ , &o. j ■ A B G . sin jr = cos-j; COS -jf sm a,