Cornell University Library The original of tiiis 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/cu31924058531959 CORNELL UNIVERSITY LIBRARY 924 058 531 959 DATE DUE j 1 j CAYLORO PRINTED INU.S. A. Production Note Cornell University Library produced this volume to replace the irreparably deteriorated original. It was scanned using Xerox software and equipment at 600 dots per inch resolution and compressed prior to storage using CCITT Group 4 compression. The digital data were used to create Cornell's replacement volume on paper that meets the ANSI Standard Z39. 48-1984. The production of this volume was supported in part by the Commission on Preservation and Access and the Xerox Corporation. 1990. KEY TO TODHUNTER'S DIFFERENTIAL CALCULUS. ^ KEY TO TODHUNTEK'S DIFFEKENTIAL CALCULUS uv H. ST. J. HUNTER, M.A. FELLOW AND LATELT MATHEUATICAL LECTURER OF JESUS COLLEGE, CAHBBIDQE. "Eonlion : MACMILLAN AND CO. AND NEW YORK. 1888 [The Right of Translation i» reserved.] M. CTHmfariBBe : MlNTEl) By C. J. CLAY, M.A. AND SONS, AT THE UNIVERSITY PRESS, PKEFACE. The solutions in the Key will, I think, be found suffi- ciently full, and no pains have been spared to ensur(.' their accuracy ; they are also so worded that figures can be easily supplied by the student to enable him more readily to follow the reasoning, when geometrical. Figures are however given to the curves in Chaps. XXVI. and XXVIII. To make the book more useful and complete, I have, in some instances, adopted improved methods — chiefly in the Solutions to some of the examples in Chaps. XL, XIII., XV., XX. and XXII.; and I am bound to acknowledge some obligations to the last chapter of Mr Turnbull's Analytical Plane Geometry. H. ST. J. H. The references, unless otherwise specified, are to the tenth edition of the Differential Calculus. KEY TO DIFFERENTIAL CALCULUS. CHAPTEB IV. 1. See Arts. 29, 33, and 47. Or, from definition -^ = It. of ax r{,-Jx + h- ijx) = c-i-{jx + h+ Jx) = c-¥'isjx. 2. y=?-l, .•.byArt.47,^ = -^,. X "^ dx X' 3. ByArts.31and44,^ = -,-L_-(l + ^)-7r^ = ^7^|' 4. By Arts. 29 and 60, log meaning to the base e, -p=logxH — 6. By Arts. 50, 54 and 68, $-' = tan a; . ( - coseo^x) = - -r\- . dx * sin 2x C. Arts. 63, 47, and 31 lead to ^ = — ,- + -^-- = —^ . ' ''^ (o2-x-)2 (a2-x=)t {a'-x"-)i 7 So ''y ^ /^_(JKf^ 3x' <. DO -J- = 1 + ^^^^^— — ^— ■- — • dx (x-a2)if (l-x^p (l-x=)T 8. From Arts. 29 and 49, ^ = c* (1 - x^) + c"^ ( - Sx^) , dc. 9. So ^ = l.e=^ + 2e=^(x-3) + 4c^ + 4xc- + l, dx = <;=' (2x - 5) +4e^ (x+ 1) + 1. 10. Here ^ = 2 ..e=^ + 2e"(2x-i:) + 4e^ + 4 (.r + 1) e', d-c. T. D. C. K. 1 DIFFERENTIAL CALCULUS. IV. 11. \ogy=nx . log - , .•. (Art. 72) - . -p! = n. log- + 7u; . - . - n y dx n X n = nlog5 + „, ... g=„ (log^+l) • y. &<=. 12. 13. dy n . x"~' dx (l + xY (l + x)»+i (l+j;)»+i* dx " e' + e-* (e* + «-==)» ~ (e' +e-^)^' 15. ^ = ?2^ + _??L _ *y dx X a + x h-x =x(a + x)''(h-xf(2a+xh-x + Zxl>-x-ixa+x), &c. 16. ^=m(a+x)'»-i(6+x)''+(a+x)".n(6 + x)»-i, <6c. ^_ dy_ m n _ dx~ (a + x)'»+i.(6 + x)»~(o+x)'».{6 + x)''+i~ 18. -r = tan^x . Bec'x - sec^'x + 1 = tan'x sec'x - tan^x = tair^. ax dx (x+Jl-xY I 2 ^i-isj v^l-x»(x + Vl-xT 20. Art. 69 leads to /^ = 2x. tan-i'? + (a2+a=) . — L. . -, ... &c. dx o * ' , x^ o a' „, _1 , dy _ 1 , ^~x Vax^ + tx + c, .-. ^- -^ ^ax^ + bx + c 1 1 2gx +6 _ -2(air' + 5x+c) + x(2aa!+6) _ +fe»;+2c ^ " 2 ' Jax'' + bx+c ~ 2x' Jax' + bx+c ~ -ix' Ja3?+ bx+e ' 22, ^ = ;; — ; ;— -r . p-— . n6x"~', bv an extension of Art. 63. dx log(a + i)x'') (a + ftx") '' ='-2--G-2)-'(M)-2=-777T— ^ ' ' ' Bm(-+i) DIFP'ERENTIAL CALCULUS. IV. 25. y(Jx+^a)=Jx + a, .: ^(Jx+Ja)= -y .-—=+ 2 s/x i >Jx+a dx' Jx+a 1 Ja(Jx-Ja) , &c. i iJx(Jx+Ja) 2^Jx + a 2 Jx{Jx + ^a)Jx+a 26. 21ogy=log(;-±--^,..._^^^ = ,-^+J-. \l—xj y ax 1 + x 1-x dy 1 /r+^ ^ 27. Logy = llog(l-i«)-|log(l+a^); l . "■^dx" 6 + cx- ~(6+cx=)^' ■'• "• dy -xBm~'x , , , J- = — , +1-1, Ac. dx ^^l_x2 A« VVl-x* (1-x^p 1-*^ 2Vl-xV^ :&C. DIFFEHENTIAL CALCULUS. IV. ' C7. ULay=x+ jr^, .-. Bee'y.^ = l -^ , .Q ztana dy 68. Biay = ~p==, .: cos«^ Ja'-x^ " dx , ( 1 ar* ) a=tana ((a»-x2)4 (a*-x^p (aS-x'ji _di/ .^g''' - x' flee''' a _ — 'T~ . ; I •'• IXC. "^ (o2-x2)4 dy 1 1 /6^^2 ( -2a 2a (a' -a') ) -(68_x'')a(6!'-a.°) 70. y =tan-i f tan | j = | , .-. Ac. a cos a + -; — — 7 — 9 T Q 1 . _ 6 + gco8a _ b b g'-i' 1 «. "°^~a+6cosa~ a + 6cosx ~ ~ 6 "a + icosa i' dy_ 0^-62 6 sin a _di/ ^/a' - 6' ■ sin a ..ooflj/^— jj ■ (a + 6 cos x)2 "" dx a + b coax ' 69. ,-. Ti ■ *"=• DIFFEEENTIAL CALCULUS. IV. 74. cosy = 2x2-l, . . -siny . -^~=ix= --^ .2xjl-x', &o. dx dx' dx x' + ^ + xI'-^JY^^^'^Jl + i^ !>?■ \~ Vl + x"' VI .. ay_ 2i+ ^2' _ 2a:- V2" Ja--f d^ " „ d"->i d^i „ . hence (Art. 80), differentiating n times, 10. If «=(a-2)e^+x + 2, then u=0 when x = 0, and ^ = 6'^ + (x-2)«^ + l = (x-l)«"^ + l, .-. =0\vhen x = 0, and j-.^-e' + (x-\) e^^xe', .: j-^ is always positive when I is so, . . ^- must increase with x (Art. 89) when x is positive, and dx " = Owhenx = 0, .•. 3- is always positive when x is bo, .-. u increases dx ax with x; but u = when x = 0, .". u must he positive for all positive values of X. DtFFERENTIAL CALCULUS. VII. lo CHAPTER Vn. 1. If /(x) = e«.(3-x)-4a;.e=-x-3, /(0) = 0; f'{x) = e^(G-2x-l)-ie'{l + x)-l, .: /'(0) = 0; /"(a:)=e"(10-4a:-2)-4e'(2 + x), .-. /"(0)=0; /'" (x) = «» (16 - 8x - 4) - 4c' (3 + x), .-. /"' (0) = ; /"(x) = e»(24-16x-8)-4c'^(4 + x), .-. /"(0)=0; /'(x)=e»(32-32x-16)-4c=(5 + x), .-. /'(0)=-4; - 4x* .•. first term is — -.— , \1 2. If /(x)=log(l + c'),/(0) = log2, /"(0)=J^;and/"(x)=^-^-^.j,. {1+e')'^ [l + e'f (1 + e^f (l+e^'f 1 + e" &e^ 6f' f^ ••• /"'(0) = 0; and /"(x) = ^^-;-^, - ^.^— ^ + .•. /''(0)= -^. .'. &c. by Maclaurin's Theorem. 3. If /(x) = «*""', /(0) = 1, and f{x) = f{-x), .: there are no odd powers of X, .-. /'(0)=0=/"'{0) = iS:o., and /' (x) =/ (x) (sin a; + x cos x) ; .-. /"(x)=/'(x)(sinx + xcosx)+/(x) (2co8x-X8inx) (1); .-. /" (0)=2, and differentiating (1) twice by Art. 80, /" (j) = 3/" (x) (2 COB X - X sin x) -/ (x) -;- (3 sin x + x cos x) + terms in /' (x) and/"'(x), ,•. /"(0) = 12-(4cosx-xBinx) when x=0, or /"(0) = 8, , 2x2 8x< ... e"i"x = l + -- + — + ... 4. If/(x) = e'.secx, /(0) = 1, /' (x) = c^ (sec X + sec X tan x) =f (x) (1 + tan x) , .-. /' (0) = 1, /" (x) =/' (x) (1 + tan x) +f (x) . sec' x ; ,-. / ' (0) = 2 ; /'" (i) =/" (x) (1 + tan i) +/' (x) . 2 sec" x +/ (x) . 2 Bec= x tan x, .•./"'(0) = 4, .-. &c. 16 MFFERENTIAL CALCULUS. VII. 5. If /(x) = (^^)"./(0) = l;/'(.)=|.«.(l±^)"-\ .■./(0, = ^,andr(x)=^.e^(l±£!)'-V!L(!L=i)i?(^£!y^^ 6. If /(x)=+^/(l+4x + 12x=), /(0) = 1, and /'(x){l+4a; + 12x2)=/(x)(2 + 12x); .-. /"(x) (l + 4x + 12x=)+/'(x)(4 + 24x)=/'(x) (2 + 12x)+/(x) . 12; .•./"(0)= -8+4 + 12=8, &o. 7. Let f(x, 7i) = (e' + e-^'', then /(x, 71)=/ (-x, n), .*. there are uo odd powers of x, and /(O, 7i) = 2", /' (x, n) = n(e^- e-') (e' + e-==)"-i, /" (x, n) = n ((^ + c-»^)» + 71 (ji - 1) {c» - e-"^)!" (e= + e-^)"- ■- = n2 ./(x, 7i) - 471 (n - l)/(x, 71 - 2) ; 2" . . /" (0, 7!) =7i« . 2» - 4n (71 - 1) . -J =71 . 2», and /" (x, 7!) = n'f" (x, 71) - 47i (n - 1) /" (x, 7i - 2) .-. /"(0, 7j)=n3.2''-47i(7i-l)(7i-2)^=2».7t(3n-2), &o. 8. If/(x,7!) = cos'>x,/(x,7i)=/(-x,7i), .•. there are DO odd powers of xj / (0, 71) = 1, /' (x, 71) = - 71 C03"-^ X . sin X, /"(x,7i)=7i(n-l)coa"-2x. sin^ x - 71 cos" x =71(71-1)/ (x, n-2)-n'.f(x, n), .: f" (0, n)=-n. and /'• (i, 7i)=7i (71 - 1)/" (x, n - 2)-n^f" (x, n) ; . . /" (0, 7l)= -71 (7t- 1) (78- 2) +n'=7l(37l-2), and /" (x, 7i) = 7!{ji-l)/''(x, n-2)-7jy"(x, n), which gives /"(O, n)=-7i{15(7t-l)2 + l), .-. &o. 9. If / (x) = - log (cos x), f (x) =/(- x), .-. there are no odd powers of x, and/(0) = 0, /'(x) = tanx, /"(x) = seo2x, /" (0) = 1, /'" {x) = 2 sec^ X tan x, DIFFERENTIAL CALCULUS. VIL 17 or if"'{x)=f"{x).f'{x) (1); ••• if'(')=f"'(x).f'{x) + {f"(x)]\ .•./"(0)=2,andbyArt. 80 from (1), ir («) =r («) ./' W + 3/" (x) . /" {x) + 3 {/'" (X))' +/" (x) ./" {X) ; .-./-(0)=16, and J/"" (x) =/'" («) ./' {X) + 5/" (x)/" (x) + lOT (x)/"' (a;) + 10 f/" (x)}» + 6/'" {x)f' (X) +/" (x) . /" (X) ; .-. f^ (0)=2 (5 . 16 + 10 . 4 + 16) = 16 . 17, &c. 10. If /(x)=e'^°",/(0) = e,/(x)=/(-x), -/(x)=+/(x).sinx (1), -/" (x) = +/' (x) Bin X +/ (x) cos x, .-. /" (0) = - e, and from (1) by Art. 80 -/" (i) =/'" (x) Bin X + 3/" (x) . COB X - 3/ (x) sin x -/ (x) cos x ; .•./"(0)=4e, and -/"(x) =7^ (x) . sin X + 5/" (x) cos x - 10/'" (x) sin x - 10/" (x) cos x + 5/ (i) sin X +/ (x) cos x ; .. /"(0)=-31e, .-. &o. 11. By Taylor's Theorem if / (x) = sin-' x, Bin-i(x + ft) = Bin-ix + 7i./'(x) + ^/"(x)+... and here /'(x)=— i= , /"(x)= — - — Vl-x2 (l-x2)f {l-x^)i (l-x2)J (l-x2)^ _ , , ix 5x (1 + 2x2) 3i (3 + 2x2) and /"(x) = -.+ — 5 r-= — ^ r • ■ • *"'• (l-x»)-i (l-x2)J (l-x=)i 12. Let / (x) = log (1 - X + x2) , then / (0) = 0, and /'(x)(x2-x + l) = 2x-l, .-. /'(0)=-l, and /" (x) (x» - X + 1) +/' (x) (2x - 1) = 2, .-. /" (0) = 1, /"'(x)(x=-x + l) + 2/"(x){2x-l) + 2/(x)=0, .. /"'(0) = 4, /" (x) (x' - X + 1) + 3/'" (i) (2x - 1) + 6/" (x) = 0, .-. /" (0) = 6, /' (x) (x» - X + 1) + 4/" (x) (2x - 1) + 12/"' (x) = 0, .-. /' (0) = - 24. T. D. C. K. 2 18 DIFFERENTIAL CALCULUS. VII. 13. Let /(i)=log { Va2 + x» + x}, then /(i) +/(-«)= 2 logo, .•. there are no even powers of x in the expansion, and fm=\oga, f (x)=(l + -j^=J\^(JW:^^+x) = -f4=^, \ tJa'+x'J i^a'+x' .■.f{0)=\, a and /"W= ""' , . .-. (a'+x2)./"(x) + x./'(i)=0, (0' + x2)5 .-. (a» + x=)/"'{x)+/"(x).3x+/'(x) = 0, .-./'" (0)=-l; and (a^+m?) /" (x) + 5x. /'" (x) +4/" (x) = 0, .-. ^a2+x2)/'{x) + 7x./"(x) + 9/"(x)=0; ..r(0)=J. .-.Ac. 14. If /(x)=log(l + sinx),/(0) = 0, /'(x) = j^^^,/'(0) = l, •' ^ ' l + Binx (l + Binx)2-l + sinx' ••■' ^"'- ^• and .r'W = (j^;^„../"'(0) = l,..*e. 15. If /(x)=etan-.x /(0) = 1, /'(x) (l+x»)=/(x), .-. /'(0) = 1. and /"{x)(l+x') + 2x./(x)=/'(x), ../"(0)-l, and /'" (x) (1 + x=) + ixf" (x) + 2/ (x) =/" (x), .-. /'" (0) = - 1, and /"(x)(l + x») + 6x/"'(x) + 6/"(x)=/"'(x), .-. /"{0)=-7, .-. = " dj, "* di ' , ^^ ,1 ^ , <*'" „ , I du du and -^-(,by-ax)=au-b, .. ^by-ax)=-b j-^ + a j^. (6) g=log(l + xj,) + j|£-=log(l+xy) + l-j^^; d?u y y , du -' <2x(22/ 1 + x^ (1 + xi/)' ' dx 1 + xj/ ' u 2 '^" ^'(x=+y2)^ ^'(x' + sr*)*' d°tt _ 8 xy d^ d=u 2 X 8 X 2x dx» ""dxdj/ 3 (x' + y')^ " (xHr)* 9 (x= + «/=)* 1 du , , , du . -b-d-x=("-')di'*'=- 7. f!^= 2e' . yz" + 2x23^2" ; . •. -^ = 6e'yz" + 4xh,z ; ay <^yoz 22 DIFFERENTIAL CALCULUS. VIII.. du d^u 8. - =^ye^', ... _=^e-»'+x» . y.e^', and ■ . , = e'»' (l + xyz + 2xyz + x^y'z"), or I dxdy-i + x'\^l_^^,_^y,^i (l+^2 + j,s)ti Also ^^<^y' (X + x^ + y^)^' " ^^^y" (1+x^+y'^^' 11. Here Ja'-x-' .^=Ja^-x^ ^a'-y^Jofl^^-xy ^a'-z* -zxija^-y'^-yzjc^-n? = V"" -y^ •-j- = 's/"" - ^^ ■ J- l"? Bymmetry. Also, .-. Jd'-y'' . Jd'-z^ . ^^= -yja'-x- . j:^^^ -X Ja'-y-'. Ja^-z^ + xyz-zJa'-x^.JaF^, and .-. 'Ja?-x'^'Ja^-y''Ja'-z'' dxdydz =xy>/d^-z'-»/d'-!,?.>J^^^fa^-zi+yz>J^:^ DIFFEEENTLIL CALCULUS. Vlli. 23 12. (1) «"=x»+y»+z'-3«/z, ana^=8(a!2-i/2)«-", and T-3Z=-^' • e— - 9 [x' - yz) (y^-zx) e'"; + 54 (x^ - yz) (y2 - zx) (z« - xj/) e"'" „ff« dii du „ ox ay az ,„, da du du „ , „ „ , , 3 (2) ~+ — + — =Se-'(3?-yz + y^-zx + z--X!/)=— — -. ' dx dy dz \ s ;> ;>/ a;+y + s (3) Differentiating the result of (2) with respect to x, y, and z and adding, +y3) = {x» + ^» + zT +(!/■'+ ^* +'«!/)''- 2»M3/'2 + ■se + a^) - 2y' (yz + zx + xy ) - 2z' (1/2 + zx + xy) = {a?+y' + z'-yz-zx-xy)'; ^ ^ d°M_ 3 ■■ dx2'^dp'''dz2~ (x+y + z)"' d'u d'u /d^u d'u d'u'N (6) d'u /d°u d'u d^Y ■ ••• ~didydz \dx^ '^ dy' "^ dz^J ' dx^dydz 3.2.3.4 72 •"• ^(x+y + z)»~(x + y + z)» 24 DIFFERENTIAL CALCULUS. IX. CHAPTEE IX. 6. y=a + x\ogy, .•. in Lagrange's Theorem 2=0, and (3/) = log ?/, / {y) = Bin (y) ; 3.2 (J . . sin j/=sina + a;.eo3a.loga+-j^ -j- {cos a (log a)'} + igit ^n — 1 7. «(v)=2;''.ew, /(y)=j/'».e"»; .-. J/™, e"" = z" . e" + a . zP . c'' . — fz™ .6"') + |r ■ di' + T-:-^^jri )«''•«'"•«'"•«"(— +«)[+ ••■ *<:• 8- (1) /(y)=siny = 0(y), .•. Biny=Binz+x.Binz.008z + T7J 3-(am^z . cos;)+ +— .;=——, (sin" z COB z) + . (2) /(!/) = sin 2y, ^(!/)=siny, .•. Ein22/ = Bin2z + x. Bin2.2cos2z+ +— 3-^;::^ (sin" z. 2 cos 22) + . 9. Here in Laplace's Theorem f{y) = e'', '. of-^l:;=l. X 2. =lt. l-^nx»-'=-, whenx=l. n 3. =lt. of {e' + c-"=)-7-coaa:=2. 4. =lt»(«» + c-*-2)-j-(l-coBx)=lt. (c'^-e-*)-=-Binx =lt. (6*+e-^)-T-cosx=2. 5. =lt.ri--ri^W3Bin'xcoBx=|l-U + -x»j|-T-3x', neglecting higher powers of x, = - g • 6. =lt. of.a*loga-6^. log 6=logo-log6 when x=0. 7. =lt. of (sec'x-l)-H(l-co8x)=lt. 2 sec'x tanx-r-sina;=2Bec'x=2. 8. Expand sinx; orthus: lt.=lt. of (l-coax)H-3x'=lt.8in:t-r6x = lt.0OBX^6=T;. o 3 9. =lt. of 3co33z-=-(l-3coB.2x)= -g. 10. =lt.of(-l+iU-^^=-^^^5^=_l. 0. =lt. of(-l+i)-^ j2x-7? X. 11. Lt.of ,^^^=lt.-l-=-i=-l. logx X K. .Lt. = («»-e-«+2sinx)4-(sinx+xcosx)=lt. pf (^+""'+2008 1) 2coax-xBinx 4- DIFFERENTIAL CALCULUS. X. 27 ,„ ,. 2co8 2a;+2Bin2x-2oo8a; -48in2x+4co32x + 2sinj; . IB. ljt. = : = . = ^ ;r =4. Bin 2x - sin x 2 coa 2x - cos x 14. Lt. of ( xsinx-^ )-i-cosx=It. (Binx + xcosx)-j-(- Binx)= -1. 15. Lt.={e»(x-l) + l}-r3e='(e^-l)==lt. of {e'^(x-l) + l)-f-3{c*- 1)= = It. of xe'-r- Oe» (e"^ - 1) = f -^ 2 (e» - 1) = 5 4- 26"^ = 5 , O o D or expand e'. 16. Algebraically, expunging the common factor x^-S, the quotient reduces to that of 2 quadratic expressions, and the results nK obtained by maldng x=3 and -3 succesBively. Diflerentiatmg, It. (when x=3)=lt. of ^_^,^,_■^^^^gy = (162 - 42) -=- (135 - 123) = 10, and when x= -3, this fraction=(- 135 + 123)-f-(- 102 + 42)=—. __3f 56)81 ~ 2 I 2 5J 20" 18. =lt. of I? Ji+l(x-l)i'^Mix(x'-l)l-\]=-\. 19. =lt. of {|xU|(x,l)ij ^-j§= =iifB(.U^|J) = 0. 20. Expanding sin X, &c., the It. is = (m* - m) —~ (wi2 - 1) -;7 = 3 , or, differentiate thrice. 21. =it. of — ^=lt. -^ = ^. TOsmmx m-'cosntc nr 22. The fraction = cos o . sin x-r- (- sin a . sin x) = - cot a. : or diflerentiate. . nsec'nx-nseo'x cosx+cobtix 23. =lt. of = — 5 s — =2. 71 COS X - n COS BX cos" X . COS'' TUB ?8 DIFFERENTIAL CALCULUS. X. 24. Thebaction = {^a+x+Ja-x)-i--T=(,Ja.'+ax+x^+Jax)_ ijx + a, I ,- ^2 1 2^i 2Va V2a 2 + cos2a!-8ina!_^ - 2 sin 2a - coa ae _ a; 8in2x+a;cosa~ ' sin 2a; + cos x+ a! (2 cos 21 -sin a;)"" ' w — 2x 1 1 and It. of TT^ — s-=lt. of -2-7-4 cos 2x = ~, .•. It. reqd.= --. 2 sin 2z 2 4 27 If 2* be very large, sin (^j ^^ = <^-^2»~"' or, differentiating. It. = that of cos ^ . a . 2~* log 5 -r- ( 2-"^ . log 5 1 = a. 1 - a'-l 28. Ifx=-, (a'-l).x = . .-. It. when J/ = 0, is 2^ y a''loga-f-l=log a. 29. f^+ lY=e''°'(i*^) , and It. when x=oo of logf^+lUl=lt.of-^,.J--('-i) = ^=«. X X .-. It. of (| + iy=e«. 30. (1) = It. of {m" log (m) . sin (nx) - n= log (n) sin (mx) -m . m'" cos nx - m . n' cos J7ix}-f- (n sec' Jix - m sec' mi) = 1. (2) X is arbitrary, and differentiating with respect to m, the It. = (x .m^~i .sinnx-x .n'cos»nx)-=-(-xsec'?;ix), or when m=n, It. = n*-^ (n cos nx - sin tix) cos' nx. 31. (l+i V=e"°'0+^), and It. of log (l + Jii) - J DIFFEBENTIAL CAICDLTJS. X. 29 32. (^)L^'«'^\andU.of?2i*5^-l2i5 ,. , a; 2a;-sm2a! ,. , 2-2coa2a; = U. 01 , = ■ „ =lt. of -7- 1 xBin2x sin2x+2zco82x 11 » 4 sin 2a; . ,. , „ , =lt. of -: 5 . . . = 0, . . It. reqa. = e'>=l. 4 COB 2x - 4 Bin 2z . a! 3- •(-^)''=e"' ' ,andlt. of (logtani-logx)-^!^ =lt. of (cotisec'x-- |-f-2a= „ „ . „ \ xj 2x'sm2x ,. , 2-2coa2x ,^ , 4Bin2x =lt. of - — ;— -T — ^ — = It. of =lt. of 4xBiu2x+4x'cos2z ' 4Bm2x + 16xcos2x-8x'Bm2x 8cob2x _1 24 COS 2x+ vanishing terms ~ 3 ' .-. It. reqd. = 6*. (i \ 1 1 , tang tr£f j5=eS"«-i-, and It. of (logtani-logx)-:-x5 ,.,/'. - 1\ „ , 2x-Bin2x =lt. of ( cotxsec=x-- )-f-3ir'= „■■-. . \ x/ 3x'sm2x and Bin2a;=2x- -^, .•. It. reqd.=e° approximately 35. (cosmx)»=«»'°«<«"'"l+', and It. of nlog(co8mi)-^x=U. of-m7itanntx = 0, . . It. reqd. = c"'=l. 36. (cosmx)*'= e"'''«(<»»™''+"', and It. of n log (cos mx)^x' = It. of - mn tan (?nx) -:-2x = It. of - m'^ sec' (mx) -h 2 = -—--, . . reqd. lt. = e 2 . 30 DIFFERENTIAL CALCULUS, X. 37. (coa mx)^= e^ , and It. of nlog (cos mx)-i-x' ,, , mntan(»na;) m^nx , . ,, = lt. of ^-| — ■ = - -=-j- -when x is very Bmall; mitt .-. in the It. when x=0, It. reqd. = e s^ =e-"=0, n being supposed positive. If n be negative the It. = oo . rtJ cot^ X 4- sin X 38. =lt. of 3ar'oot''i-2ar'cot!«:coBec«a:+oosz x* x' = 3 cos* a;. -:— = 2coBx.^— =— +C08a:=3cos''x-2cosx+oosa;=2, Bin'' X Bin-* X ••• It. of a;-i-Binx=l. 39. Differentiate 4 times ; or, expanding np to terms in x', lt. = thatofi,(2.+|+y-^^(n-|+) =lt.ofi('«^-*^)=-|. x*\ 6 2 / 3 1* 1 . (1 1 X 11. 1_, 40. 41. /sinx)'»"*=e'-\iiiioB»ini and It. of cotx =lt. of cotx-^ -coseo'x= -Binxcosx=0, .. reqd. lt.=e<'=l. ._ ,, -sinx-cosx , . „ ,, Binx-cosx 42. =lt. of— TT — -^ — or (■.• Bin2x=l) = — 2cot2x ^ ' 2cos2x =lt. of -(sinx+cosx)-T-4Bin2x= j^. 43. Lt. of ^* is 1 when 6=0, ( = «-?p) , and Va"^^. cot l^ V^^! = '- . J^ . cot j'- J^l . ^±±^) , V (^ 'V o + x) 2 A' a+x |2 V a+x$ t * .-. its It, =that of - (a+x) = — » r X ■ DIFFEEENTIAL CALCULUS. X. 31. 44. =lt. of (l-I)Bin-s--4-co8- / . IT* -i 1 IT ira:\ 2 TT . TTX — Bin — 2 2 1 10J£ 46. ar»-^=e'-',andlt. ofloga;^(l-x) =lt. of --= - 1, .•. It. read. =e-'=- . X ^ e 46. x*~'=e^''-^°'', andlt. of logi-r-x", when x=0, = - oo -i-(+0)= - 00 , .-. a^~°=«— =0. Bat if a be negative, loga!^i'=^^-^ . =— s=0, and .-. x'~''=l. sin jj- X . (1 - x) 47. =U.of|sec(|x)tan(|x).(-,-l,)=-^.^^ 008^ Q I IT , IT . = --r(l-xl-i-COS's- x= — T—7-TrBUlvx = ao . 4 < ' 2 4 2 48. (4x» + + )i = <.i'°«'^++i^ and It. of the power of e = U. of =0 when i=go , m being a positive integer, ,*. required Ax H — \- lt.=e<'=l. 49. =H.ot2^i\(2+-^^M±^) 1 (\ V<« + ''x+x'/ 2X + 6 + 2 Vax+ftx + i' _ ° ^ ) ,y/ax 6 + 2 i,Jax\ 50. The expres8ion=i | - 1 + \/y^\ = ^ '~ -^ + ^^ " ^^'* ^^ ~ *^"*'' .-. expanding by Bin. Theor. =5- (- * +0)= ~ ^• Or differentiate the form {^1 -5x- ^1-x] -i-2x ,Jl-x, die. 32 DIFFEEENTIAL CALCtJLUS. X. 51. Lt. = that of- » Bin (xd)^ {- 2x66''^} = "°''gj^ . (1 \ \ M / 52, ^■■' + = -gwhen 1=0. 53. Lt. = thatof |e*(sma!+cosa:)-«'' V2cosU-jU-4-(e»-c°) =lt. of e'.2cosac-He*=2cosowhenx=(i. 54. =lt. of c'"^'*(^^', andlt.of {log(ax^++)-log«}^-^ =lt. ot ,^ {''^''°g°'(-^') + +K-^, and a,»=l when i=co, 0/+ + .-. It. reqd. = e""?'''+'°8«a+ + 55. Expanding, the lt.=that o{\x+x--r^+ -il-^- qT^+ j\ 56. =6" say, -where y=-\og( —2- j = { log (log a) - log x } -=-x =lt. of -r^^ -=0, .-. It. reqd.=e»=l. X log X X 57. If X = 1 - 2 where z is small and ultimately = 0, thelt.= ](-z-J-)%(2.-.^)i|^(-z + |-)^ = {z^+ higher powers) -r (z's + higher powers) = 1 . DIFFERENTIAL CALCULUS. XL 33 58. If «=a*"-i-6*", logit=x'».loga-a;''logi, and as a, b are each> 1, log a, log 6 are both positive, .•. log u is oo or - oo as m> or <;n, when x = ai , and accordingly u=ao or 0, 59. If ;r = i, x-^Mog fi+i'i =3'-::i2l(LL?)=u. fl-^r^V y \ "J y^ \ 1 + yJ 60. Let — ^ = y, and — =v, then vy = tan-i y + log {y + JT+ip] ; .-. when j/=0, i; = — =2oru= — , and when 2/ = oo , t>=0=«. Also x=\, .-. 5- = ^ , o' ay a' . d« c du 2ci/ 1 1 and y^- = -y-3-- —§ = -V + , ; H — ; ; "dy a" dx a> l + i/ ^l + l/ .•. when v=oo , ■=- = —,=0, and when w=0, 20" du , , y , y iy 2c»du a? dx' = -tan-'y-log(y+ Vi + r)+j|^= + ^iU^; \ _1 1 I 1 V . 1 ^—^^Z^fl \~l+y^ ^r^^^l + f (l+yT Jl + f (i+j,2)^i ■ -^ = -] ,.. .„„ + --^3=-!, when y = 0; •■ di^~2c^' CHAPTEE XI. 1. ^ = J!*.^ + ^.ji,and21ogu=log{^-j,«)-log(z= + /): dx dy dx dz dx 2 da _ 2i 2z_ 2 du 2ii 2v " u'dz~^^^~^+y'' u' dy~ z--y' z- + f' ^. du 2u I, .dz jidy) f dz dy\ ^ ^""^ di=^^ r'di-^^dxH'-''-VTx-'dx) ■ jwT^^;^)^ T. D. C. K, 8 V PIFFERENTIAL CALCULUS. XL. 2. sin u= - , .-. cos « . — = - . ; . T ■ , »/ , dx y dx y' dx , (??t / rfz dy\ , """^■'■d-x^y^dx-Txj^y-'Jf-^'- 3. e''!'(l + n2/)^ = max™-i = -.e»i'.y; 4. y log a; = a: logy, .-. nogx--j J^=^°Sy-'-; ■"■ ^ = ^(*^l°ei/-V)-^{2'l°S«-a:)- 5. {x+. dx dx dy ' dx du' dx dx dy ' dx du ' dx~ ' .■.p\^^ .p^-^ (p-i)\ + p .^-p ('/-i\=q;&o. dx (du dy da \dy J) dx dy dx \dy j 10. X and y are connected by the equation ^ (a^ y)=x (?') > *n^ -r > ~^ mean total and partial differential coefficients. Taking y as independent .,,.., du , , . dx dd) dip dx variable, then, :r = X « -t" =t^ + t^ • 3-; dy ' dy dy dx dy .•.|(2-x'(.,) = -x'W.g. 11. (1) — = (a)^ . yx»-^ . log + 1 vseo xy . tan xy, (2) ^=-1, .•. -5^ = o*" . yx"-! . log a + 1 ajsee xy . tan xy - a^" . log a.x" . log x - y \/sec xi/ . tan xy. 12. From Ex. 11, 0=0"^". ^x"-' . log a + 1 ^/sec xy . tan 2 xy + ^ ( a*" . ai'log o loga; + - ^/Bec kw . tan x^ ) &c. ■ dx \ i J 13. Here when x = = ?/ , (Ax)^ + 2a Ax^ . Aj/ - aAi/' = 0, ...inthelimU 2^= (|)'. i.e. g=0. or . ^2^ 14. Here when x = 0=2/, Ax*-oAy'+2aAxAy'' + 3aAx". Ay = 0; .-. Ax-a (^)'+ 2a (^)' + 3a^=0, ••. in the limit (iy-(i)"-2-»''-2=»-(s-=)(i*')=»' &c. 36 DIFFERENTIAL CALCULUS. XL 15. When x = 0=y, aAi' + Ai' . Ay-aAy3=0, whence 1 = ( ^ J dy . i. e. the real value of -~ is 1. ax 16. Uy + h = z, y^ = f-,and •' dx dx a-* (2- 6)''=2Ma''-2^"*h=22(a»- 62 + 262-22), .-. when 1=0=2, Ar^ (^_ j)2=^2(a2-62 + 26Az- A2'), ' dx Jd^^^' and in the limit *''=(«''-*') (^^1 ' 17. ■Whena = 0=j, in thelimit, 2(A^'-Aj;») = 8Aar', .-. ^=±^: If x-\ = x', y-l=y', Ar=Ax', Ay=Ay, .: ^ = ^. ana {77l>-(x' + l)»ix'(x'-|)=2{(!/' + l)2+(x' + l)2-2(^+l)}=, .•. approximately - Ax' {Ay' - Ax') = 6Ay'^, it)' dy , n dy I^n/B dx dx lb 18. If 1 = 0, i/=Oor ±1. "When j=0, -Ai/2 + 3Aa;A2/-2At' = 0, If ,,=fl=2, ^ = ^,and(z±ir-(2±l)= + 3x(2±l) = 2x'; •' dx dx .-. when 2=0 = x, ±2A!±3Ax = 0, and.-. ^^=-2" 10. y, 2 are functions of x, ,.i(y+x^j)+i.'y-y^=o. xy \ dxj X dx x^ dy y-x y ' ' dx~ X ' y+x' 1 d2 1 dz „ dz 1-xz z - . -, +x-,- +z = 0, .-. -J- = =— — .-, z dx X dx dx 1+xz x du d?/ dz „ ^^^ "dx + ^ + 2/5^+'di='': dn _ y ^x-y) ^ (X2-1) •■• "di- ''+x(x + j/) "^x X2 + 1 ' Differential calculus, xii. 37 20, Here z is taken as a function of the independent variables x and y. ™ X z dz ^ 11 [dzy z d-^z „ d«8__£»jl 1 C« ^l__^/'i ^'N •'• dr"" 2 [aS "^ is • a* ■ z^ j ~ a i> \, J-V * „ d'z c* /, a;'\ ,, y z dz . So T-2=-r2-3(l--2l- AUo j^„ + -. .— =0, dz dz d^z „ ax dy dxdy d'z _ 1 c'xy dxdy z ' a?Vz^ ' CHAPTER XII. dx „ , du du 1. x = ev, .-. — =e»=a:, and .-. :r = ^ j-> dy dy dx d'u d f dv\ „a'u au a-u . „ J (Pa du _ d-it dx'-' dx' ' ' dy^ dydy d9 1 ^ ora+x!t^ = ^- dj;~d9'da: (1 + a:'') * d9 ' ^ ' dx dfl' ...(I + x»,g + 2x(l + x=,g + .=g.,=0. _ dt dy_dv X . fl-\ IVj.*" ^ ''" di' •• dx~d« '2' •• dx-'~2'd« "^4"dt''' x' dhi dy . , d^j/ dy • 4 df^dt ^^ df ^d£^** / t JiTv'V 1 • 38 . DIFFERENTIAL CALCULUS, XII. and g=t(l-«=){gt(l-t=) + |(l-3t»)l=j,t=(l-t=)-. • . dt All dy 5. l=-smt.-^, .•. ^= -^ .cosect, ax ax at , ■■■ Pi = ?! eoBec= t - P, cosec« t . cot (t) ; . „ d'y dy „ d'v d^y . dx^ dx at'. at' x^-y ^rsmecoae^ + r>coB'fl-rsin#coae^ + r'Bm2ej 3 6 ^cosfl|^-rsin9j+rBineUin9^ + rco8ej r cos ( ay 7. I + y;;p-T ; JT *" coseT3-rBm( cLu and :f'^^y (asin Ex. 6) = =/Z^(coBe.J;-rBme), r' (cose^-rBinflj ■ •••(-+3'l)-^(^i-2')=?S-' 8. ^=oBint, -T^io{n + co3t) at at dy m + cos'f _ d^y _dt d / n+cost \ '■ dx~ siuf ' ■'■ d3^~dx' dt\ sint / 1 I (71 + cost) cost) ncost + 1 Binl ( ~ sin^ t 1 o . sin' t ' a sin I .DIFFERENTIAL CALCULUS. XIL „ 11. e=*=tan (!),.-. ;j^=2e=^-^Bec=t = and -r^ = dx ■ e-- + £-sJ!' dj/ _ dy 2 ■ ■ dx^dt" ■«"+«-'"' ■'■ dx'= ~ dt" ■ (e" + «-")'^ dt * («■•" + e-2i)= ' 12. g = BOC=, = l.x=.-.J = g(Ux,. d'« d^ .-. original expression = (1 + ar^)' ^ + 2x (1 + x^ ^-, +^(l + x''){2 + 0x'-8x= + 2x=}=0, ...g(l + x=p+2x(l+x=)g + 2 2 = 0. 40 DIFFERENTIAL CALCULUS. XH. IB. AM„A..X98.g=-g.(5y. dx" + \dxj \dy dy^J ■ \dy) ' 14. ^; = l + 2^.-.^ = ^'(l + 20, dt ' dt dx^ d'u (f« „ „.„ ~du d^ ,, , . ^du -=^,(l + 2t)= + 2- = -(l+4x) + 2_. , „ dd) , dd> dtp Hence ^=a,. ^+l,^+c,^^, quantities, -whence ~ -t + jt + JT reduces, by virtue of the relations among , X -3!c' + 3x+7, -^=3a?-6x+3, whieh is Batiafied by x=l, ax and then —^=Gx-6=0, and j3=6, .•. neither mazimom nor minimum ; of course u is a maximum when x= + oo , and a minimnTn when x= ~ - 50x) = 30 . a: (2x2 - 25) ; thus x=Z makes ~^ negative, and gives a maximum, x=-i positive minimum, x=i positive, minimum, x= -4 negative, maximum. 2 8. u = i o-l, .'• is a maximum or minimum as l + x-x^=i', say, X + x-jr , dw , „ = {a+x) (a-x)- (a + 2x) (a - Sx). When x=-a, ■-—^={2a)'(-a).ia, .■. a maximum. When x=a, -^-^=0, ''"* dx3 '^ ' ■'• °^'*^'^- ^^®° *= ~2' d^' ■ 2' V TJ ■■ 2"' ■'■ mmunum. \Vhenx=^, ^— .,= -3 . -r- • I -rr I . -rr . • • a maxminm. 3 dx' 6 \i J o 13. -r-=u I V — r- 1 . ."• 2 (x-a) = 3 (2x-a) or x=- , and then dx \a-x a- Ix] > / » 4 ' ii_du / 3_ _^ \ _ j _3 4 _ 1 _ _ « ( 3 4 \ :=~diA a-x'*'a-2xy " ((o-x)^ (a-2x)2J~ aO/3Y~/lY[- « /IG ,.V 4a V!')=0, .•.c^y = ii?, .-. y*-i{c'y)^ + {c'y)^=0, .-. 2/«=27c', i.e. y=J::C !JW, and .:x=±c^3. Also (y3-c«x)g+3»i'=0, .-. g= -3x=-4-c3(±4/{27}J=F^3); .•. x=c ^3 gives a TnaTJmnm, and x = - c iJS gives a TninimTiTn , 27. If he be at vJ, B the nearest point of the beach (straight), C the point to be reached, X the landing-place, \SX=x, t the time along AX and XC, then t=-;- +-?- = r- iJx'^+Q+ !-■=, . . for minimum oft, 4 4 ■ = = ^,i.e. 25x2=16(x» + 9), .-. x=±4, 4 ;^x= + y 5 corresponding to the positive and negative signs of Jx' + ^, . . x=i, only, applies; and this gives CX=1 mile. dH y 3? 9 Also 4 ;q-j = — r = J , which is positive as the posi- <*^' V*2 + 9 (x2 + 9)^ (x»-l-9)^' "■ " ha " tive sign of the radical has to be taken, and .-. the solution gives a 28. If a side of the curcnmscribing rectangle make an angle 9 with a side a of the given rectangle, the sides of the circumscribing rectangle are (acosfl+ftsinS) and (a sinS + ftcos fl), .-. the maximum is required of (a cos 6 + b sin $) (a sin 6 + h cos 6) which = (a' + 6^) . sin . cos 9 + ab, and this is greatest when 25 = = , and .•. the sides of the outer rectangle are — — each, 2 ^2 and it is .-. a square. When ^=0, there is a minimum, i.e. the rectangle itself. T. D. C. 4 50 DIFFEBENTIAL CALCULUS. XIII. 29. If a;=the Bide, the content of the box (of depth x, and without top) is x(a- 2x) (6 -%x)=u say, ;•. ~=(a-2x) {Ji-2x)- X {2 {J)-ix)+2 {a-ix)] = 12x^ -ix(a+h) + ab. a + bJrJa^-ab+h- 6 f a + by fa + by ab , . . for maximum I x -— 1 = I —rr- I - Tn > ^^°- "'— Also -— „=4 (6x-a + 4), .•. the negative sign of the radical gives the maxi- a + h- J a' -ab + b^ mum 01 u, 1. e. x= ^^-g — . 30. If r= radius of semi-circle, h the height of the rectangle, o=the perimeter, and M=the size of the window; then o=2ft+2r+7rr, «=-^j- + 2rA, .•. u=-^ + T{a-2r-irr), Of ,■ . for maximum 0= 7rr+a-4r-2irr, .-. r= , , ir + 4 .-. 2h=a :.a = ., .•. h=T. Also ^^ = - 4 - ir, 7r + 4 jr + 4 di" .'. the result is a maximum. 31. If PQR be a circumscribing equilateral triangle, P corresponding to A &e., and iAGQ=e, «= height of PQR; then 6 sin (9 + ^) , , CQ= ^ ' , CP. Bm'^=aBia(^+ir-e-cY sin ^ and .-. u={CP+CQ).Bm'^=bBm(e+'^)+aBm(0+C-'^j, .•. for maximum ^=0=6.oob( 5+^ j+acosf *+C-^) , .•. squaring and adding, w' = l2+a' + 2o6coB ( <^ --s-) =:a' + 6=-2o6 COB fc +| j . Msoj-^=-bemfe + ^j-aBm(e + C-y\=-u, and is .-. negative, .- . the result is a maximum. X)IFFEEENTIAL CALCULUS. XIII. 51 There is a minimum of u when one side of the equilateral triangle is wholly or partly coincident with a aide of the given triangle, d then having its greatest or least value consistent with the question and the nature of the given triangle. 32. The axes are supposed rectangular, and OA, OB both positive, as also a and b. _ (1) See p. 193. If OB might be negative (or OA) then OP would give a minimum. (2) 04 + OJ? = a + 6 cot * + 6 + a tan e, and clearly e = or ^ would give a maximum, .-. for minimum b cosea' = a sec^ 8 and tantf= . /-, the negative sign being inapplicable. (3) OAxOB = {a + bcote)^ .tii.ne=:a'ta.ne + 2ab + b-cote; .-. for mini- mum a'seo'e=i''cosec' 9, and tan(? = -, as in (2), gives the minimum value: and fl=0 and — give maxima. (4) OA + OB + AB=a + bcote + b + atane-i — ■. — „+-^: .-. for mini- sin 8 cos mum a (sec' 9 + mne sec' e) = b (cosec* 9 -l- cos fl . cosec' 6), or asin''e(l-|-sine) = 6cos''tf(l + cose), .-. a(l-cos*) = 6(l-sine), .-. (a-b)^ {tan'> 8 + 1) = {a- b tan e)\ or tan= $ (a« - 2db) + 2ab tan 6 + b^-2ab = 0, .-. tmi0{a^-2ab)=-abJ=j2cJ>.{a-b) = (a-fj2^) {bJ=j2ab), the ambiguities corresponding, .-. tan 9= — , and the lower signs may a±j2ab b + ^2ab Tr make tan 8 negative, .-.for minimnm tan 8 = — , : fl = or - correspond a + j2ab ^ to maxima values. (5) 0.<1 X OB x4B = (a -1-6 cot*)', tan 9 sec 6; .-. for minimnm [a + b cot 8) (sec' * -H see 9 tan'' 8) = 36 cosec'' 8 tan 9 sec ff , or (otan»-t-6)(2tan»9 + l)=36tan2ffcosec»e=36(tan''«-t-l), or 2atan'e-i}tan''e-)-atantf-25 = (1). The roots of (1) are all positive, if real, and 9=0 or - gives a maximum. Thus there is one minimum, m there are 2 minima values and another TnnTiTTiTiinj ai one or 3 roots of (1) are real. 4—2 52 DIFFERENTIAL CALCULUS. XIIL (6) Here for a mmimnm (a + b . cot «)»-i 6 . cosec= 9 = (a tan 9 + 6)»-i . a sec' $ ; 1 . . tan»+' e = - , and the real yalne of tan 8=1-1 a \aj 6=0 and ^ give maxima valnes. 33. If ABP be the triangle, the angle APB being constant, P lies on a fixed circle through ^, £; and AB being constant the area is a maximam ■when P is at the points G and D where CD bisects AB at right angles, and G is equidistant from A and B ; eo also is D. Practically if C be farther than D is from AB, the maximum area is ACB, 34. Let the quadrilateral be ABGP, where iAPG=a, AB = a, BC=c, and from Ex. 33, CP=AP=x say, and let / ABC = 0. Then AC^ = 2x^(1- cos a) = a' + c' - 2ac cos 9, and twice the area = x2 sin a + ac sin 9=ac sin9 + „-;r (a' + c^ - 2ac cos 9); 2 (1 - cos o) .. for maximum C03fl + :; sin9 = 0, or tan6=-tan", .•. B=ir--, 1 - cos a 2 1 .-. 20 = 2Tr-a = e + lBAP+BCP, or ABC=BAP + BGP. If now BP^AP, BP>CP, thus I BAP + BOP^ABC. So if BP<:AP, lABC>BAP + BCP, .: BP muBt=AP=GP, i.e. P is equidistant from A, B, C. 35. If any circumscribing ellipse be projected orthogonally into a circle on the minor axis as diameter, the parallelogram must project into a rect- angle, and the area of the ellipse is to that of the parallelogram as the area of the circle is to that of the rectangle, .-. this latter ratio is least when the ellipse is least, but the area of a circle to an inscribed rectangle is clearly least when the rectangle is a square: let i= radius of the circle in this case, then the ratio required is irr° : 2r''=ir : 2. Aliter: analogously by aid of the auxiliary circle. Also the diagonals of the parallelogram bisect each other, .-. they intersect in the centre of the ellipse, and the diagonals of the square are at right angles, thus the diagonals of the parallelogram are conjugate diameters of the minimum ellipse, whence it can be constructed ; for if 2r, 2/ be the diagonals, a the angle between them ; 2a, 2b the axes of the ellipae, 6 the z between 2a and , , r* cos^ 8 r' sin' . . 2r, then a, 6 are known, and j— H rr, — = 1 gives 0. ' ' a^ 0^ 36. Let C be the centre of the ellipse, and the tangent at P (eccentric angle ip) meet the axes of x and y in Q and R; then the equation of QR is 'LS21z. + ?—--?=l, .-. CQ = a sec ^, CB = 6cosec^ (or from the auxiliary a b DIFFERENTIAL CALCULUS. XHI. 53 ciicle), .•. QB'=a'Bec'0+!>'coeeo'^, and as Qii is a maximum when 0=0 or ^, for a minimiiTn a? sec' ^ . tan 0= 6' cosec' (p . cot^; ..,.!^ = £2!V = J_ and (3JJ» = a(a + 6) + 6(a + i), .-. Qi?=a + i; 6 a a + b \ y i /. i and PR : PQ : : a coa ^ : a (sec ip - cos 0) : : cos' : sin' tf> ;: a -.b, .: PR = a, PQ = b. 37. If Qli be the chord the vertex P is clearly snch that the tangent to the parabola at P is parallel to QE. If S be the locus, ilf the centre of Qli, and a the angle between it and the axis, the area of the triangle = (3ilf.PJf.Bino, and QM^ = iSP.PM, .-. area = QJf'. sina-T-4SP=QU3Eina-7-32 . SP. 38. The corresponding miuimmn triangle on the circle on the minor axis (as diameter) is eqnilateral, and the heights are the same, as corre- sponding tangents meet on the minor axis, .•. height required is x, where X • =6 or a;=35. o 39. If r be the radius and 6 the angle of the sector, the perimeter = r0 + 2r=a say, and the area = -^ = ^ (a - 2r) ; . . for maximum or minimum ?=2r, and then rfl=o-2r=2r. Also t itself may be a maximum or minimum. If 9 be restricted to 2t, the leist value of t=^ r-r , and then area=^-j^^,, but when |=2r, the area=r'=^ which > -^^^^^-^^ , (t>3); and when r is greatest ^=0, and area = 0; thus r=j gives the greatest area. Negative values are excluded. 40. If PSp make an angle 9 with the major axis, SP=Z-eSP cos 9, 21 1 sin $ Pp=, ^ s-;, , and . . area= s HS .PpsmO 'x — ; „ ^^„ - . ^ l_g2gos=9 2 l-c'cos-e sin 6 ^ - - If then u=, „ i-, , for mazmium or minimum 1-e'cos'd du cos e Bin 9 . 2e' sin cos B He" l-e'cos'fl (1 - e^ coa^ »)- cosg(l + .'cos'9-2e=) . <, = -„, ,o,= 5=2- 1,. ~ (l-e'cos'ff)* 2 e- the latter of which is only possible if 2e'> 1. Now as * increasing passes through the value - , ^ changes sign from negative to positive or the reverse as 2e'> or not >1, and .-. 0=-^ gives a TnJTiimTitn or irmiiTTimin accordingly. 54 DIFFERENTIAL CALCULUS. XIII. Also if 2e' > 1, as 6 passes through the least value given by cos 9 = + »/ 2 - -^ , — becomes negative, . . there is a maximum. So by symmetry if cos e= - / 2--^, And if e = or JT there is a minimum. 41. Of. Todhunter's Conies, Oh. Till. Ex. 20. Thus the length of the 2 s normal chord at (x, y) = —^ i^^a' + y°)' ; .. for minimum, as the vertex and points at an oo distance give maximum lengths, and .-. the Iength = ^^ {12a=)*=| . 8 . 3 v'3 = 6a V^- Also the distance of the point of intersection from the vertex is f8a^ + r\ I 1 faa'+yy j' V^rMio^^V y ) ^ i • and such distance is a maximum -when (x, y) is the vertex or infinitely 8a^ + 7/2 1 distant; .-. for a minimum of — — f- {{y' + 8a'')'+lGa'y'']--' - ^ {{y^ + 8aY + lOaY-) + V ■ {{y- + 8a^ ■ "iy + Wahj] =0, or c"{(y- + c^f + 'ich/] = {ii^+c')y-(y^ + '2c% or {ij- + c-)''{o'-y-) + chj-{c'^-y'^) = a, i.e. y'^=(?=8a' is the only real solution, and .•. &e. It may be easily seen from a figure that the shortest normal chord inter- sects an adjacent normal chord on the curve, and in one of the 2 points where the curve meets its evolute : and the evolute being convex to the axis, any other normal chord meets the curve in a point farther from the vertex. See Chs. XXIV. and XXV. Ex. 5, and cf. Art. 337. Now (Todhunter's Conies, Oh. VIII.) the normal at (x', y') is y= -g-x + ^' + |-s, and if this pass through (h, h) a point on the parabola, •' tia' 2a ■ 4a ' thus one normal from {h, i) being that at (ft, k), the equation -l=^--^j- - has equal roots in y', hut 2/'2 + j,'7j + 8a2 =0, .-. P=32a^ DIFFERENTIAL CALCULUS. XIII. 55 or i=±4a\/2, andJr'=-|==F2a^/2, and .-. if the length of the Bhortest chord be z, or '=^-^ {16a= + 8aM^ = Gaj2.^=6a^. 42. If D be the distance of the ships at any (say subsequent) tinie + 1, D^={a + ut)'' + {b + vtf-2{a + ut)(b + vt) COS e, for TniniTT i nTn of D, •.• t= ± oo gives a maximum, {a + ut)u + (b + vt)v=u(b + vt) coaB+v (a+ut)co8 9, a+nt b + vt and .-. ±D = U COB 6-V U-V 003 5 (_ av- bu \ ~ 2uv cos e-v'- u-J ' 2ui> cos e-v^-u? X { (u cos d-vy + (u-v COS S)' - 2 COS 6 (u COB -v) (u-v cos S) ji = ., ~., r / . { »' Bin2 e + »= sin= fl - 2uv cos 5 (2 - cos" 9 - 1) ! *, or D, being considered positive, (oo ~ bu) sin fl D=- {u!'+v^-2uv cosO)-! ' Aliter: if at the given time the ships be at A, B, and the velocity of the ship at B relative to that at A make an angle with AB, then obviously the least distance =^B sin ^ &c. 43. Let A be the vertex (a, 0) say, APQ the straight line meeting the ellipse in P and the circle in Q, FQ = r; then AQ = 2acos 0, d?.AP^sm^B + b^(a-AP cos Bf = a^ft^, or AP(,a?sai^0 + Vco??B) = 2ah^coae, and . . r (o^ sin" 9 + 6" cos« 8) = 2(i cos 6 (a= sin^ ff - &« sin" B) = 2aVsin"flcos9; .-. for maximum of r, r . a»e" . 2 sin 9 cos S=2aV (2 sin e cos" B - sin' tf), or r cosS=a(2cos"<'-sin"S); .-. 2a' . e" sin" 9 cos" 9 = a^ (2 cos" 8 - sin" ff) (1 - e" cos" «), or 2e" (cos* e - cos" 6) = (e= cos" e - 1) (3 cos" e - 1), 56 DIFFERENTIAL CALCULUS. XIII. or e2cos''«-cos=9(3-e=) + l = 0; now the product of the values of cos^tf fromthis = -j which>l, .•. the greater value is impoBslble, and the lesser value, given by 26^ cos^ e=S-e-- J(ii-e'y-'-ie'= (3 - e^) - ^(9-6^1 (!-«"), is possible if Q{l-e')-<(3-e^f-4e\ .: if 6e*<:8e^, which is true. When 6=0 or -, r=0, and is . . u minimum. Thus the above equation must give the 2 maxima values, there being 2 positions of APQ equally inclined to the major axis. 44. If the ellipse be ah)' + l^x^=a-h^, and the circle x'' + [y + b)''=r', then d^if' -b'(y + by'+ i V - aW == has equal roots, or 0=a2(r2-a=)-6V, i.e. 62=a=-^, and area = 7ra6; if then u=a'b'=a*--, , for a maximum area, 4a'=— i- , . •- 2r' fi 4\ -•. a = which gives aline ellipse area = 0, OTa^=:--^ , and . .V=r-[^ — -^\ , , /2~2 2Tr= and .-. area=5n-'' V g-g^OS * ^' Also a may be a maximum, and . . =r, and the ellipse a line ellipse area=0, thus (1) gives the maximum area. 45. If a triangle be drawn with the same vertex, and base in the same straight line as the base of the given triangle ABC, and circumscribing the circle on the axis of the ellipse which is perpendicular to the base, then the area of the circle is to that of the described triangle as that of the ellipse is to the area of ABC, stXxA the first ratio is least when the described triangle is equilateral, and then its height which equals that of ABC is 3 times the | axis, .-. &o. The ellipse is a minimnm when it is a line- ellipse, and the finite axis coincides with either the base or axis of ABC, Cf. Ex. 38. 46. Let a=radius of the circle, 29=angle of sector, r=radius of base of cone, 20 = its vertical angle ; then a = slant height = r coseo 0, surface of cone = Tri'^ cosec =a? (it - B), and volume of cone = a say = -J m^ . cot . Thus r=aBinc6, tt — ^=irsin0, and — 7, = sin^ ^ . cos A, ,*, for maximum or mini- mum of 11, 2sxD.coa' =siii? i.e.2»=-?^(V3-V2). Also 6=Te gives surface and ,•. the volume = 0, a minimum. DIFFERENTIAL CALCULUS. XIII. 57 47. If PSp, QSj be the chords, and PSjp make an angle with the major axis, "/>=; z + 5 ;; = -i s T-^t .-. Q(7=i ., . ., „ , .•. the Bum— 5_ ' "l-e^Bin'ff' ■ l-e2 + eJBiu*9coB'i«' is a mazimnm or mimmum as sin $ cos & is a minimum or maximum, 1.8. as 9=0 or .-. 4 Thus when the chords are parallel to the axes their sum is a maximum, and when they make equal angles with each axis their sum is a minimum. 48. Consider a plane section through the axis of the cone and cylinder. If X be the radius of the base and y the height of the cylinder, y -.biia-x -.a, and.', the Tol. of the cylinder =«=xx''.- (a — x) which is a minimum when 2 x=a or 0, and .•. for a maximum of it, 2x{a-x) = x', i.e. x=-a, o and.-. voI.=^.^a3.1 = A^6a=. 49. The convex surface = 2irx . 2/=2t . - . i (a-a;), .•. for maximum a-x=x, i.e. x=-, and surface = 2x- . ^ = -^ : the surface being least when a;=0 or a. 50. "Whole surface = 2rxy + 2irar» =2tx- + 2ir - x (a - i) = u, 2ax .•. for maximum -r^ + a-x-x=0, b •■• ^=,r-;i It but xa or x is negative. If a6-!-2 (6-a) 2o, which includes the other condition. If then b>2a there is a maximum, for -;— = 47r : also x=0 and a give minima values, dx^ a X being then a minimum or maximum. If b<:2a the solution does not apply (there being discontinuity owing to the restriction of x being not^a), but then the solution indicates that u increases with x, so that the real maximum is when x=a; the minimum dhi being, as before, when x=0. Also here the sign of ^-j is no guide, as the variable x is limited and ^ is not=0. In this case the maximum and ax ynmimiim yalues of X itself Correspond to those of u. 58 DIFFERENTIAL CALCULUS. XIIL 51. Consider a plane section through the axis of the cylinder (passing through the centre of the sphere). If 2x be the height of the cylinder, its volume = 2x . IT (r^ - ar"), and the minima values are when x=0 and t, •: for 2r the maximum r^ - x^=2x^ or the height = —7= . 52. If height = 2x, convex surface =2x . 2ir ,^7^ -x^=ijr ^u say, the surface is a minimum when x=0 or r, and for a maximum of u = x-{r-~x''), a(r2-a;-)=a:'or x^=— , .•. height =r ,^2. 53. The whole surface = 4irx y^j-* -x- + 2jr{r'-ar') = 2xu say, / — ■ — 2x2 .-. for maximum 0=2 Jf' -a?- -^=^ - 2x, Vr'-x2 or r=-2r'=xv''^-x2 (1), and . . 5x*-5rV + r* = 0, or x==J(5±^/5) (2), Jl /5±^/5 5=Fx/5\4 and substituting m (1), we get=F-g-= + ( — j^ j^ j , '■' " mist be positive, .-. the lesser value of x= in (2) alone satisfies (1), the other value being introduced in the process of rationalization. Thus and . . the height=r . J2( 1 ^U , This gives a maximum as 1=0 or r give minima. 54. Consider a plane section of the cone through its axis, which passes through the centre of the sphere. If x be the height of the cone, 20 its vertical angle, then x sec 6=2r cob S, and the volume 1 8 = -7rx' Aa.ii' 6 = ^Tr^sm^8eo^ 0, o and there is a minimum when x = or 2r, .-. for maximum sin cos'fl is a 2 4r maximum, .-. cos' 9 = 2 sin' fl cos « or cos^ 9 = - , and .•. x=2rc032e= — . o o 55. (See Ex. 54). The convex surface = cosec . TTx" tan' 6 = iirr' sin cos^ 0, 4 .. for maximum cos' 9 = 2 sin- e cos S, and x=^r, as x=0 or 2r gives a minimum. DIFFEEENTIAL CALCULUS. XIII, 59 56. The whole surface = v3? taii= e. (cosec 6 + 1)= ivr' (sin e cob' + sin' 6 cos!* d), and 6=0 or ^ give minima values, . . for a maximum, 0=coB3e-2sin2ecoBe + 2sinflcoae(cos2e-Bin2«), orif x{=2reos^e) = 2rz, 3z- 2= -2^1^(22-1) (1), .-. IGz'- 232" + 80 = 0, and 32z = 23±^/i7, and substituting in (1), and observing that sin 6 and .-. ijl - z is positive, C9±3^/r7-64= -2 Vl^(4G±2,yi7-32), and the upper sign is .•. inapplicable, as it does not satisfy (1), whUe the ^/'-^' lower sign gives 3 ^/l7- 5=4 (7 -a/i?) V tjy or 6 ^-10={7-^Vi) {Jrf+1) which is true, .•. for a maximum x=2tz = ~g- (23-^/17). 57. If height = J/ and x= radius, irxhj = V, and u = sum of the areas of convex surface and one eni=2Txy + Trx^, 2F „ du 2r „ .". u= l-irx-, ■r-= r + 2ira;, X ax X- /V\i V .-. for minimum x=( — I , .•. x'y = — = T'B,nix=y; also — i = -Y + 2jr is positive, . . result is a minimum, u is clearly a maxi- dx x^ mum when x=ao, or »/ = a> . 58. If 26 be the vertical angle, h the height of the cone, and a the radius of the sphere, then 7i=a + acosec 0, and volume = — . a' (1 + cosec 9)' . tan' 6, and for maximum 6=0 ot —, 3 ^ .•. for minimum - 3 cosec 6 cot 6 . tan' 6 + 2Q. + cosec 9) tan 6 sec" 9 = 0, or 3 sec 9 = 2 sec' 9(1 + sin 9), i.e. 3(l-sin9) = 2, and •• '' = si°"^(3)- 59. Let 26 be the vertical angle, I the slant side of any one, then volume = - Z' . sin' 9 cos 9, and for minimum clearly 9=0 or -; .-. for 3 ^ . maxmi" Tim . 0=2 sin 9 cos' 9- sin' 9, and .-. ta.n6=j2. CO DIFFERENTIAL CALCULUS. XIV. 60. If be the given point, FOp the required chord, QOq an adjacent position, for a minimum ultimately the area POQ=pOq when Q moves up to P, .•. vT-j = l ultimately, i.e. FOp is bisected at 0. Hence it RO be the diameter through meeting the parabola in R, FOp is parallel to the tangent at R. If FOp be drawn parallel to the axis the area is a maximnm. 61. Let FCF', QCQ' be the 2 conjugate diameters, a the eccentric angle of P, — + a that of Q, and R any point (a cos 9, 6sin0) on the ellipse; then CP, CQ are xh sin a -ay cos a=0, and xtcosa + oy sin a=0, .•. sum of perpendiculars from R squared _ a^h- (oo3 tf sin a - sin 6 cos a)° a%^ (cos a cos 6 + sma sin 0)- ~ a* cos^ o + 6' sin* a o'^sin^a + i^cos-a ' let then u=sin'^ (B - a) {a'siD^a + h'''CO&'a) + cos^6- a(a^eos''a + VBin^a), then for maximum, -ji—^—~ ^™ 2 (9 — a) (a- — V) cos 2a, do .•. $-a=0, — , TT or 3jr, i. «. iJ is an end of one of the conjugate diameters ; and j2= - 2 (a* - J>^) cos 2a cos 2 (9 - o), .-. when 9-a=0or7r, -j-. is nega- ad Utr tive, and when fl - a = ^ or -^ , -^^ is positive ; thus F, F" are positions of R for a maximum, and Q, Q' for a minimum, a beings— . B itself has no proper maximum or minimum value: .•. the above forms the complete solution. 62. (1) For a small change in x,/{i) is unaltered, and .•. also {/(x)} ; and on a nearer approximation /(x) i^ diminished whether x increase or decrease ; .•. the corresponding change in {/(i)} is of the same sign whether x is increased or decreased, i.e. {/{x}} is either a maximum or a minimum. Similarly in the other case. CHAPTEB XIV. 1. If the given expression =u, -^ = 2 I ) -— — which ax \x c + xj x* (c + x)2 reduces to - c'-=-x' (c + x)*, .-. -j- is negative, c being positive, and .-. &c. ...*f=21og4-+{c + 2x)f^-J-) = 21og-^+i + ^. dx "c+x '\x c+xJ "c+x X c+x' DIFFERENTIAL CALCULUS. XIV. Gl .', by Ex. 1, -r- diminishes as x increases, but when x=tti , -r =0, .*. ^- is dx dx dx always positdve, x and c being so, .-. -r- (e*) which =£* . -r- is always positive, and .'. e" increases with x. du 3. ^=e"(l + 24;-3) + 4e*(l + x) + l, d'u .'. ^^=e'>'{22x-S + 2)+ie'{2 + x) = ie''{e'{x-2) + x + 2}, .'. by Ex. 10 p. 86, —^ is positive for all positive values of x; . . -r- increases with x, but when x=0, j-=0, .•. -j- is positive for all positive values of x; and .*. u increases with x^ and when x=:0, u=0, .'. u is positive for all positive values of x. 4. If U7=the given expression, dw e^{2{x-2) + l\+e'{x + 2+l) 3e' 2n.^(xl2)l thus -- = ^-j — —^ {c*" (3 - x) - 4x e^ - X - 3}, .•., by Ex. 3, — is negative for positive values of x. Also x=0 gives by Ex. 15, p. 144, 10=^, which is, .•., the maximum of w for positive values of x. 5. If «=(l + i)*, logM = -Iog(l + x) = l-f + |'-x + ...by Art. 117, .-. (Art. 115) u=e * » ■> I, / a: x2 x'\ If X x^ y 1 / x , V , ) = ^r+(-2+3 --4) + J2t-2+3 -) +]3i-2+-J +-( _ I X x' x' x^ x' x' \ "^1 2 "'"■3 "T"*"?" "6 ~ 48 ■*"■•■* I, X 11 , 21x'| 6. By the method of Ex. 5, (l + x)*-e=e ( 1--| -e approximately; .■, = -= , in the limit when x=0. X 2 62 DIFFEBJENTIAL CALCULUS. XIV. 1 1 '- e 7. Ifx=-, the expression =u=-(l +3/)" — ^ log (1 + y) and as in Ex. 6, 1 - e e - (1 + 7/)" = — = neglecting higher powers of y than y, and y ' y ■^ ^log(l + j,)=i(j,-|V...); thus u= — -^ h x + positive powers of y, .•. when x = a> , and y = 0,u = 0. y 2 y 2 " 8. If the given expression =u, and y=- , ...h.Ex.5.=^.e{l-|.g,-...S-?^(.-f.f...) 11 8 ... , = --6-56 + positive powers of y, A A .-. whenx=aD and?/=0, «=«. 9. Lt.=lt. of-:; r,-^]nx"-'-c''"'°«*.cos(loga;).-{ X X X / 1\ 10. Lt. =lt. of-tan^. oosec^ ^^ .^-i-f - coBec'a:+- j '2 2 '2 X x-sin^a; sin"x »« = +-^-H- .7 = =^=lt. of- = 0. Binx Ein-'x x-sin-'x x sec"x secx ,, c - tilUZ sec X t ilani/1 \ . ilani tt ctanx-j-e" 1 -sec^x ) = n8inxH-e" =Owhenx = .-, \n / 2 ' .•. «'»=0, n being finite. , , , \ ■ , n o > sin (n - m) X . see tix sec mx 12. li = (tan nx - tan mx) -i-sm (n^x - mPx) = ^ — : — . , „ :;r-, , ^ ; \ ; Bin jx(7i''— m-)} n\ _ "-»" _ 1 . ■ ■ ^ ' ~7v'-in^ 7it+n' (2) u = -TT- — i • sec'' mx=:-r- sec' mi. Also by differentiation. PIFFERENTUL CALCULUS. XIV. 6*J 13. The given equation shows that 9 is in general a function of x and h, and if /" (x) be not zero, Taylor's Theorem leads to / (X + h) -f (x) = hf (x) + ^ ./" (X + e,h) =}if' {x+eh)=h . f (x)+h .eh . f" {x+e^h): flj, ffj being proper fractions; .•. when h becomes 0, S=g' So if f'.{x) be the 1" of the differential coefacients, after /' (i), which is not zero, V (x) + p/' (* + e,h) = hf (x) + ft . ^-y^ ./r (x + e,h), 1 whence •=er- 14. If be the same for all values of 7i, must=5 by Ex. 13, and ex- tending the expansions to f"'{x), it then follows that /"'(x) = 0, .-. f"(x) must be constant with reference to x. This would make / (x) of the form a + bx + ex'. 15. Logz=Bin!c, .-. i»^ + z^ = sec''x, dr 1 dy dxj and --=-seox,.. .^ = seox^; d*t/ d« secx/ d°« , . d>j\ .-. z^ + -^ = ( secx-T^ + secxtanx-;- , dz* dz z \ dx^ dxj and .-. l = ^, + tanx.^|. 16. If x=pcoB^, j/=/)Ein ^; z^rcosd, p=rB\ne, then by Art. 205 (1), du dn 1 . du — =cos ^ ;5 sm ^ — , dx dp p dip dxi . du 1 , du — =Bai ^ -^ + - aos -T-; dy dp p dip fduy /duY fduy^l fdny (dny ^ fduy (duV'_ fduy 1 (duy i /duy •• \dij '^\d~y) '^{dzj -{drj ^r^-\dej ^i^sin-e \dipj 04 DIFFERENTIAL CALCULUS. XIV. . , ^ du du „ du . ^ du . „ dii . . ^ du du = "5-,- du du du 'dz-^''Tp = ''Tr' .-. the given expression \\drj '^f^'\dej '^ r^em^e'\d)] \dr) ' 17. If (x, y) be P, each co-ordinate plane is supposed to pass through P, and .-. X, y being variable, the co-ordinate planes are coincident, and by ordinary formulae, {=oH-a;co8a-(-2/ sino, and 7; = 6-xsina4-y coso, where a, b, a are constants. mv <'? drt . d{ . dv Thus — -=cosa, -r^= - Bina, ^ = sm a, — ^=C08a, ax ax dy dy and .■. J 2 = ( cos o — - sin o 3- J (0), (vide Art. 226) , dV • „ y' - aV - "^V, P=''y (18 - 4i - 3y ), ^=x'y (12 -2x- 3y) ; ^"=36xj,' - 12i«i,» - Gxy'=6xy2 (6 - 2x - y), Now ^ = when x = 0, ory=0, or 18-4x-3y=0, and — ^0 when x=0, ory=Oor 12-2a;-3y = 0. ay T. D. C. 5 .66 DIFFERENTIAL CALCULUS. XV. (1) If 18 = 4j! + 3y, and 12 = 2x+3y, x=3 and .-. y=2, .•. A is negative, and AC-B'=x:*y^{72-S6) is positive, .-. there is a maximum of u. If either x or y=0. A, B, C all vanish ; and (cf. Art. 230) ^ = 36y= - 24xj= - 61,', ^^ = 72xy - 24x'2, - 18;rj,', ^,=36x=-8x3-I8x^.f;=-6x3. .•. -when ^ = but x is not zero, P=0=Q but S and T do not vanish, and ■when x = and y is not zero, P does not vanish, .•. in neither case is there a maximum or minimum. If x=0=y, P, Q, S, T all vanish, and then S=-2*^==°' Jl^ = ^2j,-48xy=0, ^-J«^,= 72x-2.x=-36x. = 0,^,= -I8x»=0,|i:=0; ^v-hich does not vanish, .■. by an extension of Art. 230, there can be neither maximum nor minimum. 8. ^ = 2(a-x)(26y-3/2), ^ = 2(6-1^) (2ax-x=), .-. for maximum or minimum x = a or 2/=0 or 26=y; and y = 6, or z=0 or 2a; and .g=-2(2!,j,-y=),g=-2(2ax-x=),^^=4(a-x)(6-,). (1) If x = a and y = 6, -B=0,^= -26", C= -2a*, .. there is a maximum. (2) If y = and x=0, J=:0=C and £=iah, .■. neither maximum nor minimum. So if y=0 and x=2a, and symmetrically if y=26 and x = 0; and also if y=26 and x=2a similar results follow. 9. ^^=i{x>-x + y), p^ = i{y3-y + x); ■ ^"-12x=-4 ^"^ -i !'°"-12w2-4- and -^ = = -^ give x' = x-2/= -2/3, .-. x= -j/ and x'=2x, ax ay .-. -x= ±;y/2^and y= ^sj2, or x=0 = y. blFFBRENTIAL CALCULUS. XV. 67 (1) If x= ±V2= -y,A = 20, B=i, C=20, .•. there is a minimum. (2) If x = 0=i/, A= -4 = C, and 5=4, .•. AC^B* and the expansion in Art. 228 redQoes to -2 (ft*-2Ai + t') + i?j, and when ft, h are smaU enough this is negative, unless ft = fc ; now 3-i=24a!=0, 3-773- =0=,-r-5. j-,=24y=0, and xj = 24=t-. and the other terms in R^ vanish, .■. when h = k, the expansion reduces to h*+ k*=.2k*, which is positive, .'. there is neither a maximum nor minimum. If X or y be x> , u is a maximum, 10. ^=3(x»-2x-l),g=4(3f'-6y» + 0i^-2), .-. ^=0 gives i = l±V2, f^=0 gives (j/-2)(i/=-4y + l) = 0. ax ay (1) If y=2 and x = \i.^2, A=±G^2, B = and C=-12, .-. y=2 and X = 1 - 1^2 give a maximum and then u=(v'-4y)2+2 (jr'-4y) + » (x»-2a!- 1) - {*=-2x-l) -4x-l = 16-8-1-4 + 4^/2 = 3 + 4^/2. (2) If x=l±V2, andi/=2±V3, 4=±6 ^2, and C=12{l^j3){-l±j3) = 2i,.-.x = l + j2aniy=2±^3 give minima values, and, then u=l-2-l-4(l+ \/2)= -6-4 .^2. Tninimnni when 2x + y = a and 2y + x = b, a + b . 2a -b 2b -a and .•. x+y=-g- ,andx = — ^ , y= -g— , and .-. M = j {3 {x + y)' + (x-yf} -ax-by =j||(a + 6)»+(a-6),j-|,(a= + 6=-o5)=-^(a» + 6=-a6). •68 DIFFERENTIAL CALCULUS. XV. _ du _n 2x y du n y x ^u 2 *"'di~3~T~12'^^4~2~i2' dx^^'l' d^u 1 dhi 1 ^^ . %— i = — t:; 1 3— >= - o > •■• there is a maTimTiTn when dxdy 12 dy' 2 in—8x+y and 3n=6y + x, i.e. 12n=24x+3y = 2iy + 4x, and • 'g _ y _ ^" _ z + y + z _z ■ ■ 21 20 188 47 ~ 6 ' 13. ^ = 3{x= + aj,), ^ = 3(j,« + ax). .-. -,=6x, ^^=3a, ^,=6j,: and for maximum or minimum x'+ay=0=j/^ + ax, and .•. x=y or x-¥y=a, .-. ii?-ax + a'=0, which is imaginary; or x=y=0, which give A=0 = C, and .•. neither maximum nor minimum ; orx = y=— a, and .*. A= —6a=G and there is a maximum or minimum as a is positive or negative ; and then M=a', Also X 01 y may = ± ao , &c. 14. If u=x(x^ + y^)-3axy, ^=3ii? + y^-3ay, -^=2xy-iax, d:'a . dH „ „ dru ^ ,, — .=6x, - — —=2y-ia, -r-„=2x: and for maxminm or minimum dx' ' dxdy " ' dy^ (1) x=0 = j/, and .-. A = 0=C, B=-3a, .-. neither: (2) x=0andi/ = 3a, . . J=0=C, B = 3a, .-. neither: (3) 2y=3o and 3a?= (^\ , i. e. x= ± -g" • ^""^ *^®° A=:ta3s/3, C=±a^/3 and B = 0, aJa a Ja .: a minunum or maximum as x= — ^— or -^ , A 2 , . °'n/3,„, „ 3 3^3 3 , /5- and «= ±— ^(3)=F3a'. — j- = Tjfl' V3. 15. If u = the given expression and v=l-ax-by, dii 2x a ,. „ ., du 2y b ,. „ „. ax i; u* ' dy r e' ^ .•. - = I =r say, ana for maximum or minimum a 2ra{l-r(o» + 6=)}+a{l + r> («" + »')} =0, i.e. r'(a' + !>')-2r,-l = and .-. r(a'+i»») = l± ./T+^+P, DIFFEBENTUL CALCULUS. XV. 69" , dht 2 iax 2a' , , ,, 2 du . ""^ d?=v + Tr+i;T(i+^=+y')=i;.v^=o. d=« 2la! 2ay 2ai„ „ ., 2 , ^ , „ „ ., ar y li^l + a^ + is . . _ ^, . , thus - = ^ = — ^^^ — r5 give a maximum with the upper sign and a a a'+O'' minimum with the lower. 16. Vnien y is constant u'oo (c-.t) (x + y-c) but (c-x) + (x + y-c)=7/, .-. u is greatest by variation of x when c-x=x + y-c; so by variation of y when c-y=x+y-c, and . . u is a maximum when x=y = ^c. ,_ _, „ ■, . , a + r (icosfl + csinfl) 17. If x = rcos #and u = rBin9, «= ^ ,- , (l+r2)4 T du - or+ (6 cos e + c sin ff) du , „ ■, ■ „^ f and -r- = ^ ; I Ti = (c cos e - 5 sin 6) j , dr (i + ,ojf de (l + r»)i .•. for maximum tanfl==- and ar = Jb^ + c^, and .•. x=-, y = -\ al8o£=0 clearly, d(P Jl + r' Ja' + b'+c' and (Pu _ -3r(6co8g + (!Bing)-a(l + r°) + 3flr' ^~ (l + r^)^ ~ ^1 + rA <*'■ / o= + 6" + c=' .-. there is a maximum when ==- = -. oca 18. If u=xe^^"="'°^ -;^= - (1 + xsin «), ;^=m(1 + xco8m), ax X ay '^ .•. for maximum or minimum sin y = cos y = ± — — , and i==fV2; also^2=-(2siny + ^sm-y); .-. ^ = f«'.-i^±V2=F-^^, ^ = "(l+xco3y){2 + xsiny); 70 PIFFEBENTIAL CALCUhUS. XV. .: B=6.and t— = u {(l + a;cosy)=-a;ainy} ; .-. C= -«"•-'. x= Bin 3/==Fc''»-'. ^2; . but A = ± —pr , .•. A and C are of opposite signs, and u has neither a maxi- mum nor a minimum : {y^ means any value of y corresponding to the case in question where sin j = cos j/ = ± — j . The general value of y is 2nx + j , n being integral, when -— = = — . ax dij See Ch. XYI. Art. 239, for some of the remaining examples. 19. = dx + dtj + dz = —,.dx+-,.dy+-,.dz; x^ y^ " z' , \a „ , \b , 'Kc •••1+;;j=o=i+-,=i+,t; , . a h c , but - + -+-=1, X y z .-. sja + Jb + Jc = J^^, .•. &c. clearlj- a minimum, as for maximum one or two of the variables is oo . 20. 0=P-^.dx + i.dy + \.dz=^.^.dx+-^.dy+l^.dz; vx ay rz a he' 21. If a, 6, c be the sides of the A; x,y,z the distances from them of any point within it, xyz is to be a maximum and ax + by +cz = 2. . area of A ; hence — + -^,+ —=(i=adx + bdy + cdz, xyz 2 .". oa! = 6y = e2=i area of A. For minimum x = 0, Ac. If this point be P, and ABC the triangle, the area PBC= -^ ' ■ ■ *"• 22. If «=xj/z, 0=- + + =— 2"++, hence x' v' 2^ 1 , abc = — = — = -, and .-. u= ^ . a^ 6-i c^ g.anu.. g^^ For minimum x=0, &c. and DIFFERENTIAL CALCULUS. XV. 71 23. u=^ + q'+r', pa+qb + rc:=2 .a,te&ot A, .: pdp+ + =0, adp + + =0, p _q _r _pa+ + _2 . area of A 'a>++ a' + b^ + c' 24. If ABC be the A, D the middle oiBC and P the point required, for a slight change in P'b position when AP remains constant, P must describe a small arc of the circle centre A and radius AP, and then BF' + CF^ is con- stant and .■. also DP, ,: the circle centre D and radius DP must touch the 1" circle, .■. APD is a straight line. So if point B be the middle of CA, P lies on BE; thus P must be at the centre of gravity of the A. If AP, BP be given, CP is known, .*. only 2 of the distances are independent. 26. If ABC be the A; x, y, z the distances of the point P from the sides, the A with one side on BC=^ -. — r—-. — :^, hence a'x' + h'y' + ch' is to be a minimum, and ax+ Jy + c« = 2 area of A, thus ax=4y = «, .-. x=^ height o of A above BC, and so on, and P is .*. the centre of gravity. 26. If ^BCbethe A; rjrjrj the radii of the-fences, rj/l +r2B + )-3C=con- r . A stant, and r^^ . ^ + + is to be a minimum, .•. — '— =r, = rj = r3. 27. If X, y, z be the edges, x + y+z = a, and yz + zx + xy is a maximum, .*. dx+ + =0=dx(y + z) + + , .-. y + z = z + x=x + y, and .•. x=y = z, a cube, .■. , 28. If the edges be 2x, 2y, 2z, and a the radius of the sphere x'' + y'^ + z'^=a^, and volume = 8x1/2, .•. for a maximum volume, xdx + ydy + zdz = and - dx + - dy + - dz = y), X y z and .*. x^=y'^=z^, i.e. the parallelepiped is a cube. If any one of the edges =0 the volume is a minimum. The snrface=8(yi + zx + xi/), .•. for maximum xdx+ydy + zdz=0, and {y + z) dx + (z + x) dy + {x + y) dz^O, ... y±i+i=i±^+i=^-±y+i, X y z or x=y = z; a cube. y = Q=z give a minimam, ic. 72 DIFFERENTIAL CALCULUS, XTL 29. Let the Bide fixed in position be of length x; y, z the lengths of the other two, p be the distance from x of the vertex opposite to it ; then the volume of the double cone is - . irp^. Hence for a given value of x, the volume is greatest when p is, and .-. {y + z being then constant) when y=z, the vertex then lying on an ellipse of which the foci are the extremities of the base x. Thus, subject to x + 2y=a, the greatest volume is the maximum valueof ^ (j,»- J) = [|{(a-x)=-x^)=[|(a»-2a«), nnd .•. for a maximum o' - 2ax - 2ax=0, or a; = t , , a a Sa . 2 2 and .-. y = --- = -,:.e.x=3-j/=3Z. The minimum volume is when x=0 or -, which are the maximum and minimum values of x. 30, If a, i, c be the co-ordinates of the given point ; p, q,r the edges of the parallelepiped, the plane containing the point is - + - + - = 1, p q r and .•. - H 1- - = 1, and the diagonal = u = {p'' + q'^ + r')i, .-. as a maximum of u' would be given byy = ao &c., for a minimum, pdp + qdq + rdr = 0, and -, , there must be some inters mediate value of x for which u is a minimum, iS;c. 9. x^ .dx + yHy + :?dz = 0=c>(— + ^ + -\■ \x y z J .•. x*=y*=^i', .•. a!= ±!/= ±2= Ac, and .•. u=x* + y* + z'=3c*. If any one of the variables be±oo , u=ao , and .*. 3c* must be a minimum. 10. Let It = sin" X . sin" y . sin's, where x+y + 2= 2ir; thensinx, sin j/, sin z are all positive, and i<=0 if any one of the variables = or v; thus for a maximum = mu cot xdx + nu cot ydy +pu cot zdz, and dx+dy + dz=0; tan X _ tan y _ tan z _ tan x . tan y . tan z _ , m ~ n ~ p ~ m + n+p ~ ' where tan x. tan « .ta.nz = V.mnp, or k-= , mnp and .•. x=tan"Mm( -] > U QU TU 11. If «=x''.«'.z', 0=-^dx + --.d« + — .(b, ' X y ' z and 0=ldx + mdy + ndz, ^ Ix _ my _nz _ a * * ' P ~ 9 ~ >■ ~ p + q + r' - ••■•=c-TiT-,r-(?)'-a)*-(«)'' if JP. 3> J" •'6 *11 positive, u is thep a maximum as in Kx. 6. 74 DIFFERENTIAL CALCULUS. XVI. For the parallelepiped, p = l — q=r, &iid u= I -) . ■ — (taken positively, \ o/ linn if one or three of the quantities I, m, n, a be negative and the remainder positive). Here u=xyz and lx + Tnij + m=a, and if any one ot the quantities x, y, z vanish, u = 0, but if they be all positive u is positive, .•. the value found above is a maximum : u may also be =t ao for infinite values of 2 or more of the variables. Here and in Ex. 10 when one or more of the quantities p, q, r is negative, u may be correspondingly a maximum u minimum or indeterminate. Similarly in Ex. 10, u may be impossible. 12. = xdx + ydy, and = {2ax +by)dx + {2cy + bx) dy ; .•. 2ax + 6y=Xx, 2cy + hx=\y, and multiplying these 2 equations by x and y and adding, 2/=Xr2, and .-. 2x ('^ - o j = by, and 2y^^;-c^=ix, and .-. iU^-a\ (■^-,-c\ = h'', i.e. {b^-icu:)i* + if (a + c)r^-ip=0 gives the maxima and minima values of r, which are the - axes to the conic ca?- + l>xy + cy^ =f. If V^>i2a). For minimum x=2a, or y = 2a or z = a, or two of these cases. 15. xdx+ydy + zdz = 0=ldx + Tndy + 7idz, and dx (ax + c'y + h'z) + dy (by + a'z + c'x) + dz {cz + h'x + a'y) = 0; .-. x=\l + it(ax + c'y + b'z) and 2 similar equations, and multiplying by f, 2^, 2 and adding, r^=/i, .•. x(a--) + yc'+ zh' + -^=0, and also lx + my + 712 = 0; hence, eliminating x, y, z, \, I, m, Ji, 1 c', 6', i C, -i. -. m = C V, o', C-p, n 0; in this every term clearly involves Z, m, n in the 2"'' degree, and the coeffi- cient of J' is (ft --^ J (c--^)-a'; and that of /m is - "^ (<: - ^) + "'^' - C (c - -.2 j +«''''; and thus by symmetry the result foEows. 76 DIFFERENTIAL CALCULUS. XVII. CHAPTEB XVII. dfi dx + y^(l + ^) (c_i), but c-l = -^^ ; ■' \ dxj^ " x + y + 1' 2. y -e^ . cos x, .•. — = e" (cos x - sin x) ; .-. ^,=c^(cos x-sina:-sinx-cosa!)= -2e^sina: = 2 ( j^-2/) > • • *"• dt/ -, ^1 3 cPy dy 3. .-. x = a-,.-.l = a^„and..x-, = ^. cP i. y . e~^'"^ = a sin nx, .-, -ti'a sin nx= -r-j (y . e"*"^) - e-rai ('t?^ _ 2„i ^ + mV^ = - nS . i/c-'"^ .-. &c. \dx^ dz J d^v 5. .-. 2/=i.sin(x + a), .•. -3-^= -isin (x+o)= -y, .-. &C. 0. Here ~ {xij) = ae + he-' = xy =a ^ + 2 ^ by Art. 80. dil , „ d'v fdyY , y dy , .-. («€» + be-') (^)% ("e" - be-f) ^^=fe' + ge-^=aey + be'', (ae' + be-v) ^IJ- l| +(aei' -&«-") g=0 (a), . . Sec. DIFFERENTL4.L CALCULUS. XVII. 9. c + log^ = e'.(^); • ^_J j x ndy y\ 1 X x ^ di,) ''x \x+y\xdx iV 'c + y (x + yf {x+y)'' dxf t {x + yY.e-i=xy^-y\ y ^x = aeoB I ^ .log x + by, ■■ ^/7 ^ Jl ■<** V?'^^ Jx'ijl 77 10. s/i y JT = -«-(4^--^)-C=-|^: and 4x» d'y Sx y " 7 dar^'^di" 7 "'"V" ^' and and 11. = ax + 6(y + »:g)+cj/| (1) ; ■••»-">("2-S)-l(g)'-3i. »-'('S-3)-M-2*,g). •••(^J+'-disj(3'di-^i -"-'di-0 _/ dy d?y d^y\( dy_ ,^ 'dx ' dx\ ;)■ 78 DIFFERENTIAL CALCULUS. XVII. .-. (y-^ j^) -ni + S^fj-^) =0. ••■ the other result x-^-y=0 does not agree with equation (1). 12. Differentiating with regard to x, and with regard toy. _ ^^ . J=/'(l- ^)( - i^); 1 di J •^ = « {'(ay + hx) + ,j,' (ay-hx)] ; ' ' dx dy x + y ' 15. 3- = e^ sin y . 0' (e^ sin y ), -p = e°^ cos y' + bx" + ah/,/,", d?z j-J-=b4'' + abx" + af + abyxj/', ^^ = 2b:f/' + bhjf" + b'x4>", and the result follows. 19. ^ = «'"(nF+f") + «-'«(-«/+/'), ePu ^ = e« (n^J' + 2nF' +F") + «-« (jij- inf'+f) and -P=e'"'.F'-e-'".f, .-. ^=6*". Ji"'+«-'« ./", ax^ dy dy' 20. (1) X g=coB (log X). ... 2 + X g= - sin (log .) . 1 , (2) ^=cotx, .-. ^^= -cosecaa;= -(l + cot^x), .-. &c. i'^- di—^->''dy=y^y-t'' dz „dz „ "d^ dx '' *0 BIFFERENTIAL CALCULUS, XVII. 22. O=/(z.)+x/'(.).g + 0'(.).g, !=./'(.). | + ^'(.).|....£h-J=-/(.),...*c. by Ex. 16. 23. | + „. = 2/'.j(.-a) + (.-c)gj. g + „=2/'.j,-6 + (.-c)|j; then eliminate/', 1 dil and .-. Szc. du_d4, d^ (3z_6f\ I dd> , . d /3y\ , du dip , ,, du d0 di du ^ du dii „ '•• ^d^+2^di = -dt-^^' •■•*'=• „„ 1 du 1 1 idF „ dF , ,) u dx X F (d/i d/j "^ 'j and 2 similar equations, .•. eliminating — and --r ■ v-z du z-x du x-y du y-z z-x x-y ■- — + -r- + — •'- .^ = ^ — + — + ^ or, &c. u dx u ui/ u dz X y z T. D. C. K. 82 DIFEEKENTIAL CALCULUS. XVII. do. Taking x as independent variable, as in Arts. 196 and 197, ' di--'>'- d^'^\Tx) <^^' ) is which is the same for all values of n. . o""' , , , V v' x' n - 1 8. «'-V=(i>!^-x)— i:r,= (x'-x) -, or ^ = . °- n 3 \ I jjj,»-i ^ ' jix y nx n Hencethearea ="2 V~S~ j '"^^ "2* ""n'o"^!' which is constant for all values of 1/, if n + l=0, i.e. if the curve be a==iy; (a rectangular hyperbola). 6—2 84 DIFFEBENTIAI, CALCULUS. XVIII. D.Here cot^= -^ = fL'f , ... E^ = !!£.* = i , and .'. the normal is ^' cos ^-z' sin ^=y cos^-xsin^ =cos If) . a cos^ ^- sin . a sin' 0=a coa i, or dropping the dashes, y cos ^ - z sin =a cos 2^. 10. Both curves pass through the origin, and their tangents there are given by the terms of the first degree ; thus the parabola touches one branch of the cubic and cuts the other branch at right angles at the origin. The curves also meet where x' = axy, i.e. 3?=ay=J'ia'x and .•. a?=a^ .2 or x=a 1/2 and y = a^i: and if at this point, 8, 8' be the angles their tangents make with the axis of x, tan $=- = -r--^ ; and y J^i S {x"- ay) + 3 {7/^ -ax) tan 8' =0, .: taue' = 0, and .*. the required angle =tan-i(— — | 11. Let the eccentric angle of the point on the ellipse be 8, then the 2 points are (a cos 8, b sin 8), (a cos 8, a sin 8), and the 2 tangents make angles 52 fi tan-' m, tan-' m' with the axis, where - m = — . - cot 8, - m' =cot 8, a^ b ' .•. tan^=cote(l-- J-=-M+-cot=e), or (a-i)cot^=atanfl + 6cot9, and for a TnitTiTTinm of tf>, a sec- tf = 6 cosec' ff, and .•. tanS= \/-. thus la,n ip = [a -b)-i- {2 Jab',. When 9 = or ^, ^=0. 12. (.-.)S.(.-.)g=0...^.^=l. i. e. (x, y) lies on n°=l, and the coe£Bcient of z' gives 2mc = 2am^, .-. c= ±a and ni= ±1 and y= :^(x + a) are asymptotes; also the coefficient of y* gives x = 2a vrhen y is od (Art. 277). 2. If y=mx + c be an asymptote {mx + c)' = x* {2a - x) when x=od ; thus ni' -1-1 = 0, .'. m has only one real value, viz. -1, and 3m-c=2a, .-, the real ... 2a asymptote IS J' + ^ = -o"' 3. If y=mx + c, in'=0 and c=0, .•. y=0 is an asymptote. This is also given by the coefficient of z'. and that of y gives 2 imaginary asymp- totes, and there can be but 3 in all to a cubic, 4. If y=mx + c, {mx + c)-(am + blx + ac) = a- {mx + c^ + Vx'', .', for asymptotes j»-(aiiH-6)=0, i.e. jn=0 or — , and 2cm(am+b) + m*eu:—m'a' + lfi, 2ft" .. m=0 requires e = oo, and ajn-l-6 = gives c = — •=2a, .•. the finite a asymptote is y= — x-i-2a, the other 2 being at oo , 5. If y=mx + n be an asymptote, jn'=l, .•. m= 1 is the only real value, 2a + c and Sm''.n=-(2a + c), .•.n= —. .,,».. „„(,-.)'(>-:-)'..(.-S-y, 2a + c when x=ii>, .•. y = x —, Aliter: by last paragraph of Art. 274. 6. The highest powers of x and y give the axes as asymptotes. UBoy=mx+c gives m»-nn = 0, i.e. m= - 1 for the S"" asymptote, and 2mc + c=0 or e=0, i.e. x + y=0. 7. The coefficient of x* gives y=±a for asymptotes, and the coefficient of y^ gives 2 imaginary asymptotes, and there can be but 4 to a quartic. 86 DIFFEBteNTIAI, CALCULUS. XIX; 8. The coefficient of y" gives 3x+a=0 for one asTrnptote: for the other 2, let y=mx^c be one, then x^=a3? + {a-^'ix){mx-^ef, and .-. l=3m' or nt= ='=~7=i and 0=o + am- + 6mc, i.e. c= -a . — =-±2 Ji = t •/=. and the V3 <* 3 v/3 other two asymptotes are, . . 2/= ± ( -p = ) . \i^3 3 t^'AJ 9. The coefficients of i' and ^-.giye x + a = and.^ + &:=0 for 2 asymp- totes, and for the remaining one, ?/= mi +c gives (i + a)(ini + c)2=x»(mx + c-|-6), and .-. m==m or m=l, and a)B2 + 2j7«;=c + Zn .•. c = 6-o, and the asymptote is y=x+6-a. 10. If 2/=mx + c be an asymptote, JCmv 2) x + cj {(ma:+<:)''-a^} -a {(»B- 1) x+cF4-4a2 (x+y)=a', .-. (m-2)(m2-l)=0, i.e. TO=±lor2, and c(m='-l) + 2mc{m-2)=a(m-l)'; thus if m = l, -2c=0, and the asymptote is y = x; if m= - 1, +6<; = 4a, and the asymptote is x + j/ = 5a; and if m=2, 3c=a and the asymptote is 2/=2x-(-;- , 11. If 2/=mx + c be an asymptote, (mx + e)= (m -\\x-vcf- ai^ (m- 1| x + c) - 3aV= sn^ •'• tli6 asymptotes are y—x-v-^ (1±^13). 12. If 2' = mx-t-c be an asymptote, a? - 2 (ntx -I- c)3 - 3z (mx -h c)" - o%^- 20=2, = a', and . . l-2m'-3m2=0, i.e. l-m2=2m2{m-(-l), and .-. m= -1, or 2m--fm-l = 0, Le. m=^ or -1; and Gm-cH-67nc=0, »'. if m=-, c=0 and one asymptote is 2y=x; but if m = - 1 the 2 other asymptotes are indeterminate, and proceeding to the- coefficient of x, -6ka iJ'A DIFFERENTIAL CALCULUS. XX.' 87' • 14. The coefficient of y» gives x=±a tat 2 asymptotes; and for the other2, if y=ma; + c, <(m-l)a! + c[»(x2-o2)=a<, .-. (m-l)«=0, 2c(m-l)=0, and c'=.a^(?n-l)?, .*, c=0, and there are 2 coincident asymptotes y=x, 15. If y=7na; + c bean asymptote, {mx + cf-ix(inx-^cf+ii? + a{mx + cY+ax(mx + c) . _ , -6aa:2 + 26»x-6»i/,+c5=0, ... .-. m'-3m2+4=0, i.e. m + l=0 or m'-m + l-8m + 3 = 0, i.e. m*-4m+4 = (m-6)''=0; and 3mV-6mc + om'+am=6o,. 2a .-. if m= — 1, 9c = 6tt or an asymptote is x+y = — ; if m=:2, c (0) = 0, and c o ' is indeterminate, and then ichn - Sc^ + 2amc + ac + 2b' - 6%= 0, or 5 3c'+5c.o=0, i. e. c = Oor -jO, • o 5 and the other 2 asymptotes are y = 2x and y = 2x- ^a. o 16. By Art. 274, if y=mx + n be an asymptote, on substituting this value of y in the equation to the curve, the terms of the 3"* and, 2'"' degrees must vanish; .-. if y=0 be an asymptote, these terms must be divisible by 1/ ;- similarly they must be divisible by a;, if x=0 be an asymptote; and .". tlie equation is of the form xy {ax+by + c) + a'x + b'y + c' = Q: also if dx + by + c=0\ the same terms vanish, and .. ax + by + c=0 is the 3"'^ asymptote. CHAPTER XX. 1. tan0=-;— = — = tan6, .: with the fig. on p. 304, d> = 8. 2. i/isu maximum or minimum at such points, .*. — (r sin 6)=0 = 2cos''* + cosfl-l, .*. 4 cos 9= -1±3, .•. cos9=- or -1, and 6= ±.j or tt. 3. r*-^ = - -; . r'= -a=constant. T in the fig. on p. 308, will fall on dr r* the opposite side of 5 in this spiral. ob ^ . . . ab. , . , ,,d$ , 4. i -=««?+ be-A, ..\ .- -T-.= (aei - be-f).^. , .•. &c. T T* HT J 88 DIFFERENTIAL CALCULUS. XX. 5. Comparing the 1" figure to the ellipae in Todhunter's Conic Sections, the equation being - = 1 + e cos e, if P, T in Art, 285 be (r, 6) and {p, if) ; dr 2 '^ €BinS ecoa' .•. the locus of T may be written x=- , ,; &c, G. (1) If - = l + eeo8e, tan^r r Also eBec''^.^= -73 (cosec fl + e cot 9) = - cosec ff cot ff - e cbsec^ fl ; aS do .'. for maximum of (j>, cos 6=—e, and then e sec' . -y^ = cosec e . cot' 8 + cosec' $ + 2e cosec' d . cot fi = ^^j , sm^'tf .-. for a maximum sin 6 is negative and .-. fl = ir + cos-^e, and tan tti= - or d>=ir-cos~^e, and r== s=ai ^ e 1 - e' or the point is the negative end of the minor axis. The obtuee angle between the tangent and radius vector has the same value at the other end of the minor axis. aSftJ (2) The ellipse is a" sin= B + V coa' 9=-^-, .-. {a'-b')BUx8coB8,^=-~; dr •n ... (a=-l»)tan.^= -?i5^e^t^^*= -(a'tanff + J'cotO), ' ' Bm S cos e .: for maximum of 0, - (o' - ?>') sec' ^ . -^=0=0," sec' 9-6' cosec' 6, dd and .-, tane=±-,and then - (a' - i') sec' ^ . -^^ = 2a' sec' 9 . tan 9 + 2&' cosec' 9 . cot 9, „ h , sin 9 coH 9 1 .-, for a maximum tan 9= - , and — r— = = - * " Ja^ + b'' a' +6' .•, r'= — — , or the pomt is an extremity of an eqm-conjugate diameter, and the obtute angle between the tangent and radius vector has the same value at all 4 of such points. DIFFERENTIAI, CALCULUS. XX, 89 7. Tan^=r^ = ^^=tan|. .-. 0=| (ct a figure); and ,'. ^=rsin0=a(l-cosfl)sin -= 2a sin'* (^j : Q 6 and polar 8nbtangent=:rtan0=a(l-co8d). tan - = 2a sin^ ^ . tan - . 8. cos2#=-5, .•. taiid>=r— =:3-.— si« e? a? and sin ^= , = — by the given equation, 9. r=-a^Bin2fl4J, .-. tan0= _^-^^= -cot2«= -tang-2(») , •••* = - (1-2") = ^+2". The general formula gives 0=jwr + ^ + 29, but when 9 ia small ^<:x obviously from a figure, . . n = throughout the curve, 10. If rbe (r', &), e' + 0=^-ij>, andr'=rsin^i also l = oBec'5tan5.-^, .•. tand)=cot ;, and *~2~3' r=rcos-=oseo''g, and g= -^ ; v. r' = o sec^ g , i. e, r' (1 + cos 6') = 2o, a parabola, 11. If r be(r', ff"), e' + e=^-4>, and r'=«>8in0; (2^ also 1 = -osinfl. -V-, .•. tan^= -cotg, and ^=2+2. ••• " = - 2-.ai"ir'=rcos^ = 2o cos' 5 = 2o cos' 5- , i. e, &o, 12. If rbe(r', »'), e' + e=|-^, r'=rsin0; and (cf, Ex. 9) ^=^+29, .-, «'= - 3«, 2S' and r'=rcos29orr^=a"coB'2e=a'cos'-^. 90; DIFFKRENTIAL CALCULUS, XXI. 13. r is aj when eo8 6=0, .•. the asymptotes if any are parallel to x=0, and of the form x = e, where c is the limit when r=to , of r cos ff, whicbilt. of a cos 29 when cos 0=0, i.e. the asymptote is rcoB9=a (2 oos'9-l)= -o. 14. When r = oo , sin 9 = 0, and the asymptote is r sin fl =lt. of 6 + a sin 9 when , Bin9 = 0, i.e. rfline=fc. 15. Here r (cOB''0-8in^e)=a, .•. the asymptotes are r (cosfl±sinfl)=lt. of a-^(cos9T8infl) when cose±sintf=0, and when cosS + BinS = 0, cosff= -Bine= ±— ^, v^ : .-. T (cos 6! + sin e) = ± — r- ; bo the other asymptotes are r (cos e- sin «)= ±-— . 16. ..Here 4.rcose cos29=a (cos29 + 2cos'e), .•. one asymptote is 4rco3fl=lt.ofa(4cos'0-l)^cos29, when cos fl = 0, or 4r cos 9 = a. For the other asymptotes • „ 1 cos »= ± Sm e = dr — = , J2 the ambiguities being independentj-thus-if cos 9= sin 5, ' 4r (COB e- Bin *) = a (2 - 1)-=-(±^2) /-± -^-] = a; so if 003 9= -sinff, 4r(coa5 + Bine) = a. . . ,.. CHAPTEB XXI. , ■ 1. Near the origin the cnrve approximates to the axis of x, and ulti- mately when y = tha^t axis and the cnrve coincide, .•. i/=0 is the tangent at the origin, an^ y and x are of the same Bign, .-. there is a point of inflexion such as Q oa p. 315, at the origin, . . „ „ ■ dy 3x= 2x* Generally, — dx a' + x^ (a2 + a!2)2» {a-+x'')'.p'^+ix (a' + x^) p- = 6a=x + ix\ and, when f\ = 0, {a' + x^)'.. jK + iia\+ 12x,')|^=6a= + 12x=, DIFFEEENTIAL CALCtJLtTS. XXL SI .'. for points of inflexion (1) x=0, and then a*—^ = 6'a,-,'oi -t4 13 finite and y=0, .*. the origin is a point of inflexion : ■ . .-. 6a^=2a%i» and x= ±o JH; and then (4o»)'.g = 42ai'-40a^.?^= -3a\ .•: g is finite, and there are 2 more points of inflexion, y being possible when i = ±o jjs. . 2. ay{x-a)=ifl+ax^, .: a-^ {x-a)+ay=Zx^+2ax, I dx' ' dx ' - -. . .'. for points of inflexion, 2d{x-a)^^^{x^a)(6x + 2a) = 2{Sx^ + 2ax)-2'^^^^, or ■{x-a)^{3x + a) = (x-a){3x- + 2ax)-{x^+ax^),, '■ whloh gives (a; - a)*= - 2a' and .•. x = a{l- ^2): and then d^y dhi a J— , (x - a) = C and ~' is finite. djJ ' dx^ 3,. (o*-6<)^=8(x-a)'+12x(x-a)», .-. fqrpointsof inflexion x=a, or 20x = 8a, and .'. x=— ; s and(o*-B<)^=36([x-a)2 + 24x(x-a), .-. ^=0 when x=a, and ' ':^A«-k-^-L\ 2a , . which is not zero, when x=^, which, .•., gives a point of inflexion, unless a=b when the curve reduces to straight lines and there is no inflexion. For x=a, (a*-6*)^=96(i-a) + 24x which dded hot vanish \rh&ti"x=d,'.'. there is kid point of inflexion theii. 92 DIFFEEBNTIAL CALCULUS. XXL ,.. for points of inflexion, g)' = g = (g,)' . ^— , and ,•. 4(a — a!)=o or a:=V. "nd .•. « = — ^: * ^/3 2i/ tPu 6rt (f'l/ . . ., and then - , . t^,= - -n «' :ri3 '* finite, ^ 1 dy 2x 3 -4 , ._l a ox o'' 5 1 2tt»'dx2~(x' + oV (x^ + a^)^' 2aa'dx»~ (x^ + o*)"''' (x^+a^' .-. when^{ = 0, 3x»=a''or x=^~,B.nd,t'hen }-^.^ = j^^^.{x'^-a^) dx^ ^ 2a? (tr* (x^-j-ar which is negative or positive as x= ±—p. , ,•. there are 2 points of inflexion, \/3 DIFFEKENTIAL CALCULUS. XXL 9S 8. 2/=(2a-z)=o=x, .: 2tj ^ {2a - x) - y' = a\ ..when 3=0, .,(2a-x)g=6(g;, and (2a-x)^=2ij, .-. ii/^a'+f, and .•. (2a-i) = 3x and x=- and ~ does not vanish, ,-. dtc. . dy a^ tPy „dy a" 9. x^+y= — , .-. X5-^„+2/=--^,and.-. when dx X dx^ dx x^ 3-.j=0, 2x-2= = 2w, .. xy = —-, dx' ax XX •'' •'2 and .-. x = o«^; also ar ^ = -5- or --* is finite, .-. d-c. 10. 2/ = 2o7.xJ±2v'^, .-. ^ = |atx-t±ai.x-4, ux o '^ 4 5 _4 1 1 _j d^M 20 2 _B 3 1 _r, hence for points of inflexion 80^ - 9a4 x« = ; /8\8 , dh, vanishes when 8O0' - 8I0' x^ = 0, and .-. is finite for the value above fouad. 11 y _ "-'g . J^ (fy _ -1 2x(x-o) ■ a-~x- + a'' ■'■ a2"5x~x' + o»''"(x2+a2)2* 1 d V _ 6x-2a _ 8x'(x-n ) a^ ' dx^ ~ (x'+d'f ~ (x^ + ay ' h ^-1 - ^ _ 48x°-24ax 48x°(x- a) . a»", / V V / 4 a±;^3 = -j(-l±^/3)or3,=?(lT>/3), x-o(2+ fs) y-lc'--'^^^ and the straight line joining the 2 last is ^ — _—' = —^——^—^ , ia ^6 _ o I- or a-a(2 + ,y3) + 4y-a(l-^)=0, i. e. x + iy = ia which is satisfied by (- a, a), and .-. the 3 points of inflexion lie on a straight line. N.B. Generally if there be 3 points of inflexion on a cubic curve, they lie on a straight line. Cf. Ex. I, 12. Hereau = l-^, .. a ^« + — j=l-- - ?^, ,-. for a point of inflexion e''-e=-6=0=(fl'-3)(ff' + 2), .-. e2=3 and r=a. |. 13. lu=e-^, .: h («+^) = ff-'+n (n+ 1) «-<»•«), and .-. for a point of It inflexion 9^= -n(n+l), and r=6 . e"=6 {- n (n + l)i». For real valnes of 6, 11 lies between and — 1. ■• , ^x . dy , , 14. -j-=aBin^, -^ = a (n + cos ^), dtp dtp ■-■ -,—, = — r^.^r- (ncoseo0+cot^); ax' a Bin 4> d=a'x', .*; there is a double point the tangents to the 2 branches being y= ±x. Cf. Art. 347. 2. 11) ^=-Binx=Oat {0, 0), and^= -oosa;= -1, .-.there is a point of inflexion. ' (2) '-3^=cos x-asini, .-. — ^= -2 Binx-xooBa: = at (0, 0), d^y and 'SZ3~ -3'!0Ba;+iBBinx= -3, .•. a point of inflexion. (3) |^=sec2x, .-. ^ = 2sec=x.tanx=0 at (0, 0), dhi and d^—^ sec'xtan^x+2 sec(a) and changing the origin to (a, (p (a)) the curve is i/ + d)(a) = d(x + a)±x ^ .F{x + a), .•. x must be positive, and expand- X? ^- ias,y=x'(a) + -^ 0"(a)±x'« . {F(a)+x .F' (a)}, .: there is a cusp at the new origin the tangent at which is y=x ,^' {a), and the cusp is of Iho 96 DIFFERENTIAL CALCULUS, XXIL Sp+l 1" kind it x^ . J" (a) is of a lower order of small quantities than J.«."(a)i.e.if?|±i<2. ''' - 2p + 1 assuming that x't .F (a) is of a higher order than x . 0' (a), i. e. -^ — > 1. If -^2 — > 2, the cusp is of the 2"'' kind (cf. Ex. 3 (3)) 5. Making {a, 0) origin the curre is y^ = x'+{a-c)x^, ,•. x=0 is a double tangent, and near the origin ar'3i' {a^ - ax), i.e. if x be of the same sign as a, .-. the 2 branches are on the positive or negative sides only of the axes, as a is positive or negative, and thus there is a cusp of the 2°'' kind. 9. The tangents at the origin are given by y'—O, the homogeneous terms of lowest degree, .•. there are 2 branches which are real or imaginary as a-x*5- or < 4x* (a'^ - ax), or near the origin as a^> or <4a-, .■. the branches are imaginary and (0, 0) is a conjugate point, 10. Here (x' - a^)^=2ay' + 3a^y', .-. making (±a, 0) origin, the curve is {a? ±i2ax)^ =2ay^ + 3a^y^, .-. tangents are given by 4a=x2=3ay, or ^= ± A/f = ^ here : and there is a double point. When j/= -a, (x'-o°)''=a'', or x^=Q or 2a-. Changing the origin to (±a sf^, - a) curve is (x=±2ax V2 + a2)a = 2a(y-a)3 + 3a!(y-o)=; DJFrERENTIAI, CALCULUS. XXII. 97 .•. terms of lowest degree give ± 4a'x ^/a = o^y ( + C - 6) = 0, or i=0 is a tangent. Also changing origin to (0, - a), carve is (:c2-a'')2=2a(3/-a)3 + 3a=(y-a)«, and tangents at new origin are given by -2a=x==aV(-6 + 3)org = 2,,.|=±y^. 11. Making (a, 0) the origin, thecurveisa7^'=x^(j;+a- b), .•, the tangents are given by ay'- = {a — b)x-, and they are impossible or the point is conju- gate, if a<& numerically and a and 6 of same sign. The tangents are real and different, if a>b, and are coincident or there is a cusp if a=b, viz. ay*=3i?, of 1" kind. 12. At (0, 0) the tangents are given by ay'^ + bx-^O, .-. conjugate point if a and 6 have same sign, otherwise a double point or a cusp if either a or b = 0. Also for point of inflexion 2a,g = 3^-2.x,.-.2.(|y=C.-2..(...g=0); .. (12-9)x=(16-12)iorx = --''. ^-S-^Ki/-^^"^S=«^-^^' „ dy d-u „ oint3 of inflexion, ± -=^ = - x4 . (1 - a-) 2 _ - . x^ ± d'^ -^ '' ' 2'(l-x)*' j-=cp, and at the lowest point a:=0, .-. y=c, and .•. /)=(; or &c. 2. The tangent at the origin to the carve is given by the terms of lowest degree, i. e. y=0, and the curve approximates to y= - 18x', .■. y is negative, and if (x, y) be also a point on the circle of curvature at (0, 0), from a ^2 1 figure it will be seen that the limit of — = 2p, and .-. /)= ^a • — y 30 3. $' = G + 10« + 3x», ^,=10 + 6x, dx dx' .-. when x=0, /) = — — — , .-. iSrc. by logarithms. Also generally /) = {! + (6 + 10i+3:(;=)2}^-r (10 + 61), and is .-. 00 when a; = oD 5 or -3. _j d^^d^ dy dx ' dy ' dx~ ' dx^ dxdy ' dx dy^ ' \dxj dy ' dx'~ ' /d^Y_d;0 fd(t>y d?=ii and p=rsin0=2asin' = , dr dr dp „ • ,^ • - o • 2* B ia . 6 d« ^ , 2cose-l 7. By Art. 32D or thus: r^^ = tan ^= -g^;^ • 2cos«-l , r r' .*. Bincose 3. If they touch at (x, y), -^ is the same for each, and if Af^ ^!' = 0=^ + — .--, dx dy' dx dx dy'dx' and eliminating -r the result follows, for the particular values of x and y. dx 9 ^f.-\ £^ = 1, # = ?x-J. ^■=3«-i, .-. they touch at (x, y) if aj,4 = l)x4; and on the straight line where ^3 = |sBach=^^-_j--j„; and on the curve where ^ = '^ each equals (a^ + h"-)-\ and ••• ^=| = ^i-i=' ""^ '^« same values of * and y satisfy the line and curve, and .-. they touch. 102 DIFFEREJTTIAL CALCULtfS. XXIV; 10. ^ 13 a minimmn or maximum and p=r sin (p, dp . dd> . n ar ^ ^ dr ^ r .-.-=- or (fee. r p n. p= ±_ , COS i,=^. Sin ^=^-1 ; .-. 2aco,^=~ + y, ^ ds \ y^J ds p y"- 2a?/2 ,, J., ^ , X, , dx 1 !)» + «' and p= - , 5 . Also the part of the normal = y-i-^- ; .. - = -^ — " , "^ y^~b' ' ' ds n 2ay^ and .•. - ±- = -, ±asi/''>or mast be altered. 12. r=ocoB 9 is a circle of radius ■n=P- 13. 5- = - =eos ^, .-. ditierentiating as to«, - sm^. ( ± - ) = ^ - ; .•. =Fp=-j • - or, taken positiTely, p= - . ' - 14. EromArt.820.1.g|%,g=.g. and ^=2x+2,£, .-. ^=2 + 2 (^Jj +23,^ ., dhi dV „ •■• ^* dS" = dF • -^y symmetry . d^idV dj/2 ~ d^^ ' IC. (x-a)' + (y-J)==p2, .-. (2-a)|?+j/-6=0, and , , d^x /dx\* , . , „ dx ^ ■ d^a 1-. jr- =-— + — + 1 or o=2m+3i; 2m 2ni m and ,=, + (._„)|. = |.(_2.)=-?|^, and .-. p» = (a:-^)sfl + J^,\=4(m+x)=^?orp=^i!^ll DIFFERENTIAL CALCULUS. XXIV. JOS Hence the circle of cnrTatnre at {x, y) on y'=imx being (x'-a)= + (j,'-6)»=p» cuts the axis of 2 in points given hj x'''-2ax' + 0.^+1' -if = 0,h\xt in m m Trhich is negative unless z=0, .'. &c. IG. -a+(.-.)|=0..-.l.|V(,-.)g=0. and So • f. M-B)v M-B)y' (B-^)V " BG • 17. .•. tan^ = -, .•. sec'ii. — =--;, B a» $- or p=- .Bec2f=- (1 + -^) = . 18. "When j/=0, k=0 or 3a, and the curve is y^ (ia-x) = ax{3a-x), .•. at the origin the tangent is 2=0, and (cf. Ex. 2) y» 3«= . 3a ■^ X ia 8 If the origin be changed to (3a, 0) the curve ia given by 3/2 (a-x)=aa:(3a + a;), and at the new origin 2p=lt. of ^ = — , .•. p= -^ • ■^ X a 2 ds 19. — = p=n» . cot }f/, which can be expressed in terms of s or ^ by the given equation. 20. x = is a double tangent, and the 2 branches are y-= ±xijia?-x'', and 2p=lt. of — = ±2a, and thus the 2 circles of curvature at the origin are x^-2px + 7/=0,i.e. ar'=f2aa; + ?/»=0. - 104 CIFFERENTIAL CALCULUS. XXIV. dy y dy dy' y^ and ... „=(^»^^«=(£!+_2'_l^. '' y^ ' y"^' ay 22. For the circle '2-S-('-¥)g=».-"(i'i)' dV , d'y i , dhj 24 5^ = - 1> t4 = - . and T-i= - -f dx di^ o di' a* Forthe parabola —= + —;=.-/ = 0, or -^i = l-^; ^x Jy ^ ''^ V« . d?y_Ja d?y_ 3^ , . /a a\ ••di'-^rri^""5x^' U'4;' — = - 1, T-.; = - and -3-4.= ^ , and .-. &o. dj; ' dx-" a dx' a^ ' 23. r=aBec'2, .-. - = .'. p=rsmerbola through the origin. 35. From the figure in p. 349, the radius of curvature being a tangent to the evolnte, there is an asymptote to it when the radius of curvature is go ,' dhj for a finite point on the given curve, and .•. -,-|==0, and .•. in general tlie asymptotes to the evohite correspond to the points of inflexion on the given curve. Here j-=osec'x, ■r;^ = 2aBec2'x'. tanx, .•. tanx = corre- sponds to the asymptotes, i.e. y=0 and x=nTr, and the asymptotes are ftj-Fx=Hjr, 7i being integral. ' ■ 36. a^ . y= ±x^, and the 2 branches are similar; for the positive value of y, a^ . -z- = t: x^, a* . 3"^ = -r ^ > •'• i^ (^'> V) °° '^® evoliite correspond ax £ dx^ 4 108 DIFFERENTIAL CALCTTLUS. XXIV. , , , , l(°' + T''7 xi 4 /a' 25 r.\ to{x,y) y'=y + —^ -=-^+ — ;r{-r + T'^')' — x^ i or approximately y': 15 Jx 5i* «' + X'^ 2 and x'=x -. r=i--g nearly; 2aif 15 s /- ^ i .•. 3a! =x= I =-;^, I or xy^=c This fis y' may be negative includes the evolute of the other branch : and a may be negative and then x is negative, and c^ also. 37. If a* = - , J/ = c V + c'x^, .•. ^ = 2c2x + ^ cxv, 6' ax 2 and 2=2=' + x-c'-*: da?' 4 hence, if (j;', y') be the corresponding point on the evolute, ^\dx) „ „ , . l + 4c DIFFERENTIAL CALCULUS. XXIV, 109 and here co8- = e ... . a a ax ^=oot?? , and .-. f^= -cot ?^ .i . cosee=?, 4x a dx' a a a ncot - and ,-. x-;i=- " fl + cot°-^=-a, cot^.cosec^^^ ">' a a and the chOTd =ia. For the evolnte x=li-a, approximately = - J^ nearly, and ,; -''J-=(—\ ; hence sec ( — ] — ) =e " . 39. Let the common tangent and normal be taken as axes of x and y, then at the point of contact ^- = 1, -^^ = 0, and by Art. 95, if on the curve as tU x=f(Ss), and y = (t>(i»), i=/(0) + S*./'(O) + ^'./"(0) + andso j,=o + a..-' + ^.^ + ^.^f + ; l_'(i=.V ix _(Py , l_d2x dy _1 ^£-^.^^_^ rf'x , fdx\^ _d?y ■"■ p''de~d? ' ds ds^ ' da'' ' \d8 J ~ ds^ ' , 1 dp_cPx . d?/ d^x d'y^/dy\^ ^'^—n '^-'di~dF'^cb~d^'di''^\d^)'"ds^' , 8s< d°°t°=^^'^°"°"°°°'^, a(l-c'Bm''u)i vi-nta 36^£'' cos2u(l-e28in=u) + 5e»sin'>wcos=w and Keot'a.e'''°^'^= . 5 . " (l-e«8in''w)* Hence cot o (1 - e' sin' w) = Se' sin a cos u, and cot o (1 - e'' sin' u) sin a cos a — cos 2ii) (1 - «* sin' w) + Be' sin' a cos' u, .• . 3£' sin' u cos' w = cos 22=2j/2, and .-. c*=4xy. - = -, -5^ = r- , and .-. X=2; a ¥ b 4. If thelinebe -+^ = 1, na + 6 = c, a b .-. 4, da + J^d6=0, and nda + db=0; a-" if' Xx Xw 1 , , X .-. -= = n, -,| = 1; .-. X=na+6=c; a* 0* .•. 0'= — , b-=cy, and .•. Jcnx + ijcy — c, n or (nx + y - c)' = 4nxy , a parabola. 112 DIFFEEENTIAl CALCULUS. XXV. 5. J/ = m (z - 2a) - am', and m being the parameter, 0=x — 2a — 3am'; .'., eliminating m, y=^ Y''-^(x-2o).|, .-. 2701/2=4 (x-2a)3. 0. The equation to the normal may be written y cos

= a cos 2ip, .-. 94 being the parameter, 3/ sin ^ + z cob ^= 2a sin 20; hence y=a{shi sin 20 + cos 0), x=a (sin 20008 + Bin0), .•. a: + ^=a (Bin0 + cos0)^, a;- j/=a(8in0-coB0)^, or ( — ^1 =sm0 + cos0, ( 2j =Bm0-oo80; squaring and adding I ) + ( ) = 2 or &c, j;2 «5 ^ da db „ 7. — .da + ^d6 = 0, and — + — = 0; a^ M b an hyperbola with the axes for asymptotes (rectangular if the axes be so). 8. The normal at [x, y) meets the axis in {x+2a, 0), .-. the straight line perpendicular to the normal is y'=m,(x! -x-2q) where m=— or 4=c2, the envelope would be the plane of xy. Hence with the above restriction (equivalent to 2 extra conditions), i' , V^ „ r. , ij,i ^^ ^'/' . -,.da+\-.db=0 = ada + bdb, .-. -^=a, ~ = b, a" b' a' b' .•. X=c^ .•. «'= ±cx, Zi'= ±cj/, and . . ±cx±ci/=c-, or X ± y = ± c. Any 2 other independent conditions would bring the question vrithin the definition of Art. 334. T. D. C. K. 8 114 DIFFERENTIAL CALCULUS. XXV. 15. If (h, k) be a point on the ellipse, the corresponding line is T + I = 1 ; thus for its envelope p . d/i + ^^ dfe =^ j and .''il' + — = - ^~ = - ^y-''. . IG. If {x' , y') be on the 1" ellipse, the corresponding chord of contact is Tx' t/tl' -^ + '-TT = 1, ■-■ for the locus ^^.dx'+l.dy' = Q = %^dx' + y^; a-' h- •' ft2 ^ ft'-i ' Xj; x' X« «' , , , , h- ■'■ 'T = Ti' T! = rii • • X=l, and .•. a; = — ,x, y' = Tiy' *°^ •■• the locus is — j- +^=1. u a* 1/ 17. If (/t, /.) be its centre, such a circle is x' + y'=2hx+2kii where . . forthelocus xdh + ydk=0=a'kdk + h^ {h- a) dh; .-. a'h=\y, b^(h-a) = \x, and .-. x= + ,/=?^+2?^ + 2«x, or X = a2[,2 (x2 + y2 _ 2ax)-~2 (a^x^ + iSyZj^ .-. a*b*(x' + f-2axy = ia*b*{a^x^ + hhj-), or (x2 + y2-2aa:)2 = 4(aV + 42i(S). fc- it' 18. If {h, k) be the centre of the 2"'' circle — + ^.=1, and its equation a' p is x^+y'--2hx-2ky = 0, .: the common chord is x(h + a) + y {k + b) + c=0, .-., for its envelope, xdh + ydk = 0= „- + — -r ; a' IS' . . AX=-, , XW— -;, and ... X-{x-a--t:y''^-) = l and X(xV+y'j3=)+ax + 6y + c = 0; .•. (ux + by + c)- — x'a* + y"^. DIFFERENTIAL CALCULUS. XXV. 115 19. -yco8fl + xsin9 + c = c8inelogtan( j + - J (1); .-. j/8in* + j;oo8«=ccosfflogtan ( j + - ) + 2SmOcot(^j+^J8cc-j + - ., . /ir e\ sin 9 = c cos e log tan ( - + - + c ^ ; \i 2/ cosff ,-., eliminating the log. / ^ sin^eX y—c\ 008 ffH 1 or « cos 9 = c, \ COBfl/ "^ . . by (1) ^=log tan (j + . ••• «° = j cos tf e« + e « = sin e' cob' 9+ ( 1 - sin 9)' _ _2_ _ 2?/ cos 9 (i - sin e) ~ cos 8~ c ' 20. If (see fig. p. 304) S be the pole, and P, Q two adjacent points on the epiral, and the perpendiculars to 5P, SQ meet in 0, is ultimately a point on the envelope, and SO is the diameter of a circle through P, Q, .•. if P be (r, «) and O be (/, fl'), / OSP + y + 0=Tr, i.e. OSP=^-<,, and .-. 0' = 9 + --^, and rsr" cos(--0j; but r"-'.cosn9=r". sinnfl-;-, .•. tan d) = cot Ji9 ; .. |-^=n9, .. 0'=(n+l)e, and r=)^co8Ti9=r'cosm9'= ^i (cos nS)" . . r'(co8Bie') "=a, or r'" COB {mff) = a". 21. If an ellipse b€ y' + ar'(l-e')=a'(l-e^, and the given directrix be ■j:=c, then a=£c, and the ellipse is y^ + x'{l-e°) = eV{l-e''), .: for the envelope - A = c'e {l-e'-e'}) .•. for real values of x, «'>o, i.e. the envelope does not meet those ellipses in which e-^—p.. 8—2 v'116 DIFFERENTIAL CALCULUS. XXV. If e'-J^, y-(l-..)(oV-x=)=(l-'-^') f-^-x=); .-. 1/= ± — - — , i. e. 2 paiabolas. 22. If the given line and point be axis of x and origin, the ellipse la a-(yJ=b)° + Jy'x'-=a'b'-, as the ellipse is on the negative or positive side of the axis of x, and the equation is y-J-iby + {1- e^) x^=0, subject to a = cc, or h' = cV (1 - e'), .-. y-J^2yce Jl-e^ + (l-e')3?=0, and ±2«c , | -2fx'=0, \ Jl-e^-J and with the upper sign y is negative, .•. for real values of x, 2e->],,i.e. c > -^ ; and so for the lower sign by symmetry. 2#»a:2 (1 - f-1 Also when 2e'=-l, y^+(l-e^)x"-==- 7^2-. / • or y=(2.«-l)=xMl-e'), .. «==^„ and 1-^^ = ^^-,; • • V^ +X2 + 2/; -*" 2/ • (a:. + 2y5)2 , ■■ ' +y -c. 23. If S be the focus and P, Q be 2 adjacent points on the conic, S is one point of intersection of the circles on SP, SQ and the other point is the foot of the perpendicular &om 5 on PQ, and .-. ultimately the locus is that of the foot of the perpendicular from 5 on the tangent at P. Hence the locus is a circle or a straight line, as the conic is a central conic or a parabola. 24. If a pair of tangents at right angles be drawn from (h, k) to and the chord of contact is a''yk + b'xh=a'lr', .-. for the envelope of the choii, a^ .ydk+b' .xdh = 0, and hdh + kdk=0; .-. a'^ = \k,b^x = \h; aV 6V a'6^ and .-. -7 J- + -- ;- = -.i- ,-5; if this be written b' a' a? + b' .'. the envelope is a eonfocal ellipse. DIFFERENTIAL CALCULUS. XXV. 117 25. xcos39 + j/sin3fl=a(oos2«)'; .-. -xBin3e + yco83ff= -o(sin2e) {cob 26)^ ; and j: = o Jcoa 25 (cob #), y = a ^cos 2S Bin 8 -, .-. a:=- 2/2 = 02 0052 29, .•. a^ {x^ - y') = {x- + 1/-)-. 26. As in Ex. 23, the locna is that of the foot of the perpendicular from the centre on the tangent ; and if ^ be the eccentric angle of any point, the tangent is + -= 1, and the perpendicular on it from the centre a X sin

-9\, r'=r sin A, but -t;: = c sec" .tan-, \2 / de n « ,'. tan d)=r -y =cot - or 0==- - • , ^ dr n ^ 2 n and .-. - 9' = - - 8 or 6= = , and r'=r cos - , n n - 1 n a ot .-. r'=csec"~' - = CBec''-' = . n n- 1 28. If a be the an^le of the spiral, its equation is (Art. 354) r^&e'*^'", and if (r', 6') be any point on the parabola which corresponds to (r, 6), then r Aaa=r' +t' eoaie' + ^- e - a\ , or r' + r* sin (ff + o - e') = 6 Bin oe' '"* ", is the equation of the parabola, $ being the parameter, .'. , for the envelope, r' cos (9 + o - 6') = 6 cos a . e and .'. C08a=Bin(9'- tf), .•. 6' -6 = -^ -a, cota(e'+a-^) . e ^ *', ^coBeca.e^""^)"*".«*'«"«=ce*'<»t-, i.e. a similar eqiiian^lar spiral. fi cot a and .*. r' .sin 2a = & COB a or ^ o ^^^^^ ^ ' ^ 118 DUTERENTIAIi CALCULUS. XXV. 29. As in Ex. 23, the envelope is the locus of the foot of the perpen- dicular bota the pole S on the tangent at Y (Art. 284) to the locus of Y. If the locus of Y be given by p', r' and j)=r sin a be the equation to the spiral, by Art. 329 P'=— , and p=r', T .". |>'=r^-=-/coseco=/sino, .■. the locus of Y and similarly the envelope is a similar equiangular spiral. 30. If the parabola be y'=iax, and (ft, X:) any point on it, the cor- responding circle is {x - hy + {y -k)''={h+ a + cy, Bxib]ect to k'=iak, .: for the envelope {(x-h) + (h + a + c)\ dh + (!i-k)dk=0, and hdk= iadh. :. k(x+a+c)+2a(i)-h)=0, 2ay . „ i,_ y{x + a+c) _ -,.-» • • If ~ "' — ~; ' , ^ * or i= - and h = ^ — r-j,, but {y-k)'' + x^-2xh={a+c)^+2h{a+c); ■■y (x-a + cf*" (a + c)-^ (x-o + e)=' ' or ( a;-ha-Hc) y' ^^^ ,_ (x-a-hc)2 * ' ' or {x+a + c){y^ + (x-af-c^)=0. 31 . For the envelope ax sec tan S + 6y cosec tf cot ff = ; Bin9_ C08ff_ ±1 ' (by)^ (ax)^ \(ax)^ + (hy)^',i' :. T!(ax)^+(6y)^}'=o'-6>, .-. (ax)^ + (}ry)^=(a'-b'^)i. 32. If AP=iaeoB0, P is (2a cos* 0, 2a sin cos 0), and .-. the circle on HP (A being the origin) is (Q \ 2 / Q \ 2 y»2 x--acos-flJ -I- (jr--asinecosfl J =-2-cos'ff, or x'^ + y^-Zaxoof^e- -^^sin2e-H2a=C0B» e=0, or 2(x2-fi/2)-3aj/sin2e-3ax(l-t-coB2fl)-h2o2(l-(-cos29)=0; .•- for the envelope iay cob 29 = (3aa! - ii^ sin 29 ; sin 29 _ cos 29 _ ±1 iay ~^<^':-'^<^^~ ^(Zayf+(Zax-2a?f\^' :. 2 (x'+y^ - 3ax+2a^= J={{3ay)^+ {3ax-2ar)^\i ; .-. {2{x'+y^-3ax]'=9a'(x^+y'')-l'2a?x-6a'^{x''+y'\ + 12a^x =a'(x<'+y^. MFFERKNTIAL CALCULUS. XXVI. 119 CHAPTEE XXVI. 1. The direction of the asymptote is given by x^ + y^^O, .:, if its eqaation be x + y = c. -3c=a, and the aaymptoto is x + y=-.-. At the origin approximately y'=ai^, .: there is a cusp of the 1'' kind, x=0 being the tangent. Also for any yaloe of x there is only one real value of y, and V y' = i' (a - x), y is positive when x lies between a and - oo , and when z > a, ^ is negative ; and the curve meets ^ = when x = or a and changing the origin to (a, 0) clearly the curve cuts the axis of x at right angles. Hence (o being positive) the general description of the curve is that it has a cusp at (0, 0) where the 2 branches touch the axis of y above y = 0, the branch to the right of x=0 cutting ^=0 again, and extending nnbroken to tonch the asymptote at oo ; the other branch extends to touch the asymptote at a> at the other end. If a be negative the curve is the same as the first turned round the origin in its plane through 2 right angles, the equation being the same in form as when a is positive, on changing the signs of both x and y. As a test of the curve it may be noticed that for any value of y there are one or three real values of x. 2. 1{ x + y = b he the asymptote, -36=0, .•. the asymptote is x + y = 0. AiVhere the curve meets x^y, x=y = -^ =c say, and changing the origin to this point, the curve is (x + c)'+(y + c)'=2c', and approximately near the new origin x''+y^+c{x + y) = 0, .: the tangent is x + y = 0, and x + y is of the opposite sign to c ; thus the curve is symmetrical as to x=2/ and lies between x + y=0 and the asymptote x + y +2c=0, and there being but one real value otxory corresponding to any value of y or x respectively, the curve spreads from the origin to co to touch the asymp- tote at both ends, and having necessarily 2 points of inflexion equidistant from x=y. Turning it through 2 right angles, the curve is seen to be the same whether c and .•. a be positive or negative. 3. If y = mx + n be an asymptote, (mx+»i)'(x-a)=x- (x + a) has 2 oo roots in x, .". »h" = 1 and 2vm-m'a = a, .: n=— , and .■. j/=± {x + a) are asymptotes, as also x=a. Also if x be positive x > a, and if negative x< - a for real values of y, but when y=0, x=0, or —a. Hence the origin is a conjugate point, and the curve, which' is symmetrical as to the axis of X, cuts it at ( - a, 0). Changing the origin to this point the equation is j^' (x - 2a) = x (x - a)*, .•. near this origin 2ay''+a'x=0, or the curve is nearly parabolic, being concave towards the left of x=0, and this branch extends with 2 points of inflexion symmetrically as to y = to touch both y = ± (x+ a) at o . 120 DIFFERENTIAL CALCULUS. XXVI. When x>a, j, .■. taking the positive value of y, y (x+af x^-a?' - - " x+a the ciirve:>l, .. the curve lies above y=x+a: thus there are 2 branches touching the 3 asymptotes at oo as in the fig. p. 372, but on the right of z=0. If a be negative and the signs of x and y be altered, the equation is of the same form, .-. the^curve is the game as before but turned through 2 right angles. 4. The real asymptotes are y=±a, the curve is Gymmetrical as to both axes, and passes through the origin where the tangents are x= ^y. Also " " a'v' , .: y is possible for all values of z; so x*= ,3 , -'■ y Ue* between a and - a, and for any value of x there is but one positive value of y. 2II Hence the curve consists of 2 branches undulating through the origin, and ex- tending the one to ( + aD , a) and {- 00 , -a), the other to the other ends of the asymptotes. 5. The asymptote is x=4a, the curve is symmetrical as to y—0, an , and j/ = 6 is the asymptote ; when e = -,rit a minimum; thus from d=0 to ir, P, P' trace out 2 oo branches symmetrical as to X = and on either side of and concave to y = h, Bupposing that b>a. Also when e' = + ir, /= -ftcosecSia, and .•. the points P, P' are repro- duced in reverse order, and .'. the curve is composed of the 2 branches, while r=0 is impossible on the lower branch, •.• fc>a: but r was a factor in the polar equation, .'. is a coujugate point. If b=a, the lower branch passes through the origin, and the rectangalar equation reduces to 3?{y -b)- = y^ifibij-y'), .'. 2=0 is a double tangent at 0, and the equation is approx- imately 6x'=2y', .". is a cusp of the 1" lund. If o>6, let o = 6coseco, then from 8 = a to ir-o, P' will trace ont a loop on the left of i = 0, which will pass through O, •.• OP'=0 when 6 = a, and in this case at the equation is approximately b-x*=y^(a^-b-), and .•. is a double point: or the — -^ directions of the curve at are given by ff = o and ' — -"-= - — #=r-o, .■. there are 2 branches at 0. In fig. (1) 6>o; in (2) fc above y=0, and is bounded by j/ = a ; and by varying $ slightly from „• > it will be seen that each branch touches z = at the pole, which is, .■. , » cusp (of the l" kind) on the curve. Negative values of only produce the same curve. If o be negative the curve is ~^ — turned round ^-—0 through an angle r on to its plane again. 13. 'When ^ = 0, r=-Go; 9= , r=0, and the direction of the curve there is along x=0; when 6 = ir, r^ -(x> again; and negative values of sin 9 are inadmissible ; thus r being always negative, tbe curve consists of 2 branches forming a cusp at the pole (of the l** kind) below ^=0, and spreading to oo one on either side of a;=0 below y = 0: when 0=0, the hmit r • o ii. i « log sin 9 of T sm 9=that of — ^ , coBec U = COtS-r-(-COBCc9cot fl)= -Bin# = 0, .-. y = is an asymptote: 2ir+9 or -(r + 0) &e. give only the same curve, which is symmetrical as to x=0. 14. Corresponding to each value of there are 2 values of r, sub- ject to cos S being always positive. From #=0 to ^, r^a from to -r- ; o + mr, (n + ^jx + (-l)- [n + \).-(- 1)" then as « from to j- and then to t, - qc from 1 to and then to 1 again. Thus, for positive values of 0, the cnrve stretches to go above the azis of x to the right of x=0, and, as increases, curves round the origin ad infinitum, cutting the axis of z in the same 2 points at each revolution, the part above that axis being greater than the part below, but tending ultimately to the form of a circle, as n increases indefinitely. Similarly for negative v^nes of ff. 17. The equation gives the same value of r for J=ff, .: the curve is symmetrical as to y=0, and r cannot be negative (a being supposed positive, and if a be negative each value of r is negative, and .'. the cnrve may be supposed tnmed over the axis of 2^ on to its original plane). When 0=0, r=:0, and as r cannot be negative, there is a cusp at the pole. When tf==fc--, r=a, and when 5=x, r=2a, and thus the curve is in the shape of a heart, as cos i9 can but vaiy from — 1 to 1. 18. If - S be put for 0, r becomes - r, .■. the curve is symmetrical as to x — 0; T continually decreases as increases, and thus the curve for positive values oi 0>— forms an infinite coll round the pole, which it continually approaches and ultimately reaches. When 0=0, r=ai, and r sin = the limit of — — = a, .■. y = a is an asymptote: thus from S = to ^, the curve stretches from 00 to r= — . The negative values give a similar curve, also having y = a for an asymp- tote, but ultimately touching it at the opposite end. If a were negative the asymptote would be below y=0 Ac. 19. If P be (x, y) the tangent ib {x'-x) -f'- = (y'-y) and the normal is d0 ^ dx , , ,dy dx .(1). (2): 124: DIFFERENTIAL CALCULUS. XXVL and ~ = -(a + 6) ( cos" - 9-coae ) ; .. (1) becomes x' ( cos - - S-cosS J +?/'( sin — ^ S-slnfi | = /sin - - e-sinO j j(a + 6)sin 9-6sin"T^- si + ( cos -y- ff - cos 9 1 i{a + b)co3e-b cos ^-r— BY = (a + 6) cos , e -h-ta + b)+hcoB-0, b ' b , . a + 2b „ . a, , a + 26„. a . or j: . sin--i-- fl . sm =-j 9- « . cos— r^ — 9 Bin — 26 26 ■' 26 1.6 = {a + 26) sin=_--, ji 1 X • , • a + 25 „ , a + 26„ , „, , . a9 .: the tangent is x . sin — ;t7— o -y . cos — ;,- 9 = (a + 36) sin ^ . J6 ZO Zu Also (2) becomes x'fsin --- 9-sin9 j-yYcos — y— 9-cos9 j = - (sin— j — 5 - sin 9 J i6 cos — — 9-(a + 6)cos9{ /a-f6 \l.o + 6 1 + ( cos -,— 9-COS9 I J6 Bin —,— S- (a + b)BuiS'. \ b J ( b ' ' ) = + (a + 6) sin j^ 9 - 6 sin - 9, (I + 26 „ , . a + 25 „ aO or aces—- — 9 + « sin - -,r;- 9 = a cos — : 26 26 Lu which is satisfied by (a cos 9, a sin 9). In fact P is turning round B and .'. the tangent at P is perpendicular to BP and /. ito. 20. Interchanging x and j/, with the notation and figure of Art. 356, the curve required is clearly the locus of the foot of the perpendicular from Af on BC, and is .•. a curve within the cycloid but meeting it at A and E, and symmetrical as to DE. Also -■ = -r- (1 - cos , :, at A, where ^=0, the curve touches AD; and at E, where ^ = x, ;p=0, .', the curve DIFFERENTIAL CALCULUS. XXVI. 125 touches the cycloid at A'. Farther, /. = - . -— (sin <*) = - cob *, .-. tlie curv« dx^ a d^ ^' a is convex to AD from 0=0 to - ; and from for # in either equation, it is transformed into the other, and the point ( - r, r + 0) is the same as (r, 6). 126 DIFFERENTIAL CALCULUS. XXVI. 23. Ab cos {-i}='^4' the curve is sjinnietiical as to 0=0. When 0=0, r=a, and r diminishes till 5 = 5'' "hen r—O. From 0=0 to - -g- , a similar arc is formed, thus the curve consists of 2 loops, one within the other and touching it ; and, as any line through S meets it in 3 points besides 5 (in general), and for every value of there are (in general) 3 values not zero of cos - , .•. there can be no other branch of the o curve. When = Tr,r=SA = - If 0, at P, =3o, then at Q, = 3o + x,and at R, -0=-(ir-3a); thus tan PJS = a cos a . sin 3a-^ I a cos a .cos3o + - 1 = (sin 4a + sin 2a) -^ (2 cos'' 2a + cos 2a) = tan 2o, .-. PAS= 2a, :.APS=a and ASQ = Sa=3 .APS. Also = jsin tan S^Q = SQ sin 3a-r- f" - S<3 . cos Ba J = acos ( a + q ) Bin3oH-! T-ooos ('> + 5- ) cob3o'. (4a + |) + sin(2a-^)!.jl-cos(4. + |)-cos(2a-|)j ^4a-^ + :r)+sin^2a-0j-jl-cos(2«-|)+eos(4.-?3^)j = -tan {2a-l),. ■.SAQ=l-2a..:PAQ = l. Tan54B = C03 (?-») sin3a-f j-cos ( ^-a J oosSo + g! = jsin (^2o + 1^ + sin (^a - 1 U -H j 1 - cos f 2o + ^ j - cos f 4a - 1 J | = j-sin(2a4'-)-sin(4a-l^)j^jl + cos(2a-|)+cos(4a-:f)j = sin(2^-2a)|l + 2cos(|:-2a)j-HJl + 2cos(|-2a)!.cos(^-2a), SAR= 2x 2a, .-. QAR=^ = PAQ. Incidentally the 3 values of cos - are given above in terms of 0. The Jigure assumes that 3o<^ , whence the particular solutions. DIFFERENTIAL CALCULUS. XXVI. 127 24. la 2a =r (1 - tan 8), putting 9 - - for 0, i. e. turning the initial line throogh an angle -, the equation becomes 2a=r ( 1 -] , i \ tan 9+1/ or 7-= a (tan 9 + 1), and now changing the sign of $, r=a(l-tan9), and .•. the 2 equations represent the same curve in different poaitioas. This may also be arrived at by tracing the curves. Frbm the above it appears that to move the curve from the position given by tht 2°" equation to that given by the 1", it has to be turned round the poleiin its plane through an angle j, and then turned round the prime radius on to its plane again. Hence any radius vector to a point of inter- section is symmetrically situated as to the 2 curves, and must .•. bisect the angle between the tangents at such point of intersection. 25. For simplicity the equation may be written t/ = sina:log(m sinx), which is only altering all the ordinates in the same finite ratio, and so of the abscissae. If m be positive, sin x cannot be negative, and if ?» were negative, by changing the signs of x and y the equation would be the same in fact, and .•. the curve is the same for a particular numerical value of m, whether m be positive or negative. Taking m then as positive, x must lie between 2pir and 2px + x, p being any integer : -r^=c0sa:log(msinx)+co3x, .'. the curve ax is parallel to y=0 when cosx=0 or mainx=e~' (o); and when x=2pir or 2(p + l)ir, y= ^ = th3 limit of coseox cot X • n -= -smx = 0. cosec X cot X (1) Thus, when 7R>1, ^ is negative from x = till msinx = l, and then y=0 again, and as x increases to ir-sin-^ I — j y is positive, and=0 when a!=T-sin-' (- )) and then becomes negative till x = ir. Then there is a break in the curve till either x — lir or -ir, and so on; i.e. the curve con- sists of an number of separated equal and similar branches. If m were now negative and of the same numerical value, the breaks would all be filled up, but the curve would not be continuous. ^cf^ :^^l^ KzF\^ 128 DIFFERENTIAL CALCULUS. XXVII. (2) U m=l, msina; = l gives x=2pr + ^, and thus there are no posi- tive vahies of y. ^/ \u z^Tzr v^\J {'■i) If m 2 5(1 -xSa* (1 + 1 - arSfli) = - - a*x"i (1 - x8a*)i, 3x(i+S) 1 «* "/l-x V'l-iSai 2. If y = (sinx)''"', logy=Binalog(Binx), .-. - . ^=cosxlog (sinxl + oosx, y dx "^ for maxima or minima values of y, cosx=0 or sinx= -, or y=0, r=^ = oo, and .-. Bmz=0; DIFFERENTIAL CALCULUS. XXVIf. 129 ^=0 --^-- dx ' y cbfi 1 1 d'y 1, /IN 2 1 2 ^. ^ . .-. when Bmx = -, - . t-^= — log (")+* — = - + « — which is positive, 1 and .•. there is a minimum value of y — l-\ . When cos x=0 and i/ = i, d^ii -j-o= - 1, and -■• y is a maximum (as is otherwise obvious). When y = Q, logy= - oo =sinx . log(sin a;), and the only value of sinx which might apply would be sinx=0, but the limit of sin x log sin x is 0, thus ^=0 gives no solution. When Binx=0, i/=0''=:l, which is clearly a maximum. When cosx=0 and sinx= -1, y= -1 and -p is imaginary, but y= -1 is really a maximum, though not according to the definition in Chap. XIII. If sin X be negative, the difierential coefficients of y are all imaginary, and if the curve !/ = (sinx)''°* be traced, negative values of siux will give an oo number of conjugate points, for if -sinx be any proper fraction, in its lowest terms, with an odd number for its denominator, then y is possible. If -X be put for sin x for possible values of y, y= ± 1 - 1 as the numerator of X is even or odd, x being positive, but < 1 ; and if x be very small and give a possible value of y, in the limit ±y=(x)-'=e-'"°'='=e'' = l, or y=±l. As for positive values of sin x, j/ = ± (c)» would correspond to a maximum or minimum if such a value made y possible. Honce it may be conjectured that fractions (in their lowest terms) with an odd denomi- nator will give an algebraically greater or smaller vaiue of y, the nearer they approach - in value, as the numerators are even or odd, Thus sin X = - 5 will give a larger negative value of y than sin x = - - ; so (2-7)^?>(2-9)^S (by logarithms), &c. 3. The ellipse being a^y' + V'x' = a''b\ let one of the base angles be (x, y)\ then the area = j/ (a-x); this is clearly a minimum when ■i/=0 (and .". x= ±a), and .'. for a maximum ah/dy + ¥xdx = and (a-x)dy-ydx = 0, .: — — -i — or a'-x' + x(a-x) = 0, a-x y i.e. 2x--ax-a-=0 = (2x + o) (x-a); .-. x= - r for the maximum value, and then r; = 1 - 7 > 2 u- 4 and .•. the greatest area is -j- , >jx T. D. C. K. 9 13(| DIFFERENTIAL CALCULUS. SXVIL 4. If PQ be at a distance y from AB, and AB = 2a, PQ = 2x, the equa- tion o! AQ is = ^ — , and at li, by symmetry x'=0, .'. 11' = —■- ; u+x - y ' .1 J J < J ^^^ hence the LPQR = x -.ij f = — — , subject to x^ + «'■' = a- ; .-,' -f"r maxi- \ x + a) x + a ' mum xdx + ydy^O B.ni '2a:(a!+a) -i^^l ~^— . + -— ^ =-0, ' ' ' (x + a)^ x + n or (x + 2a)ydx + x{x + a)dy = 0, :. (x + ia)y- = x-{x + a) = (x + 2a) (a^-x"); .-. either .r= - a which gives a minimum, or x' + fj-a) (i + 2a) = 0. i.e. 2x= + ax-2a2=0, and .-. *=-l±»/r7; a the negative sign would make j: > a numerically, .'. for the maximum valua ofP(?fi/^=^-« = -^iL.i. ^ 2a ^i< 4 5. If a be the radius of the ^ circle, its base axis of x, the vertex of the A at (0, a) and its base angles at l±x, y), the area of the ligure is %xy + x(a-y)-x(y + a), subject to .T- + v'' = «'-, .'. for the maximum value, .rdx + ydy = 0, and (y + a)dx + xdy = 0, •■• x-=y(y + a) = a--y'^; but y cannot be negative here, .•. 1/ = .-;, i.i;. ic. For minimum x = 0. G. If 2S = vertical angle of the cone, r the radius of the base, and a that of the sphere, the slant surface of the cone = xr* cosec 9; and from a figure of a princ pal section of the sphere and cone it will be seen that i'cote=a (1 + cosec 9) or r=« (tan S + secff); .•. for a maximum surface 2rco9ec ffdr = r- cosec Scot 9 dtf, and dr = (i (see'- 9+sec9tan«)(/0, 9 = a sec 9 (sec 5 + tan 0) = - (l-i-cosei 2 sin » (1 -I- sin ») = (! + sin 9) co6= 9, and. .-. 7jCOt9 = aEec9(sec9 + tan fl)=?{l-|-cosec9); or sin=e-i-2sin9-l=0, and .-.sin 9= - l-i- ,,'i', the negative values being inapplicable. 7. Here, from Ex. 6, the surface = irr-(l-(-co8ec B) subject to 1 /, /, „, i, ,. '■'"'' cot 9 rcot t' = a (1 + coscc w), .'. the surface = ; PIFFERENTIAL CALCULUS. XXVII. 131 hence for maximum Zr' cot Sdr=r^coseo'ed0 and dr = ade {see' 6 + BeoB tan ff), , . r cosed* a (1 + oosec ») cosec^ » 3 cot fl .' 'a cot* or 3(l + sin*) = l + cosec9 and .-. Bintf = -. 8. If « = cos . cos cos ^, ta,n0 .d0 + tan + tan f dtp = O = d0+ dip + d = \f/=- and then u=^, clearly .-. a maximum. O o „ ,, Ac' , dy' , dii , du , du 9. Here vr ='i' j =':i' • :r = 'i T^ + '2 T-' : dx ^ dx dx dx dy {y) = Bmy, and l('(y') = siny'; then the coefficient of x'-* . x' is - ^r- ! sin'' z . sin z' . cos laz + a'z') • 2 dz ( dzdz' ^ M = — i — . — { ao' . cos (az + o'/) sin' z J da = -^ . sinz . sin 2' (a. sis\(az + a'z') sin z- 2 cos 2 . cos (az+a'z')\, 15. It equals rcos0 where COB0= — , .'. , = : — - = — = — ; ' re sm 6 ey aey -2a«6= -a'V , ^ i'l and tan 0.,=-t — r, — „. ■ .., = 5 — ^ ; and tan *,= - -.,- ; b* .-. tan 0j . tan ^3 = ^^5, = tan" ,p^. ,„ -TT i^V sin ^ . , ■ dy f dy ,\ . 18. Here -- = -; — ; — '—-^ , .: tan J'= sm o . -^^ -H ( cos o 7^ -I- 1 , and as dx sm (a - ^) dx \ dx J inArt.307,l;=^jl.2cos..|.(gy|. dhi / dy ,\ . dy d?y sm o .-,-„( cos a -^ -I- 1 I - sm a . -/- . cos a 3-^ .„ . dfli dx^ \ dx J dx dx- ^--^''■i- (llcos^.g/ ■' and .■.^slna.g.j(sin..|)V(los.goyj = sin«.g.il.2cos».g.(|)'j. 134 DIFFERENTIAL CALCULUS. XXVII. , _dg d\li _ \ dx \dxj ) "° '''~Ix^dx~ . d^y i d^u The sign of this is positive of negative, *ith sin a . i-* , to make p positive. 19. Here-jL+ 1 .^ = 0, J ax J by dx 2(ax)i 2(i>y)f Vdx/ Jby'dx' dy /by " dx~ V <"'' "* -^ '(o.T)1f axjby^ 2 {ax)'! and .-. by Ex. 18, p= f 1 - 2 cos a . ^ /^ + '^)^ . ,_'<'"'i_ V y/ ax axj {a,Jby + b ^ax)sma. , 1 = 2 (ax - 2 cos a Jabxy + by)^ . —s—- — • ^ ^ '' *' aftsino For the vertex ax -2 cos a Jabxy + 6y is a minimnm snbject to ^f + yf=l, and .-. -^+-%=0, y a y/ b ' Jax Jby and dx( a-coso a/— =! J + dy ( 6-cosa a/ ) = 0, .-. J ax ( a - cos o a/ — -■ \ = V"!/ ( * - cos a a/ — ) , or a ( tjax - cos o . tjby) = 6 ( V6y - cos a tja^c) ; itfax Jby 1 ab a cos o + 6 a + 6 cos a b a a" + 2a6 cos a + f cos d H h T + cos a a These values of x and y are finite, u, i being bo, and .•. apply to the vertex. 20. In the first case the circle of curvature of the curvie at the origin touches the axis of y, and if it meet the curve in another adjacent point the circle is determinate, and then (x, y) being such adjacent point x(ip-x) = y\ y- and .•. ultimately when x and y vanish, 2p= limit of -- . Similarly in the DIFFERENTIAL CALCULUS. XXVII. 135 21. If the axes be turned through an angle o so that the axis of x' eoincides with the tangent, 2^= limit of -,- + y' where a;'=xcoso + i/sina, »/' = y cos o - x sin a, and .. 2p= limit of ^£7= ^^— . y y cos a — x sin a In the curve y- + 2ay - 2ax = 0, the tangent at the origin is y = x; ^ 2'^'-^ y' /'> . . a= - and ±p = limit of , = '^"' = -2a Ji, . . &c. i "^ 1 y^ ^-^:ji -2a nn T» • . J i. ^ <** P <''■ dr dp 22. If j) = r8iniA, and tan rt=r^-, - = -— = ■- '- " ^ ^ dr' r dp d.e dO I ■ dr d0\ = rco^<^-^ I sin ^ — + roos0 . -j^ \ dS dd J = rcot0-^rcos0 ( 1 + j^ ) > •'• p=»'cosec0-hl 1+ j? ) . 23. Here :^ = - 2o (sin » - sin 28), J'- = 2a (cos ff - cos 29) ; aff au dy , , . e . 36 38 . ^ SB .: ,- = tan if/ = sm ^ sm — hcos ~ sm = = tan -^ , dx 2 'i 2 2 2 , 38 , fda\ , . 8 da S . 8 For the evolute, if (x', ■>/) be the corresponding point, (o a mt\ 2 COS 8 - cos 28 --a sin - . sin -- j = a(-cosfl+-co3 2ej, (R 6 2 sin - sin 20+ -a sin - cos o ^ =a( = 8ine+-sin2flj, and if for 6 there be written rr + fl, and the signs of x', y' be changed, com-. puring with Art. 360, this represents an epicycloid in which the radius of each circle=-. 24. Here -^ cannot vanish, being unity, .■. the curve has no multiple dy _ ■ points. For points of inflexion, /^ = 4x»-12j?-36x, ^=12x»-24x-36=0j dx ax' 136 DIFFEftENTIAL CALCULUS. XXVII. cPu .-. (x-3)(x + l) = 0, and then j^=24(a;- 1) which is not zero when x = 3 or - 1, which values of x, .". , correspond to points of inflexion. 'When ^=|(l±-s/5), fx-|y=5.QYorx2-3x-9=0, and .-. ^|=0, and y is then a minimum, and the curve parallel to the axis of x. 25. Here f^= -.-^.2x, ^^^ -,-'? (2-i:^), and —* = e-='^(8x-2x.4x2-2), and .-. whenx = ±— =, -,^=0, and ^ is finite, .-. Ac. dx' dx' 26. At a point of inflexion the tangent is stationary, .°, if = ?• -- , .■. as in Art. 298, if the equation of the curve be dr fir, 6) = 0,-r-. must be of the type j; at a multiple point, iff{T, 6) be rational: up but if the equation be irrational the argument no longer holds, as the dr different signs of radicals give rise to more than one value of -j- for par- ad ticular values of r and corresponding to /(r, 6)=0, and .•. jj is not dff necessarily of the form - . 28. (1). Eationalisiog the equation, {4 (x2 + j/')-ll'=27y2, .-. for multiple points, x . {4(x=' + y') -ll2=0 and 24 {4(x='+3/'')-lj2i/=54y : if 4 (x^ + y') = l, then y = and .'. x= ± = , which values agree with the equa- tion to the curve, which is 4 ( x--^ )=3j/'-4//°=3y' nearly, thus x->j and the values of y are equal and opposite, so that there is a cusp of the first lund at each of the points ( J^-, o\ , y = being the tangent. But if X = either y = 0, which does not satisfy the equation to the curve, or 24(4/-l)'=54,i.e. 3,= ±l|l±|j=±jor ±-, which do not agree with the equation to the curve from which y2-l + 3(y2-2/S)=0, and .-. y=±l, or is imaginary, when x = 0. DIFFERENTIAL CALCULUS. XXVII. 137 For points of inflexion 8x — + 8y- Sy-S = o, and Avhence (2y9 + l)' = 0, and .-. there are no real points of inflexion. This is an epicycloid, cf. p. 391, Ex. 21. (2) When y'-2xy + 2x'-j^=0, for multiple points 2(y-x)=0 and 2 4x-3x*=it/, .: 3x' = 2x, i.e. x = or -; but the given equation may be (2 2\ ^ , ^ ) is not on the curve, but the origin is, while if y - a be not zero, x must be > 1, .-. the origin is a conjugate point For points of inflexion y = x±x ,Jx -1, dy , V'^'^liV . „ 3, „ l/'3 \ i .44 .•. x=- and y is then possible and=,- ± — =, .•. there are 2 real points of 3 ** 3;^3 inflexion, as — ^ clearly vanishes for a different value of x. 29. When x=2, y= -1, and changing the origin to (2, -1) the equa- tion becomes y = (x + 2)(-x)±(-a;)', .". t. is negative and the tangent at the new origin is y + 2x=0, and y + 2x=~x^ nearly on the curve, .•. both branches lie on the same side of the tangent, i. c. there is a cusp of the 2'"' kind, y on the curve being < - 2x. 30. -r4 = (ia!-18, 3^ = 0, .-. x = 3 and « = 2, for the point of inflexion. dx' ax* ' 31. The angle of the spiral is a, and (r, 6) being a point on it, and (r", ff) any point on the corresponding parallel line, the perpendicular on the latter from the pole = r sin a cos a = r' sin (6-6'), .: r' sin [6 - 6') = sin a cos o . Aa^, and .•. for the envelope i' cos (6 - 6") = sin a cos a . J a . cot a , whence tan (»-«') = tan o, and .". 6-e' = a, and .•. t" sin a= sin a cos a . Aa , or the envelope is the curve r = ^coso.a''''"", a similar equiangular spiral. 138 DIFFERENTIAL CALCULUS. XXVIII. 32. If P, P' be 2 points on the spiral, the pole and Q the other point of intersection of the circles on the diameters OP, OP", it will be seen from a figure that, the angles OQP, OQF being right angles, Q is the foot of the perpendicular from on PP', .: the locus in question ia the locus of the foot of the perpendicular from the pole on the tangent. Thus when P, P' coincide, if P, Q be (r, $), (/, «'), and a be the angle of the spiral, and r = /ia*, r'=rsino, and $'=9- l^-a] , and .•. j' = sin a . /I . a =Aa'' . sin a . a*^, a similar spiral. CHAPTER XXVm. 1. If the radius = a, and the straight line be axis of x, and the origui the point of contact when the base is y = a, then when the semi-cixcle has tamed through an angle 9, the centre is at {a9, a) and the equation of Hie base is y-a + (x-a$) tan 9=0; .■. for its envelope atane = {x-aB)sec-e, i.e. x=ae + ^Bm2e=~{2e + Bin2e), and »/=a-aBin"fl=- (1 + C0S2S), a cycloid (Art. 358), the vertex being at (0, a) and the radius of the generating circle = 5. 2. With the figure and notation of Art. 358, if the cycloid roll along x=<), and has turned through an angle <(> when P is in contact with x=0, MD, which is parallel to the normal at P, is then perpendicular to x=0, and 0=the angle through which AD has tnmed=x , .'. the equation to the a base is j/ - 1: + (x - fc) cot ^= 0, (h,k) being D. Now AM is in this case parallel to the axis of j/, and thus h=DM + PM sin n = 2a coa - + sin - \a6 + ae,\n 8 -2a cos - . sin -' 2 2'""2r " -»v,„=2.„„2j a a = 2a cos 5 + off sin - ; and ft = arc AP- PM cos - = 2AM- PM cos = = 4a sin -^-a6 cos - , 2 2 2 2' and .'. the equation to the base is y sin ^ + X cos

-2 sin- cos = coB^, and " '- '-\^'* -I'Hm-Hm- which is the same curve as that given in the text, the axes being inter- changed. 8. If P, F'he 2 adjacent points on the hyperbola x'-y'^a', C being its centre, the radical axis of the 2 corresponding circles is the perpendicular from C on PI", and the 2 circles ultimately intersect in a second point on the perpendicular GY on the tangent at P, but twice as 'far from C as Y. Now if P be {x', y') the tangent at P is xx'-yy'-a-, and CI" is " +i=0; X y x' v' a or hence at Y, — = - ^ = —=;=.—- = -„ — „ , and .'. the loous of Y is ^ y Jx'-y" x'+y"' [x"- + y-r- = a''(x'-y^, a lemniscate, and the locus required is the similar lemniscate {x' + y^f=ia'(x'-y'). 4. (1) The lowest terms give x{y^-x') = 0, .: there are 3 branches, touching x=0 and x= ±y. For the first of these, x being smaller than y, the equation is approximately y* + 2ay'x =0, i.e. y- + 2ax = 0, and this branch is .-. approximately parabolic to the left of :r=0. For the other 2 branches y-= - ax:tx^{a^ + 2ax - X-) (the lower sign corresponding to the branch above given), L 2ai-i" 111 /2ax-x'\'i •■• 3''=-"^ + «H^ + -2TS^-2-2--2 {-^^) \ .•. y^ix^, as X is positive or negative (supposing a positive), and .•. the branch touching y = x lies on the same side of y = x as the positive part of y = 0, and similarly for the branch touching :e + ^ = 0; the results being reverised if a be negative, so that in neither case is there any inflexion near the origin. (2) The lowest terms in x and y give y* + x* + Sx^^=0,i.e, {y^+K')^+x'y^ = 0, which give impossible values, except at the origin, which is .'. a conjugate point. (3) The tangents at the origin are given by xy {ay - bx) = 0, .: there are 3 branches. For that touching y = 0, when y is much smaller than x the eqnatioa to the carve gives 3i*=ibiify or x^ — iby, which is parabolic. 140 DIFFERENTIAL CALCULUS. XXVIII. Similarly for the branch touching .t=0. For the remaining branch, if hx 2/ = — hz -where z is small compared with either x or y, /6xV , /hx\^ , fbx \ . ^ I — j +*( — ) 2- «a2* ( l-z)-x*=0 nearly, or j(-) -l|x*=462x=, and .-. ■t'' |(^) -l| =46« nearly; .•. if 6 be positive and >a, z ia positive and the curve lies on the same side of ay = bx as the positive part of x:=: ; and so on for other cases : thus there is no inflexion at the origin. (4) The tangents are x'y=0, :. there are a double branch touching x = 0, and a single branch touching y = (). For the latter, y being much smaller than x, near the origin a:°=2a'^y or satisfied the equation to the curve, but they do not: for real values x' must lie between d^ and 6''', and y being always finite, or zero (where x'=o' or 6*), there are 2 loops, symmetrical with respect to the axes, their breadth in direction of the axis of x being a~b: for their breadth parallel to z=0, y* is to be a maximum, and taking a, b as positive, j,= = ,» + 6._,= _5;^ = („_5)»_(x-^y; .-. the greatest value of y is a ~ 6, and the corresponding breadth is 2(a~6), .■• &c. Changing the origin to (±a, 0), when b::=a, the equation is (x±a)2 1(2 + (x2±2ax)'-=0, or approximately oy + 4a^x'=0, ■.• x is small, and this is elliptical in form. 6. The curve is clearly symmetrical as to both axes, and .'. it is suffi- cient to consider x and y as both positive, and then y = {6* ± x ijid' - x*}3. (0) With the upper sign, x may have any value from to o J^^, but no greater value for real values of y ; for each of these limiting values of x, y = bt and for intermediate values of x, y>b, and there is only one positive value of y (x being positive). Thus there is a curved line connecting (0, 6) and (a ,^2, b) concave to the axis of x. (j3) With the lower sign of the inner radical, x must not only lie between and a v/2, but also 6= be > x J-2a'-x', i.e. x*- 2a V + a* + b*-a*>0. (1) If !i°>a* the 2"'* condition is at once fulfilled, and there is a corre- sponding curve line joining (0, b) and (a ,^2, b) convex to the axis of x. DIFFERENTIAL CALCULUS. XXVIIL 141 Also generally at (0, 6), ~ = ± 5 . —^— = ± — - , and thus the equation, ax z () ^/2 when b'>a', represents 2 curves resembling figures of eight with y=±b for their axes. Changing the origin to (a ^2, 6), the equation becomes {{x + a^f- a=)» + (1/2 + 2byY- = a*, or {ofi + 2axj2 + a^)' + {y' + 2by)'> = a*, .-. x=0 is the tangent at the new origin. (2) If a=>62, let a* = b* + c*, so that c'^a^. Then the 2'"' condition becomes {x' -a")'>- c*, or x' — a^ — c^ and x^-a^ + c" are of the same sign, .•. either x'^^a^ + c* or ^a'-c^, and .". from the two conditions x' either lies between and a^-c'^, or between a^ + c' and 2a^ for real values of y. Now for (/3), when x^=a''^c'', y'' = b- - ,Ja* - c' = b' -¥=0, and changing the origin to (ija^d^c', 0), the equation becomes {(x+ Jll^^?-y^-a''\- + {y^-b^y=a*, or {x- + 2x Ja-^'^- ± c*)" + (j/'-" - 6')=' = oS and .'. x = is the corresponding tangent. Thus on the whole, when a?>b^, the equation will be found, on drawing a figure, to represent 2 figures, each resembling a heart (without cusp), turned opposite ways, and with the axis of x for their common axis. If a = b, the equation may be written (x" + f-a')^=2xh/, .-. x'±xi/ ^2 + y'=a^, which represents 2 ellipses, which both the previous pairs of curves turn into. 7. If the conic be lu'=l + ecos{S' -a), then iu = l + e cos (9 -o), and (cf. Art. 325) at («, »). ^ = 5^ , and ^-^, = -, , .•.-Z^^=e8in(e-o) (1), d^u -l-^ = eoos(e-a) (2). Hence ^(« + 2-') = l. andby(l)and(2), ,/ . „du „d'u\ ecosa= -I sm 9 -r^ + cos ff-r-r, I , \ d0 dti^J ( du . d^u\ :. Zu'=l-cose'.j( sinS-^ + cosS^ j 142 DIFFERENTIAL CALCULUS. XXVIII. cPu du . ,„, „. d"ii ,„, „, • • " = " + Jr- + le ■ "" <* - ") - 57^ ■ ""^ <" - ** 8. If the Btraight line be axis of x, {x, y) the centre of curvature of the rolling curve at the point of contact in any poBition, p the corresponding radius of curvature and « the arc of the curve measured from some point ds fixed on it, clearly a: = s + a constant, j/ = p= — say; and eliminating « and V from these equations and the intrinsic equation to the curve, the equation to the required locus is obtained. In the equiangular spiral -- = coso; .-. »• . cosa + con3tant=)-=ae»<=»'» = a . cC- -«>'=<" *, which is of the form s = oe''' ™' " + constant, ds . ■■• ^=acoto.«'''™'^'' = cota(s-c) say, and .. X -b = s-c suppose = y tan a, a straight line. In the cycloid (cf. Parkinson's ilXecfeanics), s = 4asin\t', .. p=iacoaifi, and -■• (x-6) = 4o sin ^t, j/ = 4acos0, and -■. (x - *)" + y- = {ia}^, a circle. In the catenary (cf. Todhunter's Analytical Stattcs), -* = -=tanJ' or «=ctan^, .. y=csetryf', dx c and X = c tan \j/ + a. Hence y = c + -{x-a)-,a parabola. D. If (/(, A) be any point on the circle {x-n)- + {y -b)-=c-, the 2 sides containing the right angle are ^ = r and {y-k) k + (x- h) h = 0, also (ft-.a)»+(J;-6)2 = c=; .-. the 2''<' side is hx + ky = 2ah + 2bk + c'-a^-V; .: for its envelope dh(h-a) + dk{k-b) = 0, and dh(x-2a) + dk(y-2b} = 0; :. h-.a + \{x-2a), k = b + \{y -2b), and .. (x - 2a) )a + X (x - 2a)) + ()/ - 2i) {b + X . (y - 26)) =c= - a" - 1-' ; .-. \ = (a^,+ b'' + c°--.ax-by)M{x-2a)^+{y-2bY), and ■ , XM(x-2a)= + (j/-26)2) = c^ .-. (a2 + i(» +<;'' - ax -hyf = c» l(x - 2o)2 + (y - 2bf} or &e. DIFFERENTIAL CALCULUS. XXVIII. 143 10; 'With the centre for origin the equation of any euch hyperbola ii c(x'-y^)-2bxy = l ." (1), and when x = a, / = 0, .'. cx = hy or y= - , and /. <• [a- '- \ -Sa^c-l, or i=(l + co=)=-c'a= (2); .•. , for the envelope, dc .(x'- 1/) = 2db . xy, and dc (3c V + a'b') + 2bdb {l + ca') = 0; .: \ (x= - 2.'=) = 3c V + a-lr' and 2\xy = -2b{l + ca^) ; .-. \=3&>a' + ca''b^ + 2lf{l+ca-)= -b^; by (2) .-. b^(a^+x'^-y'')=-3c-a'- (3), and 6xj/ = c«^ + l (4). From (1) and (4), c (a^ + x--y^) = 3bxy, .: by (3) (a- + x''-y^f= - 2'la-xhf, or a'' + x--j/= + 3(ax?/)3 = 0. ^ 11. The axes of co-ordinates being rectangular, let the ends of the chord he (0, =fcc), the focus of any one of the parabolas be (0, a) and its directrix J- cos a + 2/ sin a = j) ; then its axis is x sin o = (w - a) cos o (1 ), and the parabola is {xcosa + j'sina-j))''=x' + (y-a)-, .'. c sintt-p= ^{c-a), and c sino+p= ^(c + a), .•. either csina=±c or ±a; the former gives x=0 for the axis, which is impossible; with the latter result, xsina-3/cosa= ± c sina . cosa, .•. for the envelope x cos a + y sin o = ± c cos 2a ; .'. x= ±ccoso(Bin-a + cos2a) = ±ccos'a, and y— ± c sin a (cos 2a - cos^ a) =: =i= c sin^ a, and .■. x3 + i/3 = c*. 12. Taking the axes touched as axes of co-ordinates, any such ellipse is given by o- (y - by' + b- (x - a)'- = a-f'', and ab = c- say ; .■. a'y- + b-x^-2c''{ay + bx) + c^=0; and for the envelope bda-+adb = Q, and, da(ay--eHi) + db(bx--c''x)=0\ :: ay (ay - c-)=6x (bx - (r), or d-ir-b-x'-^c' {ay-bx), and .-. (1) ay = bx, or (2) ay + bx=c-. (1) ay = bx='-^. .: a=±c^^,and b=^c ^/^, .-. 2c°-xy-Fic' J^+c*=0, .: 2 Jxy= J=2c±c J^. or 1xy = c^ (^^±1)-, which represents 2 rectangular hyperbole, the product of tho transverse a^es of ■which=4c-, and .-. i.e. 2a= ( I - X J + (2a2 - c') y' = a' ( a- - „- ) i which represents an ellipse ot: hyperbola as 2a- > or <; c'. If 2AP'=AB-, i. e. 2a==c^ then i = |, and aBiael=fy-^\-acoa8(yA--) + c f j/±- j=0, .•. y==f- for all values of 0, and /. PQ has really no envelope but passes through ( - , =F 5 ) , i. e. through one or other of the points of intersection of the 2 circles, through the lower point corresponding to the upper sign of the ambiguity, and vice versa, 14, (1) x^-xi/- + ay'=0: il y = mx + n be an asymptote, a:^ = (x - a) (mx + »()' for 2 00 values of x, .: l = in-, and - am'' + 2in7i=0, .•. n=-— , and theasymp- 2»i "^ *^ totes are y = ± ( x + „ ) , and also x = a. Also «' = , .-, there are 2 \ i/ X — a values of y for each value of x and the curve is synunetrical asto y=0; and when X is positive, x>a; and then if y be positive „ ^ f ay 11- = > or <: X + - , •' x-a V 2/ ' o* a*^ as x=> or than that of y=x+ - for the same abscissa, /. the curve when X and y are both positive lies above y=x+^. Thus thei« are 2 DIFFERENTIAL CALCULUS, XXVI It. 145 branoheg extending to oc as in fig. p. 372 (but to the right of x = 0). When X is negative, y is possible for all values of x; and the curve passes through the origin where the tangent is y = and there is a cusp of the 1" kind, •.• r' + ay'=0 nearly, and y increasing numerically with x, there arc 2 brunches going off to - co , the upper one being above y+x+5 = 0, and ultimately touching it; and similarly for the lower branch, as the curve does not cut y=0 a^ain, but must cut the asym- ptotes where x= - - . Here a has been supposed positive ; if it be negative, and the sign of x be changed, there will be the same curve, .". it will be obtained by turning the first curve round x = on to its own plane again. (2) y' - lyx- + fii" - a' = : by the latter part of Art. 274 the asymptotes are y»-7jw* + 6i»=0, i.e. (y-x) (jr+3i) (i/-2t)=0. When x = 0, y = a, and ^tanging the origin to (0, a), (y+a)^-l (y + a)3c'+6r' = a\ .: y=0 is the tangent and the curve is nearly Say — 7x', which is parabolic and convex to y=0, .: there must be a branch here bounded by the asymptotes y + 3a:=0 and y=2x. When y=0, x=-,-— = c say, and changing the origin to (c, 0) y* - 7y (x + c)' + 6 (x + c)' - a' = 0, and the curve is near!;.- - 14xy - 7cy + 18cx + l&r'=0, which is hyperbolic, and the tangent is 7^ = 18z; thus there must be a branch bonnded by y=x and y-i-3z=0, cutting the positive part of y=0. For a branch between y=x and y = 2x, a test in, 3 . . • ' y = = z, say, meets the curve, if possible, where 27i»-84a:»+48x='=8o'= -Ox', and .'. where x is negative, .'. there is such a branch. Also if y=»ix, -=(m'-7in + 6)*, .■. for each value of iR there is only one point on the curve, and .-. there are no other branches. If a were negative, the equa- tion would become the same by changing the signs of X and y, .'. the carve would be found by turning the above in its plane round the origin through an angle x. (3) y*-n'y'-a'x' = 0: y= ±a are the tsj^mptotes, tl;e curve is symme- trical as to both axes, passes through the origin near which the curve Is nearly y*=a'x*, or y'=±ax (2 peirabolas); also y*=^(a' — y*), .'. y lies between a and -a, and thns the curve consists of 2 branches, one extend- ing from the origin to touch the asymptotes at a. to the right of x=0, and the other to the left. N.B. The branches shoold touir/i x=0. T. D. C. K. 10 146 DIFFERENTIAL flALCULUS. XXVIII. (4) a{r' + 7x^i/ + '!xy^+y')-x-y^-(i: the origin is a triple point, the tangents there being given by (x + y) {x^+!^ + 6xy)=0. The curve is symmetrical as to x=y, :• x and y can be interchanged without altering the equation. If y = mx + n be an asymptote, m^=0, and 27im = a (l + 7m + 77"* + m*), .. n=ao, and there are no finite asymptotes. Turning the axes through an angle ~ , the equation is or {x' -f-Y- = »aJ-2(2j?- x 2a, y'= - or <(x+a)=, and X cannot iLr -2a as a^-a'> or than that of the oblique asymptote, and .'. there are 2 branches stretching to 00 , eimilar to those in fig. p. 372 but to the right of x=0. So long as X < a, 1/ is possible for all values of x, and .'. there is a branch through (a, 0) stretching to - oD at both ends. Changing the origin to (a, 0) the «qaation becomes y*(x-a) = x^ + 3ax' + Sa% .: the curve there is nearly the parabola y' + 3ax = 0. The curve is the same in shape, if a be negative, but tamed the reverse way towards a;=0, as appears by changing the sign of x as well as of a. l' ^' (7) y' - ax^y - bxy^ + 1' = : for the asymptote (vtx + n)' — ax^ (mas + n) —bx (mx + ny + ar'=0 has 2 00 roots, .•. m' + 1 = or m = - 1 , and 5m*n =am + bm^, and a+b a+b .-. n=--^,.-. x + 2,= --3- is the asymptote. If a, 2> be supposed both positive there is a double point at the origin, and approximately, neglecting :r' when x is much smaller than y, y'=ax'y+bxy^, but aa^y is also much smaller than bxy^, and .'. ax^y must be neglected, and thus one branch at the origin is y'=bx; similarly the other branch is a?=ay. Also if y=mx, x[\. + m') = vi(a + bm''), .: there is in general one value of x besides zero for each value of fn, and when m is positive x is always positive and finite or zero, and so also y, thus the positive arms of the 2 branches unite to form an oval. As ni oc from 00 to - 1, x is positive and .*. y is negative, and x oc from to 00, corresponding to the lower arm of y'*=bx which goes off to go to touch the asymptote below y=iQ. Similarly for the branch corresponding to the 2"'' arm of of — ay. The curve is not symme- trical as to x=y unless a = b. Similarly if a, Ii be both negative. If a> -6 and 5 negative= -c say, then a;(l + m') = i»(a -cm"), and there are 4 branches at the origin, viz. x^ = ay, y- + cx = 0, and two which touch y >Jc= d=xija. As m oc from to . / , x is positive (and .-. also y) and « from to again, being finite for intermediate values, there is a loop above y = touching ^=0 and y V c "^^. As j« oc from to 00 , x is negative, and also y, and x oc from to again, and 10—2 148 DIFFERENTIAL CALCULUS. XXVIII. there is another loop below y = and touching y = x \/~ and ir'=0. An a from oo to ^l- X is negative and .'. y is positive (n^ being negative) and x oc from to again, and there is a 3"* loop above y = touching x = and J/=-*\/,-- As m a from • -J] to - 1, X is positive and .■. y negative, and x a from to X , and there is a branch below y =0, touching y -■J] and the asymptote at k> . Lastly, when m a from - 1 to 0, being negative, .c is negative and .'. y is positive, and x a from oo to 0, there being a branch above y = 0, touching ?/ = at the origin and the asymptote at m . Similarly for other cases (if a < 6 change the signs of .r and y i-c). (8) y^ -5ttx-//'' + x'=0: if .r + i/ = c be the asymptote, then when x is oo , (l- -x)' + x'-5ax^ (x — cp=0, .•. c=a; the curve is symmetrical as to x=y and meets it where x = near the origin, neglecting x', i. e. when x is smaller than y, and .•. near x = 0, y' = 5ax^, thus there is a cusp where x = is the tangent; so x-' = 5ay^ gives another cusp, the tangent being y=0; the positive arms of the 2 branches meet forming a loop, and the other arms stretch to oc towards the asymptote x + ?/ = a . That these are the only branches will appear thus : it y = mx, then x(l + m») = 5o(l + m'), .'. for each value of m there is but one value at most of X, besides zero, and x and y cannot be both nega- tive for then m would be positive, and .-. by the last equation x positive also. Further x is oo only when m = - 1. If a were negative, changing the signs of x and y would make the equa- tion the same in form, so that the curve would merely be turned in its plane through 2 right angles. (9) !/ = - ± (x - u) " — : X cannot lie between 6 and - b, unless a lies between b and - b, when (a, o; is a conjugate point. The 2 curves cane- .spending to the double sign, meet at ( ±t, - j , and if a>6 or < - i, they, meet again at {a, a), dy Also «^^ = 2x±L'(x' Jx'-l>' when X = .-I- 6, ^ =-- T 00 , wh«n DIFFERENTIAL CALCULUS. XXVIII. 1*9 tlouble point. and when x=a, a -fi = 2a± ^/{a- - V), and .-. tliert i^ a ■When X is very great dx (nj=x''±(x~a)xfl~'^\ =x'±{x--ax) (l = .T=± // nearly; .-. for the part of the curve corre- sponding to the upper sign the asymptote is nij=^ 2x- -ax- — , a parabola, and the curve is above or below it as x is + or - . For the other part of the curve the asymptote is ay=ax + — , which is above or below the cor- responding curve, as X is + or - . N.B. In the figure a being supposed >& and 6 positive, the branch F should cut the rectilinear asymptote and touch the parabola at x : the branch Q should be convex to the rectiUncar asymptote and ultimately touch it ; and the parabola should not pass through the origin. If a < 6 there is no loop. (10) y' {a+ x)=3r {a - x): the curve is symmetrical as to the axis of x; z + a=Ois the real asymptote, the directions of the other two being given by x^ + y^ = 0. When x=0, y = 0; when y=0, x = or a; and for real values x must lie between a and -a; and at the origin the tangents are x'=y*, and on the curve -. = , .'. when X is positive a: > ?/, thus there is a y" a-x ' •' loop from x=0 to x = a lying between the tangents X— db y; changing the origin to (a, 0) the equation to the curve becomes y'.(2a+x)= -x(x + o)', .•. x=0 is the tangent at the new origin. Also for values of x from ' O to — a, the curve extends to oo to touch the asymptote. If a be negative, changing the sign of x, the same curve is obtained turned round through 2 right angles. N.B. The loop should be an oval. = 0, y=^=0, ""-i ^^- orif^n (11) y dy dx'' when and 1, and «*' is positive whether x be positive or negative,' at tli< y has the same sign as x for any value of x; and when x = ao, y=limit of — = - = and j^ = 0; thus j/ = is an asymptote to the right of x = 0. When x= - oo , j/= -00 .«"=-«; hence the curve goes to infinity without any finite asymptote to the left of x=0. N.B. The curve to the left should turn downwards. 150 DIFFERENTIAL CALCULUS. XXVIII. (12) y =«-■", Jx^'-l: X cannot lie between 1 and -1, and y—O when x= ±1, and y is always positive (unless the - sign of the radical be taken as well, when the carve is synunetrical as to ^ = 0). Also S-- '• i;?#ri-^^^^!--^-^^=*-^— ^^ .'. the 2 branches are perpendicular to ^ = where they meet it and stop. When x=a> , y^= — ^=Umit of -^=Oand^=0, thus y=0 is an asymptote at its + end. When and there is no finite asymptote for - values of x. (-)' X X (13) e °' =Bin-: sin- must be always positive; and the least value (-Y . of e^*' is when y=0, and is then 1. Hence the equation represents an co atr 'Y m is any integer. (14) y=e""^: y is always positive, and •.• cosx=co8 (-x) the curve is symmetri- cal aa to a;=0. As a; ot from to =■, y oc from e to 1; and as a; a from ^to ■ir,y ac from 1 to - , and so on, and the curve un- e dulates regularly ad infinitum both ways, being enclosed within the lines y = e and y=- . 2 (15)- r'' sin 9 =o' cos 29; when 6=0 or jr, r = oo, and rsine= — = 0, .-. 6=0 is an asymptote touching the curve at both ends: sintf and cob29 must have the same signs, whence all cases are included within values of from to J , from -j to x, and from - j to - — , and corresponding to each value of 6, there are 2 equal and opposite values of r. When 6=±j, r=0, and also when 6 = -j-; thus the positive values of 8 give two DIFFEKENTIAL CALCULUS. XXVIIL 151 cuT\-eB nndulating through the pole and tonohing the asymptote at each end bnt on opposite sides ; 3ir and when e= - - , 1^=0-, and when fl= • T" 0. thns the negative values of 8 give 2 equal and similar loops through the pole, and symmetrical as to 8 = ^ ■ N.B. There should be a 2™' undulating curve in the figure, similar but reversed symmetrically. (16) • {8 - 1)^= a Iff' - — \ : considering first positive values of 8, when 8=0, r= -7; when 8 = - 4 2 r=0; when 8 = Tr, r=co, and r sin 0= the limit of 3«-, r is always positive, and a de d8 ' 4 (9-T)- 2g(9-x)-2(g'-^) 2x(g-^) (fl-i)' (e-xf ' .■. r diminishes as B increases from x, until when is co , r approximates to a, so that r=a is a circular asymptote. For negative values of 9, from $=0 to - - , r is nega- tive, and r=0 when = - - ; and -j^ is always negative, Z civ .: r continually increases algebraically, approaching the limit T=a. -^^ 15. If SP, HP make angles ^, f' with the tangent at P, and that meet the perpendicnlars from S and H in Q and S, and if SP=r, HP=r' and rr'=(?, then PQ=rsec^, and cos^=^, cos^'=-t--, if r increase and .•. r" i^iniiniRh as t increases, and PR=t' sec 0' ; ..,f«.-n.-...w„f......P«=P«. r52 DlfFERENtlAL CALCULUS. XXVIII. ^^- ■' \~ ' t)~^ "■"^^ "' = '^' ^J'' •'• ^"'^ *1'6 envelope, —~^ .- .da+ —1^ . ■(.„ .dl, = 0. which is a homogeneous equation of « dimensions in ^ and * (cf. p. 109, Ex. 3), and solving it for - . - there are n values of - : r > ftn' ""' • and A/1 + AB + AC + AD = 0, , AJ -AB AC -AD • ^■^'' = "' ^""^ ■■BCD = CDA = La^^ A-Bl- SO that if ^4 and C increase, B and D diminish, and t;tce versa. y be 18. Suppose ^ = Mi+ - -t- — + -.. ; then X X J.- and ^ (/i, ) = 0, and i = - ^^'\ ; (Mi) DIFFEEENTIAL CALCULUS. XXVIII. 153 thns to the 2"'' order of small quantities ^W + Q + p)0'W+^.(yV"K)+^^(Mi)+^,.fW+^,xW=Oi 2^ (Ml) Hence y = x^i + b -^-^' say, approximately, when x and y are very large ; thus when x is positive the rectilinear asymptote j/ = /Uii+ 5 is above or below the curve as /(^i) is positive or negative; and when x is negative the asymp- tote is below or above the curve, as / (^i) is positive or negative, .°. the ends of the asymptote are on opposite sides of the curve, t//(Mi) be not zero. 19. Here 0'(Mi)=O,andif ?^=M, + ('-) +- + ^+..., »0) + !»(|)4.«(l)--»' +lAx{>h)+-}=0; thus to the 1" approximation j- tfe" {jj.{) + ^ ijij) = 0; . . to the 2°'' approximation , '.Ai,p"(^^) + Ai . ^^ + ^4 ^t' W)=0: 6 1 i. c and .-. to the 3"' approximation or coefficient of — ^ , iA^ • -s • 0" (mi) + Y-'^"W + 3^^-^^+B-V''(Mi)+^.f' (m,)+xW = 0, T. D. C. K. 11 154 DIFFERENTIAL CALCULUS. XXVIII. 20. Moving the origin to (a, /3) the equation to the curve is g. 24. 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