ri DEPARTMENTS OF MtCHANtCAL & £LECTR)CAL E-NOiNEERING OCT 1930 Universi'y of California INTEGRAL CALCULUS. A TREATISE ON THE INTEGRAL CALCULUS AND ITS APPLICATIOKS WITH NUMEROUS EXAMPLES. By I. TODHUNTEK, M.A., F.RS,, HOirOEAET FELLOW OF ST JOHN'S COLLEGE, CAMBEIDGE. SIXTH EDITION. Hontron : MACMILLAN AND CO. 1880 [The right 0/ translation and reproduction is reserved.] library CTambriligc : PKINTED BY C. J. CLAY, M.A. AT THE UKIVEESITY PEESS. PEEFACE. Ix writing the present treatise on the INTEGRAL Calculus the object has been to produce a work at once elementary and complete — adapted for the use of beginners, and sufficient for the wants of advanced students. In the selection of the propositions, and in the mode of establishing them, I have endeavoured to exhibit fully and clearly the principles of the subject, and to illustrate all their most important results. The process of summation has been repeatedly brought for- ward, with the view of securing the attention of the student to the notions which form the true foundation of the Integral Calculus itself, as well as of its most valuable applications. Considerable space has been devoted to the investigations of the lensrths and areas of curves and of the volumes of solids, and an attempt has been made to explain those difficulties which usually perplex beginners — especially with reference to the limits of integrations. The transformation of multiple integrals is one of the most interesting parts of the Integral Calculus, and the ex- perience of teachers shews that the usual modes of treating it are not free from obscurity. I have therefore adopted a method different from those of previous elementary writers, VI TREFACE. and have endeavoured to render it easily intelligible by full detail, and by the solution of several problems. The Calculus of Variations seems to claim a place in the pi'esent treatise with the same propriety as the ordinary theory of maxima and minima values is included in the Differential Calculus. Accordingly a chapter of the treatise is devoted to this subject ; and it is hoped that the theory and illustrations there given will be found, with respect to simplicity and comprehensiveness, adapted to the w^ants of students. In order that the student may find in the volume all that he requires, a large collection of examples for exercise has been appended to the several chapters. These examples have been selected from the College and University Exami- nation Papers, and have been verified, so that it is believed that few errors will be found among them. The work has been carefully revised since its first ap- pearance, and additions made to it with the hope of increas- ing its utility for the purposes of instruction, and of render- ing it still more worthy of the favour wdth which it has been received. An Elementary Treatise on Laplace's Functions, Lamp's Functions, and BesseVs Functions has been published as a sequel to the Treatises on the Differential Calculus and the Integral Calculus. T. TODHUNTER. CaMBEIDCtE, September, 1878. CONTENTS. CHAP. PAGE I. Meaning of Integration. Examples 1 II. Eatioual Fractions 23 III. FormulaB of Keduction ........ 42 IV. Miscellaneous Eemarks ....... 52 V. Double Integration 77 TI. Lengths of Curves 87 VII. Areas of Plane Curves and of Surfaces .... 121 Vm. Volumes of Solids 166 IX. Differentiation of an Integral with respect to any quantity which it may involve . 191 X. Elliptic Integrals 201 XI. Change of the Variables in a Multiple Integral . . . 210 Xn, Definite Integrals 242 Xni. Expansion of Functions in Trigonometrical Series . . 295 XTV. Application of the Integral Calculus to Questions of Mean Value and Probability 323 XV. Calculus of Variations 338 XVI. Miscellaneous Propositions 395 INTEGEAL CALCULUS. 3 1 > } 4 J CHAPTER I. MEANING OF INTEGRATION. EXAMPLES. 1. In the Differential Calculus we have a system of rules by means of which we deduce from any given function a second function called the differential coefiticient of the former ; in the Integral Calculus we have to return from the differential coefficient to the function from which it was deduced. We do not say that this is the object of the Integral Calculus, for the fundamental problem of the subject is to effect the summation of a certain infinite series of in- definitely small terms; but for the solution of this problem we must generally know the function of which a given function is the differential coefficient. This we now proceed to shew. 2. Let (/) (x) denote any function of x which remains continuous for all values of x comprised between two fixed values a and b : where continuous has the meaning defined in Art. 90 of the Differential Calculus. Let x^, x^,...x^_^ be a series of values between a and b, so that a, x^, x^,...x^_^, b are in order of magnitude ascending or descending. We propose then to find the limit of the series (x^- a) (j)(a) + (x^- x^ (ji (^J + {x^-x^) (f> (a?J + when X, — a, x„ — x,,...b — x , are all diminished without 1 '2 1 ' ^ n—1 , , . . limit, and consequently ?i increased without limit. Put x^-a = h^,x^-x^=h2,...h- x„_^= \; thus the series may be Avritten \c^ (a) + K4> [x^ . . . + K-,

/r (J) - -«/r [a). The notation is that this limit is denoted by I j> (x) dx, J a SO that 1 (j){x)dx = '\^ {b) — y^ (a). J a As a particular case we may suppose a to be zero ; then nh = b, and the limit Avhen n is indefinitely increased of hU (0) + 4> (h) + cj, {2h)+...+ (b {b-Ji)\ is denoted by I ^ (x) dx, and is equal to t^t (6) — ^Ir (0). 5. A single term such as <^ (x) Ax is frequently called an element. It may be observed that the limit of 20 (x) Ax will not be altered in value if we omit a fiiiite number of its elements, or add a finite number of similar elements; for 1—2 4 APPLICATION OF INTEGRATION. in the limit each element is indefinitely small, and a finite number of indefinitely small quantities ultimately vanishes. 6. The above process is called Integration; the quantity rb i (j> {x) dx is called a definite integral, and a and h are called the limits of the integral. Since the value of this definite integral is -^(h) —-^ (a) we must, when a function ^ {x) is to be integrated between assigned limits, first ascertain _ the function i/r {x) of which ^ [x) is the differential coefficient. To express the connexion between ^ {x) and '^ {x) we have and this is also denoted by the equation ,' • \ (^ [x] dx = ■'Ir {x). !' In such an equation as the last, where we have no limits assigned, we merely assert that '\lr{x) is the function from which (ji {x) can be obtained by differentiation; y^r {x) is here called the indefinite integral of [a + (r - 1) h]. The sum found by assigning to r in this expression all values from 1 to n differs from the required area of the curve by the sum of all the portions similar to the triangle PQp, and as this last sum is obviously less than the greatest of the lig-ures of which PMNQ is one, we can, by sufficiently diminishing h, obtain a result differing as little as we please from the requfred area. Therefore the area of the curve is the limit of the series h |(/) (a) + <^ (a + /i) + (^ (a + 2A) + + (/>(&- li)]- , and is equal to '«|r(6) — '\|r(a). 8. If yjr^x) be the function from which ^{x) springs by differentiation, we denote this by the equation /■ ^ (x) dx = -^ {x), and we now proceed to methods of finding A/r {x) when ^ [x) is given. We have shewn, in Art. 102 of the Differential Calculus, that if two functions have the same differential co- efficient with respect to a variable they can only differ by some constant quantity ; hence if -^ [x) be a function having 4>{x) for its differential coefficient with respect to x, then if (x) + C, where G is any quantity independent of x, is the only form that can have the same differential coefficient. Hence, hereafter, when we assert that any function is the integral of a proposed function, we may if we please add to such integral any constant quantity. Integration then will for some time appear to be merely the inverse of differentiation, and we might have so defined it ; we have however preferred to introduce at the beginning the notion of summation because it occurs in many of the most important applications of the subject. We may observe that if (^^[x) and j)^{x) are any func- tions of x, j[cf>^{x) + ,{x)] dx =^j>,{x) dx +j^(x) + (f)^{x). Also, if c be any constant quantity \c[x) dx; or at least the two expressions can only differ by a constant. 9. Immediate integration. When a function is recognized to be the differential coeffi- cient of another function we know of course the integral of the first. The following list gives the integrals of the different simple functions ; lx'"dx = — — : , / — = loga7, J m + 1 J X * ' I a'^dx = -, , I e'dx = e", J log^a J jsinx dx = — cosx, jcosx dx = smx, f dx , f dx I — ^7— = tan X, I . „ = — cot X, J cos a; Jsurx /: = sm - or = — cos /; ^{a^ - ic') a a' = - tan - or = — cot - . a^ + x^ a " a a " a' 10. Integration hy substitution. The process of integration is sometimes facilitated by sub- stituting for the variable some function of a new variable. Suppose (f) (x) the function to be integrated, and a and b the limits of the integral. It is evident that we may suppose a; to be a function of a new variable z, provided that the function chosen is capable of assuming all the values of x required in the integration. Put then x=f{z), and let a' and b' be the values of z, which make f{z) or x equal to a and h respectively ; thus a =f{a) and b =f{b'). Now suppose that INTEGRATION BY SUBSTITUTION. 7 i/r(.c) is the function of which (f){x) is the differential co- efficient, that is suppose (p{x) =-^ — > then /, b But by the principles of the Differential Calculus, ^-i^^' = * 1/(^)1/ W; therefore t (/(*')! "t (/(<'')1 = i' ^ 1/(^)1/W d^ rb rb' fj[x hence <^{x) dx= \ j> {x) ^ dz. This result we may write simply thus j<^(a;) dx=j(f){x) ^^dz, provided we remember that when the former integral is taken between certain limits a and b, the latter integral must be taken between corresponding limits a and b'. h 11. As an example of the preceding Article suppose that is required. Assume x = a — z, then -^ = —1, and ^ax — x" = a^ — z^. Thus /, dx r 1 Ax 1 f dz f 1 dx ^ sji^ax - x^) j V(2a.r - x^) dz J VK " ^) _,z _,a — X -,x = cos - = cos = vers - . a a a a X Again, let | — t?^ — ^ be required. Assume «/ — . , *= ' J X Aj{2ax - a"") ^ 1-^ thus dx _ a , r dx _ r 1 dx , d^~ {l-zf I X ^(2ax - a') ~~ J ^(2aa; - a'} dz ^ 8 INTEGRATION BY PARTS. dz 1 f dz -I aV{2(l-^)-(l-^r} aJv(l-^^) 1.-1 1 . _i a; — a = - sill s = - sm . a ax Here we have found the proposed integrals by substituting for X in the manner indicated in the preceding Article. This process will often simplify a proposed integral, but no rules can be given to guide the student as to the best assumption to make ; this point must be left to observation and practice. 12. Integration hy parts. From the equation d (uv) _ dv du dx dx dx we deduce by integrating both members, f dv J f du , uv = u-y- dx+ V -r dx, J ax J ax therefore ju -r dx = uv — \v -r- dx. J ax J dx The use of this formula is called " integration by parts.*' For a particular case suppose v = x; then we obtain ju dx = ux ^ j^ T ^^' For example, consider I x cos ax dx. Since 1 d sin ax cos ax = ^ , a ax we may write the proposed expression in the fonn fx d sin ax J a dx dx, and this, by the formula, supposing u = - and v = sin ax, X sin ax fsin ax , = dx a J a INTEGRATION BY PARTS. X Sin ax cos ax 1 8 — . a a Again, ix^ cos axdx= I — x^ d sin ax dx dx •^ sin ax f 2a; _ X sm ax f ~ a J a sin ax dx 0^ sin ax C2xd cos ax _ x^ sin ax f ~ a J a^ dx dx 0? sin ax 2,x cos ax [2 cos ax , H 5 r, — dx J a" a a x^ sin ax 2x cos ax 2 sm aa; . . r ^, . - /"sin aa; de"' , Again, I e sm axax= j -r— ax s'max ,, fae" cos ax , e — — dx c J G sin ace ,, ffl cos asc cZe' c e -/' ,« ( --~^^~ -^^^_ ^^ c" da; sm ax _ « cos ax ,, /a" sin aa? ,^ .. .„^^™^ ^„ _ j a o.^^^ ^,^ ^^^ e"^' 2 — e^^ - r c c J c By transposing, (-?)/ e' V . a e'"" sin axdx = — i sin aa: — - cos axj , f . , 6°* (c sin ax — a cos aa;) therefore e"' sm axdx = r-; — 5 • j a' + c' Similarly we may shew that e" (c cos ax + a sin ax) I' e" cos ax dx = 2 , a a + c 10 EXAMPLES OF INTEGRATION. 13. The differential coefficient of any function can always be found by the use of the rules given in the Differential Calculus, but it is not so with the integral of any assigned function. We know, for example, that if m be any num- ^n.+l ber, positive or negative, except —1, then lx"^dx= , but when m = — 1 this is not true ; in this case we have fdx I — = log X. If however we had not previously defined the term logarithm, and investigated the properties of a logarithm, we should have been unable to state what function would give - as its differential coefficient. Thus we may find our- selves limited in our powers of integration from our not having given a name to every particular function and investi- gated its properties. In order to effect any proposed integration, it will often be necessary to use artifices which can only be suggested by practice. 14f. We add a few miscellaneous examples. Ex. (1). !^{a^-x')dx. l^/{a^— X') dx = x\J{a^ — x^)+\ -jt-^ ^^ , by Art. 12, sup- posing u=- \f{a? — x^) and v = x. A 1 r // 2 2N 7 [ a^ — x^ ^ [ a^dx f x'dx And \iJ\a —x)dx= -77-2 2\ dx = -tj—^ ir — -77-2 j. ; therefore, by addition, 2 \\/{a^ — x") dx = X \l{s/(a+bx + cx^) \/cj I fa bx ^ \JC c ) 'JcJ' dx ^Jc] /{( ,6V, ^ac-b' v^{("+s;y 4o* 1*2 EXAMPLES OF INTEGRATION. Putting oc-i ^ = z, our integral becomes, by (2) and (3), — log {2cx + 6 + 2 Vc \/(« + hx + cx')], where we omit the constant quantity —f- log 2c. In a similar manner, by assuming s = ^ + — • we may make \\J{a-\-hx-{- cx^) dx depend upon Ex. (4). C dx _ J^ r dx J \/{a + hx — cx^) ^cj I la bx ^ A Put h^ for — t-t; — and z for x—^r, then tbe integn-al 4c 2c ° 1 f dz . 1 . _ z becomes —- -777^ irr , wliicb gives — - sin ^ 7- , that ^1 c J 1^ [h^ — z') ^ sJg h IS 1 . _, 2cx — h sm ^~2o In a similar manner, by assuming z = x — -^ we may make I V(« + t^ — c^'^) c?a; depend upon Ex. (1). Put :. = i,tken|j^^^|^^=J^-^^i-^|(;y EXAMPLES OF INTEGRATION. 13 f du If chi 1 . _, 1 . _!« = — Sin - . a X -I a -I a 17 Since sin"* - + cos ^ - = -^ , a constant, we may also write our last result thus, f dec 1 _i a 77-2 T\ = - COS - . ]x\J[x —a) a X By putting cc = - , as in Ex. (7), we deduce for the required result 1 , X iOOf dx , C dx Ex. (9). [7-^, and f- . ^ ^ J{x-ay' Jx-a r dx _ _ 1 1 ]{x- a)'" ~ m-1 {x - a)'"-' ' These are obvious if we differentiate the right-hand members. dx Ex. f dx _ 1^ r/_i i_\ , jx" — <^ 2aJ\x — a x + aj _ 1 f dx 1 C dx ~2ajx — a 2ajx + a 14 EXAMPLES OF INTEGRATION. 1 , x — a 2a ° x + a a . ,— positive: .. x+a ^ x+a This supposes — -- positive; if ~ — - be negative, we must write a — X r dx 1 Jx'-d'^Ic lo or 2a ^ a + x Ex. (11). [- ^^ + bx + cx^ ' dx 1 [ dx r dx 1 r ja + bx + cx^ ~ c J (. + , ^Y 4ac-6''' 2cJ ' 4c' If — ^-^— be negative, we obtain the integral by Ex. (10), namely 1 1 ^ 2cx + h — Aj{b' — 4ac) V(6'-4ac) ^^2cx + b + ^{b'-4^ac)' If — T--2 — be positive, then by Art. 9, the integral is 2 ^ _, 2cx + b rjrtan V(4ac - b') V(4ac - b') Ex. (12). f-^pL^^dx. Ja + bx + cx Ax + ~~ + B- ]a + bx + ex' "^^ - j a + bx + ex' "^^ Af '2cx + h ,/D_^^^ dx 2c] a has been found in Ex. (11). + bx + cx' ^\ 2c J J a + bx + cx'' A 2c The former integral is — log {a + hx + cx'), and the latter Ex. (13). I EXAMPLES OF INTEGRATION. 15 dx COS a; f dx fcosxdx f dz .„ = 2 — = 1 2 , n z= sm X, j cos a; J cos cc Jl — z' , , 1 + sin a; , , /tt x\ = h loo- ^— = log cot -r — s • - ° 1 - sm cc * \4: 2/ r cIt^ CO Similarly -.-^- = locj tan - . '' jsinic ° 2 Ex. (14). f— 4^ , and f ^f . . ^ j a + 6 cos 05 j a + 6 sm a; c?a; r ^aj [ dx _ C dx J a + b cos X J f . r.x ^ " ^A , 7 / 2 a ( sm' - + cos" k J + ^ ( cos' 2 - ''" 2 = / sec*^ ^ dx a + b + (a — h) tan^ ^ = 2 I -, — ^ — iT-2 > if ^ = tan - . Ja + b+{a-b)z" 2 Hence, if a be greater than b, the integral is 2 ^ ...V(a-;.) 2 . /(«-^)fa°| rj-tan —77^ tt^ or — r-^ — n{^^^ ~' and if a be less than 6, the integral is 1 , z^/(b-a) + ^/(b + a) loff- ^/{U' - a') ° ^ V(^ - «) - V(^ + «) ' X ,^(b — a)tim~ + \/{h + a) that is -j-p 5- log — 16 EXAJVIPLES OF INTEGRATION. r 7 To find I — —J—. — assume a; = - + ^ ; thus the iuteOTal becomes — —j- , which has iust been found. Or we may proceed thus, dx f dx _ r J a + b sinx J -I a f sm 2 + cos' ^ 1 + 2b sm - cos ^ sec* X ^a; a{\^ tan' |j + 2h tan | =2/;^ , II = tan - , Put v = s + - , and the integral becomes ■^ a ft 9, (Zy and this can be found as before. Ex, (15). Let i/r {x) denote any function of x, and let •^"^ [x) denote the inverse function, so that -\/r [i/r"^ [x)'\ = u; : if the integral of "^{x) can be found so can the integral of ■^'^ (a). For consider I ^~^ (a?) dx ; jDut i/r"^ (cc) = 2;, then x = -^{z) : thus I -v/r"^ (a;) dx= jz-r- dz = zx— \xdz =zx— \^ [z) dz. In any of these examples, since we have found the in- definite integral, we can immediately ascertain the definite integral between any assigned limits. For example, since therefore "" dx / J a EXAMPLES OF INTEGRATION. 17 = log [2a + ^/[{2ay + a']] - log [a + ^{a' + a')] , 2 + V5 = log:. p 15. The integral /.«'""' (ct + hx^y dx can be found imme- diately if -^ is a positive integer, for (a + hx^f can then be expanded by the Binomial Theorem in a finite series of powers of X, and each term of the product of this series by a;'""^ will be immediately integrable. There are also two other cases in Avhicli the integral can be found immediately. For assume a + hx"^ = f ; i 1.1 therefore -=f-^j , 21 = 1^ [-T^ Hence L'""^ (a + hx^dx = L"^"' (a + hx'') 'P' Q /,p+j-i /^ '^ -1 li — he a positive integer we can expand (f^—a)" in a finite series of powers of f, and each term of the product of this series by t^'"^'^ Avill be immediately integrable. Again, J a;"'-' (a + Ix"") 'dx= x' (ax-" + h) ' dx ; and by the former case, if we put ax~" + h = t\ this is im- mediately integrable if n 1 ... tn t) be a positive integer; that is, if - + - &5 a negative integer. T. I. c. 2 18 EXAMPLES OF INTEGRATION. In the first case, if — were a negative integer the integral might still be found, as we shall see in the Third Chapter, and similarly, in the second case, if — f- - were a 'positive integer: but as in these cases some further reductions are necessary, we do not say that the expressions are immediatehj integrable. Ex. (1). \x^ [a + xf dx. Here — = 3 : assume a + x = f; the integral becomes n 2 [(f - ay fdt or 2 [(f - 2at* + aY) dt, which gives it' 2af a'f] thus jx'{a + x)'dx = 2{a + x)H^^-~{a + x)+^ r dx Ex. (2) x' (1 + xy Here ??i = - 1, n = 2, ^- = — -; therefore — + " = - 1. ' ' q 2 n q Assume x'^+l^ f : therefore x^ = js — 7 > ^i = 1 f-l'dt (f-if dx Also f ^" , = [ ^L^.dt Jx'{i+xy~ J x'{x-'+iY' Substitute for x and -^ their values, and this becomes —J dt, which = — ^ or — -!^-!^ -. X EXAMPLES, 19 Here m = l, n = 2, ^ = --, therefore —+^ = -1. Assume a^a;"^ + ! = «*, therefore «" = dx 3 5 r dx _ r dt j._ 1 r^^_ 1 EXAMPLES. r dx _ . _i 3 + 2a; • Jv(l-3.'c-a;'0~^''^ "VIS"' 2. 1 log xdx = x (log ic — 1). 3. a;" log a; Ja; = r-^loga; irL J *= ?i + 1 [ ° n + 1 j 4. f^sin^J^ = -^cos^ + sm^. 5. [^^^ = tan-i(0. G. I / [ ^^^^ — ^j dx = sj{mx + x^) + m log [Jx + V(wi + a;)}. This may be found by putting x = z^, f - 1 + x^ _ 7. a; tan ^ a3 cZx = — ^-^ tan ^ a; — ■!»;. o ffi \2 7 Sa; „ . , sin 2a; 8. I (1 — cos X) dx = -^ — 2 sm x H r— • 2-2 20 EXAMPLES. !-. {i-xy 1 - cc 2 (1 - xy C afdx _ J. 10. -. «=7r^l02f a' + x' a^'^^a'-x'' 6? . _, a; — a 11. \\J i^ax - x") dx = ^-^ ^/{2ax - a;') + ^ sin 12, 1^7^ -^ = -J(2ax-x'') + avers~^-. JJ{2ax-x-) ^^ ' a 14, -. — dx = log (x + sin x). J x + smx , ., fiC-rsiiiu; -, .<^ lo. I :^-r ax = x tan - . ic + sm a? , , X ax = x tan - + cos X z 16 ■ ^^ j a; (logic)" (n - 1) (log^a;)""' • -j^y^ / og I Qg ^j ^^ = log a; . log (log x) — log x. J X r . ^ e"'' a sm( m + n) x-{m+n) cos {m+7i) 20. e'"'sin??2cccoswa;aa;=-7^; 2~r"7 — rz^^i e"" a sin (m — n)x — (m — n) cos {m — n) x 21. [e"'' cos^ .r (?a; = i fe"'' (cos Sic + 3 cos a;) cZx = J- (3 sin Sx - cos 8a:;) + -^ (sin x - cos x). 40^ ^ 8 EXAMPLES. 21 2 22 2a 23. rV(2a^'-a'')tZ^ = ^- •'o '^ r" _ £c 24. vers ^ - c^x = 7ra. Jo « Proceed thus : let vers"^ - = 0, therefore x = a (1 — cos 6), and the integral becomes 1 a9 sin 6 dd. J 25. X vers"* -dx = — -t— . Jo « 4 2G. ^ vers - cZic = — ^7— . 27. rW6'cos'^(Z^ = :^. 28. f-^^^ = 1 logtanf|+^' J sm ic + cos X i\J2. ° \2 8 29. dx X f^{a + bx-\- cx') ' 1 y Put x = - and this becomes a known form. 30. fV(l^sm..<;.=-!^i-\^^-^-!^^ j X dX bx o This may be obtained by putting sin"* x = 0. SI. ; £7.r = ^ tan ^ + log cos 6, where sin 6 = x. J{l-x')^ 32. 1 ^ — 5 = —. (cot 6-{ — ^r— ) , where ic = a cos 6. C sin^xdx _/« + &y, _i \/a tan .t a; ja + 6 cos' a; ~ V ab' J ^^ \/(« + ^) ^' 22 EXAMPLES, [ dx _(2a;''-l)_V(ljf^ ]x^ 35 3G 41, 42 43 ^/{l+x^) 3^' [tan''" ede = 2^^ - ftan^"-^ 9 d9 + -{-lyx+i-iye, ^^n-l ^2n-3 2n - 1 271-3 X being = tan 6. 37. Shew tliat I sin mx sin wa; cZic and 1 cos wa; cos wa; dx ar Jo •'o zero if m and « are unequal integers, and = -^ i m and w are equal integers. .38. /{log ©I'd-- {log (^)}'- 3. {log^}V6.1og^-6. 39. I -IT?:; 5r dx = -^i — 6 tan ^ — log cos 6, where cot ^ = a J X- (1 + ic^) 2 o ' 40. [^^i±^./f^^.. = V(a»-a;^)-?^;^^ ] a^-x y \a^rx) ^ ' ^/{a + x) / vers ^ - -2 f/v/(2aa; — ic^) I t; = -rrq sr cos"V, if c is less than 1. Jo 1 + ccosa; V(l — c) f " e-^ cos' ede=fQ (ei" + e"*''). J -Jn- 44. I — },^ . ^ g jr- . Assume 2; = a; + - . _ [(a + Jr«")^ (7a; . . 7 n 4 4o. ^^ ^^ . Assume a + ox = .s . J a; 23 CHAPTER II. RATIONAL FRACTIONS. 16. We proceed to the integration of such expressions as A' + B'a;+C'x\..+M'. x"" A + Bx+Cx\..+Nx'' where A, B,...A\ B',... are constants, so that both numerator and denominator are finite rational functions of x. If m be equal to n, or greater than n, we may by division reduce the preceding to the form of an integral function of x, and a fraction in which the numerator is of lower dimensions in x than the denominator. As the integral function of x can be integrated immediately, we may confine ourselves to the case of a fraction having its numerator at least one dimension lower than its denominator. In order to effect the integration we decompose the fraction into a series of more simple frac- tions called partial fractions, the possibility of which we j)ro- ceed to demonstrate. Let -p. be a rational fraction in its lowest terms which is to be decomposed into a series of partial fractions; suppose V a function of x of the w**' degree, and U a function of x of the (?i — l)"" degree at most; we may without loss of gene- rality take the coefiicient of a;" in Fto be luiity. Suppose F= {x -a){x- by (x' - 22X + d' + /3=) {x^ - 2^x + 7^ + h')', so that the equation F= has (1) one real root = a, (2) r equal real roots, each = h, (.3) a pair of imaginary roots a + /8 \/(— !)> (4) s pairs of imaginary roots, each being 7 + S \/(— 1), 24 RATIONAL FRACTIONS. By the theory of equations V must be the product of factors of the form we have supposed, the factors being more or fewer in number. Since V is of the rf" decnree we have Assume l+r + 2 + 2s = n. ^ A ■ s.^+ ^. +, ^. + ^- Cx + D + a^-2oix + d' + l3' E^x + F, E,x + F, . E^x + F ^ where A, B^, B^,...C, D, E^,... are constants which, in order to justify our assumption, we must shew can be so determined as to make the second member of the above equation identi- cally equal to the first. If we bring all the partial fi'actions to a common denominator and add them together, we have V for that common denominator, and for the numerator a func- tion of X of the (n — 1)* degree. If we equate the coefficients of the different powers of x in this numerator with the cor- responding coefficients in U, we shall have n equations of the first degree to determine the n quantities A, B^, B^,... and with these values of ^, B^, B^,... the second member of the above equation becomes identically equal to the first, and thus -^ is decomposed into a series of partial fractions. If V involves other single factors hke x — a, each such A factor wiU give rise to a fraction Hke ; and any repeated factor like {x — by will give rise to a series of partial fractions B Ti of the form ^-—., |-^=i,.... In like manner other factors of the form x" - 2oix + cr -^ ^"^ or [x" -2r^x+ rf + h'')' will give rise to a fraction or a series of fractions resjpectively of the forms indicated above. 17. The demonstration given in Art. 16 is not very satis- factory, since we have not proved that the n equations of the RATIONAL FRACTIONS. 25 rst degree which we use to determine A, B^, B.^,... are in- ependent aud consistent. A method of greater rigour has been given in a treatise on be Integral Calculus by Mr Homersham Cox, which we will ere briefly indicate. Suppose F {x) to contain the factor : — a repeated n times; wo have, if ^{x) <}>{x) ^^^^ -^{ay^^ , ^(^0 Fix) {x-aY-y\r{x) {x-a)"f{x) [x-ay' Now 6 (x) — r-^ "^ {x) vanishes when x = a, and is there- ^ ^ '' yjr [a) ' Dre divisible by x — a; suppose the quotient denoted by ; {x), then F [x) {x - ay-' ylrix) f{a){x-ay' 'V (cc) The process may now be repeated on _ ^. „-i , , . , and „ (b (x) hus by successive operations the decomposition ot ry-^ ompletely effected. In this proof a may be either a real oot or an imaginary root of the equation F (x) = -, if ; = a + /3v'(— 1), then a — /3V(— 1), will also be a root of <" (x) = 0; let b denote this root, then if we add together he two partial fractions y}r [a) {x - ay -f (6j {x - by ' {x)=A^{x) + {x-a)x{^) (2). In (2), which holds for any value of x, make x=-a, then /r(a) Since F' (x) = '\lr{x) + (x— a) t/^' (x), we have F' (a) = t (a), therefore A = ^, , \ . F (a) 20. To determine the partial fractions corresponding to a factor of the first degree which is repeated. Suppose F (x) contains a factor x — a repeated n timeSj and let F (x) = {x — a)" -v/r (x). Assume RATIONAL FRACTIOXS. 27 ^(■^•)^ A , A , A , A , x(^2 F{.v) {x - a)" {x- a)"-' (x - a)""' x-a^y{r{x)' where , ; , - denotes the sum of the partial fractions arising ylr (x) ^ ° from the other factors of F{x). Multiply both sides of the equation by {x - ay and put f{x) for -^rhl {x - a)" ; thus J^ {x) f{x)=A,+A^{x-a)+A^{x-ay...+A„{x-ay-'+^^fx-a)\ Y (ic) Differentiate successively both members of this identity and put x = a after differentiation ; then /'(a) = 1.2^3, r-'{a)^\n-lA,,. Thus A^, A^, ... A„ are determined. 21. To determine the partial fractions corresponding to a pair of imaginary roots which do not recur. (b(x) Let - U. ( denote the fraction to be decomposed; and o' ± /3 \/(— 1) a pair of imaginary roots ; then if we denote these roots by a and b and proceed as in Art. 19, we have for the partial fractions F'{a)x-a F'{b)x-b' Suppose P ' = A—B v'(- 1) ; then since w^jri may be obtained from ^, . by changing the sign of /^{—l), we must have ^, . = A +B \/(— !)• Hence the fractions ai'o 28 RATIONAL FEACTIONS. ic-a-/3V(-l) a;-a + /3V(-l)' and tlieir sum is 2 A {x-a) + 2^/3 {x-ay + /3' ' 22. Or we may proceed thus. Suppose x' — px+q to denote the quadratic factor which gives rise to the pair of imaginary roots a±^ V(— 1) ; then assume (ji(x) _ Lx + M x(^) F{x) x'-px+q^ ■^f{x)' so that F{x) = {x^-px+q) ■\lr(x). Multiply by F{x) ; thus (P {x) = {Lx+ M) ^lr{x) + {x'-px + q) xi^) (1). Now ascribe to x either of the values which make x^ —px + q vanish ; then (1) reduces to (x) = {L,x + il/J f (.r). Proceed as in Art. 22, and thus find L^ and M^. Then from (1) by transposition we have <}> {x)-{L^x^M^)^{x) = {L^_^x+M^_^ {x'-px+q)y\r{x) +... The right-hand member has x' —px + q for a factor of every term : hence as the two members are identical we can di^dde by this factor. Let <^j {x) indicate the quotient ob- tained on the left-hand side ; then 01 {^-) = (^r-l ^ + K-^) t (^) + i^r-,^ + K-2) i^"- 1^-^ + 5) t {^) + + {^''-p^+qrx{^) (2). From (2) we find L^^ and M^_^ as before ; then by trans- position and division <^,{x)HL,..^'rM^_„)^^{x) + {L,_,x^M,_^){x'-px+q)y^{x)+... and so on until all the quantities are determined. ^2 _ 3^ _ 2 Take for example ^-^ _.,„ . '^ — zrr:-,. Assume it equal ^ {x- + x+lf{x + \y ^ to L^x + M^ , L,x+M^ X¥)__. {x' + x + lf^ x- + x^-l (x + iy then a;" - 3^ - 2 = {L^x -f J/,) {x + 1)' + {L^x + il/J {x'+ x + l){x + lY-\- {x^ + x + iyx{^)"- (3). 30 EATIONAL FRACTIONS. Suppose x^ -}■ x + 1 = 0; thus the equation reduces to = {L^x + ]\Q(x'' + 2x + r). Put —x — 1 for «^ ; thus - 4^ - 3 = {L^x + M^ X = L^x^ + M^x = -i:,(a; + l)+ilf,^; therefore — 4 = — i^^ + ^v ^^^ — 3 = — Xj ; thus L^ = 3, and M^ = - 1. From (3) by transposition x^ -2x-2-{2x- 1) (a; + l)** = {L^x+M;){x'+x + i){x + iy+{x''+x + iyx{x). The left-hand member is — Sx^ — 4 (11). Put a; = 0, then A,-2B + D = (12); therefore J^ = h T. I. c. 8 34! RATIONAL FRACTIONS. From (11) and (12), by subtraction, therefore C' = — f ; therefore {x-iy{x'+i) {x-iy 2{x-iy 4^{x-iy 5 1 2a; - 1 + 8(a;-l) ' 24(a; + l) 3{x'-x + l) Jix- {x' + l)dx _ 111 tnereiore I (^_ i)*(^3+ i)- S{x-1Y'^ ^ix-lf^ 4{x-l) 5 11 + g log (^ - 1) + ^ log (a; + 1 ) - - log (x' - a; + 1) . 25. We will give as additional examples the integration x"'~^ of —„ — Y , supposing m and qi positive integers, and 7n — 1 less than n. Required the integral of -j^ — zr when n is supposed even. The real roots of cc" — 1 = are 1 and — 1, and the imagi- nary roots are found from the expression cos rd±t,J{—l) sin rO, where 6 = — , and r takes in succession the values 2, 4, ... up n to n — 2; see Flane Trigonometry, Chapter xxiil. Now by {a) _ g"'"^ _^__^. F' (a) " ^-' " wa" ~ ¥ ' thus the sum of the two fi-actions corresponding to a pair ol imaginary roots is 2 cos mrd {x — cos rd) — sin mr6 sin rd n {x — cos rO)' + sin'^ rd Hence / —;, — =- = — S COS mrO log (a;^ — 2x cos 7-^ + 1) , 2 ^ . o , -1 ^ - cos r^ + - 2, sm ??i?'t^ tan — -. ^ — , n sm rU where ^ indicates a sum to be formed by giving to r all the odd integral values from 1 to w — 1 inclusive. 28. Required the integral of — ^^ — ~ when n is supposeo odd. The real root of 5?"+ 1 = is in this case — 1, and the imagi- nary roots are found from the expression cos r6±\/{— 1) sin r9 IT where d = — , and r takes in succession the values 1, 8, . . . ur to n — 2. Hence we shall obtain / af-^dx (-1)*"', , ,, — 2, cos Wi?'^log (x'—^x cos r^+ 1) + - S sm mrd tan — -. — ~y- . RATIONAL FRACTIONS. 87 29. AVc will finish tlic Chapter with some miscellaneous remarks on the decomposition of rational fractions. 6 (x) . I. Suppose we have to decompose the fraction -p— - into partial fractions where (f) (x) is not of a lower dimension than F(x). Divide (f) (x) by F{x) ; let (}>^ (x) denote the quotient, and ^2 {x) the remainder; then (x) = ,{x)Fix)+4>,{x); therefore |^ = '/'i (^) + #(^ " Accordingly we have now to decompose ^", '. into partial fractions. It should be observed that we shall obtain the same values for the partial fractions whether we apply the methods of Arts. 19, 20, 21, 22, and 2.3 to %Pc or to-%-p( . Take, for example, the case of Art. 19: since, by hypothesis, F(a) = 0, and cj>{x) = <^^ {a:) F {x) + (j).^{x) , we have <{)(a)^(f)^(a). II. From considering the values of ^j, A^, ... in Art. 20 we see that the foUoAving result holds: let r stand for any integer from 1 to n both inclusive, then A^ is equal to the coefficient of Jif"^ in the expansion oif{a + h) in powers of h. Accordingly we may obtain A^ by ordinary algebraical processes. For example, suppose we have to decompose , ir-, rrs into partial fractions. Denote the required {x — a) {x — by '■ partial fractions by — '^^ h^. + + — ^ {x-ay {x-af x-a '^ {x-by^ (^x-by-''^ x-b' Here/(cc) = (a;-6)~^; therefore ^, is equal to the coeffi- cient of A'"' in the expansion of (a-J + A)"^ in powers of //. 38 RATIONAL FRACTIONS. The expansion can be effected by the Binomial Theorem; thus we obtain A = p(p + l)...{p+r-2) J-jr' r | r-l ' {a-b) p-tr-l ' Similarly if s stand for any integer from 1 to p, both inclusive, then B^ is equal to the coefficient of h'~^ in the ex- pansion of (6 — a + li)'" in powers of h. III. Suppose that and F{.) = [l-^)\l-^,) ^1-^.J; here 6 {x) and F {x) are of the same dimensions. By decom- posing: 1, ' { we obtain the term ^^ together with a series of ^ '^ F{x) h partial fractions, a pair of which may be denoted "by x — p x + p' where p stands for -j- . Then, by Art. 19, ^^-F'{py ^^ F'{-py Let h be less than 1c, and suppose n to increase indefinitely; then the term j^^ vanishes. And, by Plane Trigonometry, Chapter xxiii. we have , , , sin/^aj r,. > sinA;a; RATIONAL FRACTIONS. 39 therefore ^ (p) = , , and since sin kp = 0, we have a; + /3 /i 2r7r sin a; - p a; + /3 A cos kp \x — p x + p hrir 7 7 ( 1 ^"'^' A/j cos rir \x jj- Hence finally, if h be less than k, . hrtr . , r sm — ,— &VQ.hx ^ ^ ^* = 27r2 sm kx cos rTT [k x —rir ) where ^ denotes a summation with respect to r from r = 1 to r = oo . The method of this example may be applied in other similar cases. IV. Some additional information on the theory of the decomposition of rational fractions will be found in the first volume of Serret's Cours cTAlg^bre Saperieure, 1866. Sug- gestions which are intended to diminish the numerical labour involved in the process of decomposition will be found in the Cambridge and Dublin Mathematical Journal, Vol. iii., in the Mathematician, Vol. ill., and in the Quarterly Journal of Ma- thematics, Vol. V. 40 EXAMPLES. EXAMPLES. , f dx 1, (x-iy 1 ^ _i2a;+l „ Cx'-l , , /a;-2\i .2 K ^1 ^_^ r2a;'-3a' , 5 ^ ^,x 1 , a^-c — 7 ^4- ao; = 77- tan 7- ^og — — - J x*-a* 2a a 4:a °x + c j{x' + l){x' + x + l) 2 * x' + l ^ ^/S V3 a; + a. ^- '' ■^ + l)(a;''' + a; + l) 2^ x' + l ^"^ ^ f x^dx 1 , X — 1J2 _i X 7- j^q:^^ = 6^^«^+i + -F*^" V2' 8. . . . . . dx = - log - /: x' + x'+l 2 ='a;' + ^ + l' 9- -7 Tv^ ^dx = x + \og {x-l){x-2) ^ ^iC-1 — - tan ^ a;. EXAMPLES. 41 , . r a;c?a; 2 1 1 , /i . n ^*- j(l+a:)(l+i>^)''(l+aO~5 14-2^ 2^og^-L+^V /■ x'dx _ 1 , a;'^ — ic \/2 + 1 J Z+1 ~ 4V2 ° ^N^^2T 1 + ^ [tan-^ {x V2 + 1) + tan"^ {x v/2 - 1)}. + ^ {tan"' (2a; - V3) - tan"' (2^ + V3)}. ^ Y " 17. L,,-, "^ o. . Assume 1 — ?/' = vV. 18. It- c-oTTi is ?r-^ • Assume y = -, . J (1 + rr) ^(1 + 3^ H- 3«;-) -^ 1 + « 42 CHAPTER III. FORMULAE OF REDUCTION. •SO. Let a + hx"" be denoted by X; by integration by parts we have L~-^X^ dx = ^^ - f^ pX'-' ~ dx J m J 771 ax ^^^-^Mh'^^n-^XP-^dx (1). m m J The equation (1) is called &, formida of reduction; by means of it we make the integral of ic™'' X'' depend on that of a;'""^""^jr''~\ In the same way the latter integral can be made to depend on that of a;'""^^""^X'^^; and thus, if p be an integer we may proceed until we arrive at a;"'*"*"^X-^~^, that is a;™'^"''"\ which is immediately integrable. From (1), by transposition, ♦ r m+„-i^p-i ^^ ^ <^ _ ^ L--^x' dx. J bnp onpj Change m into 7n — n and p into ^ + 1 ; thus {x'^-X'dx = f^^^'l _ '^~\, U-^-X^^' dx...{2). J bn{p + l) hn{p + l)J This formula may be used when we wish to make the integral of cc"'Z^ depend on another in which the exponent of X is diminished and that of X increased. For example, if m = 3, w = 2, and ^ = — |, we have r x^dx _ X 1 r dx FORMULA OF EEDUCTION. 43 The latter integral has already been determined, and thus the proposed integration is accomplished. Since L^-^Y" dx = fa;"'-'X^-' (a + hx'') dx = a lx'"-'X'-' dx + 6 [*•'"+"-' X"-^ dx, we have by (1) ^^' - ^ [a;"'"^"-' J''-' dx==a \ x^'-'X^' dx + h [x^'^-'X^'dx, m m J J J therefore ( x^'-'X^'' dx = ^^ - ^ ^'"^ + ""^^ f a;™-^8-» X'-' dx. J am am J Change p into ^:) + 1, and we have (x-^X^ dx = ^^^" - ^ ^^^^ + ^^ + ^^ f o;-"-^ Z^ ^^ (3). Change m into m — n and transpose, then f.2;"'-^X^£?a;- ■^""'""^''' (m-n)a f„.-n-^j, ^^^^u), J b [m + np) b [m + np) J • • • v >• We have already obtained from (1) by transposition J bnp bnpj also [a;"'-^Z^ dx = a f cc'-'X-^"' which may r x"*~^^dx be written I -r-^ r ; if in equation (4) of the jueceding J Y yZa -^ Xj Article we make h = — 1, n = 1, p = — \, and change a and m into 2a and ??i + \ respectively, we have /■ x^dx x^~^s/{1ax-x') a {2m -1) f x'^'^dx J \j{2ax - a") "" m "^ 7?i J7(2^^^^^ (3), which of course may be found independently. 32. In equation (6) of Art. 30 put a = c\ m = 1, n = 2, 5 = 1, and p = — r; thus [ dx _ X 2r — Srdx J{x'T^~2{r-l)c'{af + c'r''^2{r-l)c'J{x' + cT'' This formula Avill serve to reduce the form {Ax + B) dx k 32\r > {^x'-2ojc + a^ + ^y wdiich occurs in Art. 18 ; for this last expression may be written thus, ^ A {x — a) dx . . _ /* dx that is _ A 1 .. p, r dx ~2{r-l)[{x- af^^Y' ^ 1 {(^ -°-' + ^')Y ' "By putting cc - a = cc', we have r dx _ f dx' and thus the above formula becomes applicable. 46 FORMULA OF REDUCTION. 83. These formulae of reduction are most useful when th( integral has to be taken between certain limits. Suppos( <}){x), x{^)> '^{^)> functions of ic, such that i(f){x) dx='x_{x) + j-^ (x) dx, then I cf){x) dx = x {b) —%(«)+/ '^ (^) dx, J a J a as is obvious from Art. 3. For example, it may be shewn that n ({c' - x'fdx = ^^^'-fy + J!^ [(c^ _ ^2) f-^ dx ; ti . . . - suppose -x a j^ositive quantity, then x (c^ — x"^) ^ vanishes both when x = and when x = c. Hence {c' - x') ' dx = -^ (c' - xy dx. The following is a similar example. By integration by j)arts jx^' (1 - xy-' dx = - ^^^" or' + ^ L"-^ (1 - xfdx. Hence f a'"' (1 - x^ dx = ^^ I x"^' {1 - cK)"tZa:. Thus if r be an integer we may reduce the integral to r^ 1 (1 — xY^'~^ dx, that is to ^ ; hence Jo ?i + r-l' 34. The integration of trigonometrical functions is faci- litated by formula; of reduction. Let (sin x, cos x) denote FORMULA OF REDUCTION. 47 any function of sin x and cos x ; tlicn if wo put sin x = z, we have c C clx (f) (sin X, cos x)dx=\(f) [z, V(l - z^)] ^ ^^ For example, let ^ (sin a;, cos x) = sin" a; cos' a; ; then [sin" X cos' xdx = U" (1 - z'^^^-'^dz (2) . If in the six foraiulce of Ai't. 30 we put a = 1, h = -l, n = 2, j3 = i {q — 1), we have /"s-'-i (1 _ z'y^^-'^dz m m J 4+1 2 + 1 J ^ ^ .-(1 -/)'"" ^ m±£±l r «« (1 _ ,=,.„_„^, ??z + 2 — 1 m + q — lj 2 + 1 g + 1 J ^ ^ If we put m=p + l, and ^^ = sin x, the first of the above equations becomes f • V n 1 sin^^^ X cos'"^ a; , 0' — 1 r . ,^,, 0-2 j sm" a cos^ a;c?a: = ^ h ~ ^- sm''+- a; cos" xdx^ J _?J + 1 ^ + lj and similarly the other five equations may be expressed. 48 FORMULA OF REDUCTION. 35. The following is a very important case : r • [ d cos oc /sin" xdx = — I — -T — sin""^ xdx = — cos X sin""^ x-\- {n—l)\ cos'' x sin""^ xdx = — cos X sin""^ a; + (n — 1) I (1 — sin^ a;) sin"~^ xdx. Transposing, we have n I sin" xdx = — cos x sin"~^ a:; + (n — 1) Isin""'^ xdx ; ,, „ f ■ n 7 cos o; sin'^'V-r ?i-ir. „_2 ^ therefore sm xdx = 1 1 sm xdx. n n From the last equation we deduce, if n be positive and greater than unity, L sin" xdx = 1 sin" ^ xdx. ?i ^0 Similarly, if n be positive and greater than 3, /, in- _ w — 3 r^"" . sin" ^ icc?^; = sin""* ircZ^. n — 2i '' -^.' Proceeding thus, if n be an even positive integer we shall arrive at I dx or ^tt ; if n be an ocZcZ positive integer we •' shall arrive at sin xdx, w^hich is unity. Hence, if n be a -'0 positive integer, I sin" xdx = — — y^ / .. »' Z (n even), Jo n{n-2){n-4!) 3 These two results hold if we change sin x into cos x, as will be found on investigation. 36. From the preceding results we may deduce an im- portant theorem, called Wallis's Formula. FORMULAE OF REDUCTION. 49 Suppose n an even positive integer ; then rsin"^^^ = '^ 71^3 n-5 8 1 TT * • n-i J n-2 11-4 n-6 2 . sm xdx = r-. — ;, . — — -. (2). 1 w — Iw — 3?i — 3 /, Now it is obvious that I sin""' xdx is less than Jo sin" ^ xdx and greater than I sin" icc?^ ; because each Jo Jo element of the first integral is less than the corresponding element of the second integral and greater than the corre- sponding element of the third integral. And it has been shewn that sin" xdx - ^^ - 1 sin"~^ xdx n I sin" xdx -, Therefore ~ is less than 1 and greater than . I sm" xdx Jo Hence the ratio of the right-hand member of (1) to the right-hand member of (2) is less than unity and greater than : thus n TT. • , ,, 2.2.4.4.6.6 (n-2)(w-2) — is greater than - — - — - — = — ^— ,= -. ^rz-, ^\ . 2 *= 1.3.3.5.5.7 (?i-3)(w-l) 2.2.4.4.6.6 {n-2){n-2) n and less than - — „ „ ., — ^-^ > 1-..~, , v t ■ 1.3.3.0.0.7 (w - 3) (n - 1) ?i — 1 EXAMPLES. n 1. (d' + xydx=-^ r-^ + -v-r (« + ^ ) f^-^- J ^ ' 71 + 1 n + l J T. I. c. 4 50 EXAMPLES. r ,, , X-" ' V'lax - m + 2 j ^^ ^ 3. \x sf(2ax -x'')dx = -^ (2ax - x'f- + a j^/{2ax - x^) dx. 4. a; i^[zax — a; } aa; = -^ . Jo ^ 5. [ xV(2aa; - a;^) ^^ = - 1 {2ax-ar)^ + ^-^jxj(2ax - x') dx 6. x^ J{2ax — x") dx = JO r2a 77ra* 7. I ic' ^(2aa; — x^)dx = — q- . 8. fa;" (log xr dx = "^"^ ^^^^/''^"' - ^ f^" (log a.)'""^ dx. J ^ ° 71+1 W + lj r x"'^^ ( 2 2 9. J^" (log o.)^ c^o. = ^^-^ |(log xY - ^^ log ^ + ^^^:^ 5«V 10. i^\ec'ede = i ^^- Jo v(«+^) U v"- 1 2. [sin' ^ cos' 9de = -l cos" ^ + i cos' ^. 13. [ f = 3 (tan 61 - cot 6') + i (tan' ^ - cot' 0). J sm ^ cos a fsm''ed9_ sine ^. 1 - sm ^ ^ • J "^s' ^ ~ 2 cos'-' '^ ^ '° 1 + sin 6* • EXAMPLES. 51 15. I ' (cos 2^)^ cos ede= ^"L^J^ . Assume J {2) sin 6 = sin . IG. p(a-^^)cos-^^x=(l + ^y^. 17. fvers"^ -J (Za; = (tt- - 4<) a. /••'' sin'a;c/^ c'^-l, ,, , v 2-c 18. V = — s- log (1 + c) + -s-r • Jo 1 + ccosa; c* ® ^ 2c' 19. If ^ (/O = I (1 + c cos a;)"" (fe, shew that {n - 1) (1 - c') (x) dx={b- a) (j){a + d{b- a)]. J a Similarly if yjr (x) retains the same sign while x lies be- tween a and b, we may prove that f J a ^ (x) -^Ir (x) dx = (f) [a + 6 {b — a)] j i|r (x) dx. •I a 41. The truth of the equation rb re rb I (ji{x)dx=j {x)dx = -r {x)dx=l j){a — x)dx (3). MISCELLANEOUS REMARKS. 55 For putting a — a: = z wc have j (f)(a — x)dx = — j (J) (z) dz, therefore I (f){a — x)dx—— cf) (z) dz J J a = ["(/, (.) dz, by (2). ra fa Of course 1 (f>(z) dz— j if) (x) dx, since it is indifferent whe- •' ''0 ther we use the symbol x or z in obtaining a result which does not involve x or z. We have from (1) I (j) (x) dx= \ (f){x) dx+ I cf) (x) dx. J J -la The second integral on the right-hand side, by changing X into 2a — x , will be found equal to 1 (2(X — o:) dx or (2a — x) dx. •'a J a Hence I <^ (a;) cZa; = I {^ {x) + ^ (2a — x)] dx. •J ■Jo Hence, i£ (f) (x) = cf) {2a — x) for all values of x comprised between and a, we have [ (l){x)dx='2l (x)dx (4), •'0 -^ and if (f> (2a — x) = — <^ (x), we have /•2a <}>{x)dx = (5). For example, sill' d d9= 2 [^\m' Odd by (4) n •' 56 MISCELLANEOUS REMARKS. rir and I cos'edd = by (5). •'0 42. Such equations as those just given should receive careful attention from the student, and he should not leave them until he recognises their obvious and self-evident truth. /, cos' 6 dd is by definition the limit when n is infinite of the series h [cos' h + cos' 2A + cos' U + cos' (w - 1) h], where nh = tt. Now cos' h = — cos' (/I - 1) h, cos' 2/« = — cos' {n — 2)h, ; thus the positive terms of the series just balance the negative terms and leave zero as the result. In the same way the truth of sin' dd6 = 2 sin' 6 d9 follows immediately from the definition of integration, and the fact that the sine of an angle is equal to the sine of the sup- plemental angle. Suppose h greater than a, and <^ {x) always positive be- tween the limits a and h oi x; then every term in the series 2^ {x) Aa; is positive, and hence the limit 1 <^ [x) dx must he a positive quantity. 43. All the statements which have been made suppose that the function which is to be integrated is always finite between the limits of integration; for it must be remem- bered that this condition is included in the word continuous of the fundamental proposition, Art. 2. If therefore the func- tion to be integrated becomes infinite between the limits of integration, the rules of integration cannot be applied; at least the case must be specially examined. r dx ^ . . Consider I ' — r ; the value of this integral is 2 — 2^/(1 — «). Here the function to be integrated becomes infinite when ^=1; but the expression 2 — 2^/(1— a) is finite when a = 1. Hence in this case we may write MISCELLANEOUS REMARKS. 57 •1 a clx V(l - ^r) = 2, provided that we regard this as an abbrevia- te j? tion of the following statement: " , ,,^ , *= 'oV(l-^) is always finite if a be any quantity less than unity, and by taking a suffi- ciently near to unity, we can make the value of the integral differ as little as we please from 2." Next take dx ; the value of this integral is - log (1— a), which increases indefinitely as a approaches to unity. Hence in this case we may write \ ^ = go provided that we X reo-ard this as an abbreviation of the following statement : fd fly " I increases indefinitely as a approaches to unity, and by taking a sufficiently near to unity we can make the inte- gral greater than any assigned quantity." Next consider 7- rr,; the integral here is :; [l—xY ° l — x If without remarking that the function to be integrated be- comes infinite when x = 1, we propose to find the value of the integral between the limits and 2, we obtain —1 — 1, that is — 2. But this is obviously false, for in this case every term of the series indicated by S (x) Ax is positive, and therefore the limit cannot be negative. In fact . ' ,. and .-. _ .^ are both infinite. This example shews that the ordinary rules for integrating between assigned limits cannot be used when the function to be integrated becomes infinite between those limits. 44. In the fundamental investigation in Ai't. 2, of the rb value of j) (x) dx, the limits a and b are supposed to be finite as well as the function ^ [x). But we shall often find it convenient to suppose one or both of the limits i7ifinite, as we will now indicate by examples. 58 MISCELLANEOUS REMARKS. Consider I .; 5 ; the inte;]jral is tan"*j?. Hence ( Jl + aJ ^ Jol+i = tan"^ a ; the larger a becomes, the nearer tan~^a approach( IT to - , and by taking a sufficiently large, we can make tan~^ TV differ as little as we please from - ; hence we may wri1 as an abbreviation of this statement d. X IT l^x' 2 • Similarly I — - = log (1 + a) ; and by taking a laro enough we can make log (1 + a) greater than any assigne quantity. Hence for abbreviation Ave may \vrite /. dx z =CO. l + x 45. Suppose the function ^ (x) to become infinite om between the limits a and b, namely, when x— c. We cannc then apply the ordinary rules of integration to 1 (p [x] dx; bi J a we may apply those rules to "C-;u. rb (x) dx+ i (j) (x) dx for any assigned value of fjb however small. The limit of tl: last expression when yu, is diminished indefinitely is called I Cauchy the principal value of the integral I (x) dx. J a For example, let cf) {x) = c — x ,, , '1^ dx , c — a then = log - — , a c-x "^ /J, , I - dx [^ dx , h —c and I = - - — ~ = - log MISCELLANEOUS EEMARKS. 59 ence the principal value is log log , that is c — a f dx X 46. The value of 7^^ ^r is sin"' - ; hence jJia—x) a I dx itudents are sometimes doubtful respecting the value which 5 to be assigned to sin"' (1) and to sin"' (-1) in such a result s the above. Suppose we assume ^=asin^; thus the integral lecomes \ dO or 6. Now x increases from — a to a, hence he limits assigned to 6 must be such as correspond to this ange of values of x. When x = -a then 6 may have any alue contained in the formula (4/i -1) ^ , where n is any Qteger. Suppose we take the value (4w — 1) ^ , where n is ome definite integer, then corresponding to the value x = a ve must take 6 = (4:n - 1) "^ + ir ; this will be obvious on ixamination, because x is to change from — a to + a, so that t continualbj increases and only once passes through the value ero. Hence -77-2 T\ = '^^ As this point is frequently found to be difficult by begin- lers we will consider another example. Suppose we require]^ ^2-^;^,-^. ^'^ f sec'ddO 1^ _i/tan^\ We have - ^ , . 2 a = - tan -— I ; J a + tan U a \ a / md as the integral is to be taken between the limits and tt, 60 MISCELLANEOUS REMARKS. we must determine the values of tan"' ( ) in these eases, , /tan ^\ \ a / Suppose 0, 6^, 0^, 0^, ...$„, TT, to be a series of quantities in order of magnitude. By the nature of integration I tidd = I ' ud6 + / ' udO + I \idd +... + ( ^icW. Jo Jo J di J 3j J e„ Now each of the integrals on the right-hand side can be made as small as we please by increasing 7i and making two consecutive quantities as 0^ and ^,.^^ to differ as little as we j^lease. Hence we see that the symbol tan"^ |-^- — J must be so taken that tan~^ (— — ^^j — tan"' ( ^j shall diminish indefinitely when 6^_^_^ — 6^ does so. /tan ^\ Therefore tan"^ [ — — 1 must increase continuously with 0, ; and it can only pass once through an odd multiple of ^ while 6 passes from to tt. If then we take mir for the value of tan"' ( ] when ^ = 0, we must take {m + l)7r for the value when ^ = TT ; and thus the value of the integral between the ' assiojned limits is - . A common mistake with beginners is to take the second value the same as the first, instead of taking the second value to exceed the first by tt; thus the value of the proposed inte- gral is made to be zero, which contradicts the last paragi-aph of Art. 42. . . . f (a - c cos 6) d9 Agam, suppose we require J^ ^. ^. ,. _ 2^^ cos ^ ' r (a-c cos 6) dO _ ^ [L . a^ - c" \ .g ja^ + c'-2accos^~2cJ| "^ a' + c' - 2ac cos ^j ' TT a^— & /■"■ Thus the required integi'al is ^; — | — ^ — — ^ ^ 2a 2a Jo a dd + c""— 2accos^ ' \ S^ow r di MISCELLANEOUS REIMARKS. 61 de ac cos 6 [ sec^edO __2 -ifa±c. ^ .\ 2 TT When taken between the assigned limits this gives —^ 2-^ 2 TT . if a is greater than c, and 5 2 „- if a is less than c. Hence the value of the proposed integral is - if a is greater than c, and zero if a is less than c. 47. The Integral Calculus furnishes simple demonstra- tions of some important theorems relating to the convergence and divergence of series. If (j) (x) continually diminish as x increases tvithout limit from the value a, then the infinite series (a + 1) and is greater than ^ (a + !2). Proceeding in this way we can shew that the integral 1 <^ [x) dx is less than <^(a) + be greater than unity, and is infinite ifp be equal to unity or less than unity. Hence the theorem follows by Art, 47. 49. We now proceed to investigate rules for determining whether a proposed infinite series is convergent or divergent. Let there be an infinite series 1 denote the sreneral term by . , , . It is obvious that the series is certainly divergent unless -^ (x) increases indefinitely with X : we will suppose that -v/^ {x) increases indefinitely with X. MISCELLANEOUS EEMARKS. G3 I. Suppose, as x increases indefinitely from a certain value a, that , . is always less than -j, where G and 'p are constants, ja being greater than unity; then the proposed series is less than a certain series which is known to be con- vergent by Art. 47: therefore the proposed series is con- verirent. 1 G If -r-,—s^ is less than — . , then x^ is less than G^ ioc); and, y {x) x^ loo- (3\|r (x) taking logarithms, we find that » is less than — —^ — ^ *= ° ^ logic The last expression assumes the form — when x is infinite; by the ordinary rules for evaluating such an expression we iCvIr ix^ obtain — p- — - as its equivalent. Therefore if the limit of ^ {^) tc^ (X) , , , , when X is infinite, is greater than unity, we can find a ^{x) ' ' '=' -'' quantity p, greater than unity, such that x^ is always less ' than Cyjr [x). Hence the proposed series is convergent. In a similar manner it may be shewn that if the limit of : , / , when X is infinite, is less than unity, we can find a ylr{x)' ' /' quantity p, less than unity, such that x'' is always greater than Cyjr (x). Hence the proposed series is greater than a certain divergent series, and is therefore itself divergent. xyl/* (x) II. Thus if the limit of , /, -, when x is infinite, is Y[x) either greater than unity or less than unity, the nature of the series is determined : but if this limit is unity, further investi- gation is required. Suppose, as x increases indefinitely from a certain value a, 1 C that , , , is always less than — . , ,,„ , where G and n are ■^{x) '' x\K{x)Y constants, p being greater than unity; then the j)roposed series is less than a certain series which is known to b'^ convergent by Art. 48 : therefore the proposed series is con- vergent. C4 MISCELLANEOUS EEMAKKS. 1 . G If , , . is less than — ,^ , ,,„ , then lX(x)\'' is less y [x) X \k{oc)Y ^ ' C-dr (x) . . than — L-\J- ^ and, taking logarithms, we find that p is less than — ■,.,,, — , that is, » is less than — ^ — \.\ , . ^ . X {x) ' ' ^ X^ (x) The limit of this expression when x is infinite is the same as the limit of X(x)\ ],/ — Ih. Hence if the limit of \ [ir{x) J _ this last expression is greater than unity the proposed series is convergent. In a similar manner it may be shewn that if the limit of the last expression is less than unity the proposed series is divergent. {X'U/^ (xj I , ■ , — Ir , when x is in- finite, is also unity, further investigation is required : the general term of the proposed series may then be compared •tl 1 xX{x){X'{x)]^' Proceeding in this way we obtain the following result : ^^* ^^ = ^^^' ^"* ^^ = ^ ^"'^ (^» - ^)' ^'^ ^^ = ^' ^^^ ^^^ ~ ^^' and generally let P^ = X"' (x) (P^_i — 1) ; and supj)0se that P^ is the first of the terms P^,P^,P„,... which has its limit, when X is infinite, different from unity: then the proposed series is convergent or divercjent accordincj as the limit of P is greater than unity or less than unity. We have supposed the general term of the series to be denoted by ; if it be denoted by % (^) we have to put r instead of i/r (x) in the preceding result : hence A. V / MISCELLANEOUS REMARKS. 65 we find that Po = - ^^. "^ , and that this is the only modifi- X W cation required. 50. Another form may be given to the result. "We know by the Differential Calculus that % (a? + 1) - % (a;) = %' (.t; + 6), where 6 is some proper fraction. Hence therefore the limit, when x is infinite, of '^,\ is equal X i^) to the limit of x\l ^ , i\ r • Thus we may put I X{^ + ^)) p =,^\. Xi^) _ i\ in the result of Art. 49. The theorems in Arts. 47, 48, and 49 have been derived from De Morgan's Differential and Integral Calculus ; there is a valuable memoir on the subject of convergence by Bertrand in the seventh volume of the first series of Liou- ville's Journal de MatMmatiques. An elementary demon- stration of the theorem of Art. 48 will also be found in the Algehra, Chapter LVI. 51. Required I logsina;^?^. J By equation (3) of Art. 41, j log sin xdx = [ ' log sin \^-x\dx=^\ log cos xdx. Hence, putting y for the required integral, 2y = (log sin X + log cos x) dx = log (sin X cos x) dx J T. L C. 6G MISCELLANEOUS REMAEKS. P'f , sin 2x 7 = I log —2— «•* =: I (log sin 2a3 — log 2] dx -' /•if . 1 = I log sin Ixdx — ^tt log 2. But putting 2x = a;', we have log sin 2xdx = ^ j log sin a;' c?a;' - '' = logsina;d!.r, by equation (4) of Ai't. 41 ; TT therefore 2_5/ = ?/ — ^ log 2, therefore 2^ ~ 9 ^^S 9 • Again, I ^logsin^cZ^= (tt- ^)^logsin ^fZ''{x-h)-(a + /0 = (a)+7.f(«) + ~<^"(«) + (^ "(«) Iji IIL + Thus the excess of (f>(a + h) over the sum of the first n + 1 terms of its expansion by Taylor's Theorem is expressed by the definite integral }- ( z''6''^Ha + h-z)dz. 5—2 6cS MISCELLANEOUS EEMARKS. By means of the first result in Art. 40, we may put for this definite integral <}>"^'{a + h-eh), where ^ is a proper fraction. By means of the second result in Art. 40, we may put for this definite integral or where ^^ is also a proper fraction. 53. Bernoulli s Series. By integration by parts we have I ^ [x) dx = X(f) (x) — I X(j>' {x) dx, jxcji' (x) dx=^(l>'{x)-j'^ I/)" {x) dx, \x^(f)"{x)dx = -^(j)"{x) — I -^^"'{x)dx, Thus \(f){x)dx = X(}){x) - =— Q (f)'(x)+j^"{x) Therefore, [a ff fj3 j^(j){x)dx=a(f>{d)-^—^^'{a')+^f{a) hi \n Jo mSCELLANEOUS RKAIAEKS. 69 This series on the right hand is called Bernoulli's series. In »me cases this process might be of use in obtaining I cf) (x) dx ; Jo r example, if <^ (.r) be any rational algebraical function of le (?i — 1)'" degree, " {x) is zero ; or it might happen that k''(/)" (x) dx could be found more easily than 1 ^ [x) dx. Or ;-ain, we may require only an approximate value of I ra (f) (x) dx and the integral / x"(p" {x) dx might be small lough to be neglected. 54. By adopting different methods of integrating a func- 3n, we may apparently sometimes arrive at different results, ut we know {Differential Calculus, Art. 102) that two func- ms which have the same differential coefficient can differ dy by a constant, so that any two results which we obtain ust either be identical or differ by a constant. Take for ample I {a^ + h) {a'x + h') dx ; tegrate by parts, thus we obtain {ax + hY . , , fa' , j.^ - at is {ax-^hYJa'x^- h') a' {ax + b)' 2a Qa' If we integrate by parts in another way, we can obtain {a'x + by{ax + b) a{a'x + b'Y 2a 6a" ' lerefore (ax + by [Sa (a'x + b') - a {ax + b) ] . (a'x + b'Y [Za' (ax + b)-a (a'x + b')} 6a" 70 MISCELLANEOUS REIVrARKS. can differ only by a constant. Hence multiplying by 6aV^ we have a' [ax + hf [Sa {ax + h') - a (ax + h)] - a^ {ax + by {na{ax +b)-a {ax + b'}]= 0, where C is some constant. This might of course be verified by common reduction. We may easily determine the value of C; for since it is independent of x we may suppose ax + b = 0, that is, x = ; then the left-hand member a becomes (ab' — a'b)^, which is consequently the value of C. Similarly from I {ax +b) dx+ I (ax + b') dx = {{a + a) x + b + b'] dx we infer {ax + b)\ {a'x + by {(a + a')x + b + b'Y- , , ^-— — ^ + ^ — -^-^ = ^-^ ^ + constant. 2a za 2 {a + a) Multiply by 2aa' {a + a) and then determine the constant by supposing x = 0', thus we obtain the identity a {a + a) {ax + b)~ + a {a + a') {ax + b'f = aa! [{a + a)x + b + b'Y + {ba' - b'af. If we integrate a function between assigned limits the result must be the same by whatever method we proceed ; and in this manner Ave may obtain various algebraical identities. Take for example I x'"{l — x)"dx, where w is a positive integer. We have, by integrating by parts, fx'^ (1 - xY dx = ^"'"'(^-^)" + _J^ [^-+1 (1 _ ^y-^ dx ; J ^ ' m + 1 m + lj ^ ^ therefore [ a;"' (1 - xj dx = -^^ f x'^^' (1 - a;)""' dx. MISCELLANEOUS REiLiRKS. 71 Proceeding in this way wo obtain J, ^ ^ {in + l){}n + 2) ...{m + n + 1) ^^ Again fa;"' (1 - x)" dx = f^'" jl - wa; + — ^^-^^ a;* - ... 1 dx _ 1 n 1 n{ii-l) 1 iNn 1 /9N m-\-l 1 ' 7?i + 2 1.2 m + 3 ^ m+w+1 Therefore the expressions on the right-hand side of (1) and (2) are equal if n be any positive integer. 55. By l(/) {x) dx we indicate the function of which (^{x) is the differential coefficient; suppose this to be 'v^(.r). Then we may require the function of which '^{x) is the differential coefficient, which we denote by \-^{x)dx, or by \\ (j){x) dxdx, and so on. For example, the integral of e'^ is j e'" + C^, where (7^ is a constant ; the integral of this is j-^e'' + C,x + C^; the integ-ral of this is 'O c . where -^ being still a constant may be denoted for simplicity by B if we please. Proceeding thus we should find as the result of integrating e^' successively for n times ^ + ^,x"-^ + ^,^"-^+ +A^_,x + A^, where A^, A^, A^ are constants. u. u„ 72 MISCELLANEOUS REMARKS. It is easy to express a repeated integral in terms of simple integrals. For let u be any function of x ; let t/j = ludx; let 21^= lic^dx] let Wg = \u/lx; and so on. By integration by parts we have = I u^dx — xu^ — \ X -^ dx = X ludx — Ixudx ; = I u^dx = l\x { udx — I xudx)- dx ; therefore by integration by parts, 11^ = -^ \ udx — I -^ udx — X \ xudx + Ix'udx = -^ I udx — X jxu dx+ ix Ix'udx. The general formula is \n w,^^j = x^ I udx — nx^~^ Ixudx -{ — ^j — ~ ^""^ Ix^udx — +(-ir "^"-^^--^ ^"-±^^-fruc^^+ + {-iyjx"udx. The truth of this formula may be easily established by induction ; for if we differentiate both sides we obtain a similar formula with 7i — 1 in place of n. MISCELLA^^EOUS EXA3IPLES. 73 2a > MISCELLANEOUS EXAMPLES. I r x'-dx 57ra^ . r [a? - e'x^) dx _'rr^(. _e\ dx ''+x'){h'+x^) 2ab(a + b)' ). If (f){x) = (f>{a + x), and n is a positive integer, shew that rna ra I <^ (a-) cZa; = ?i I {x) dx. Jo J !. Shew that I (^ (a;) cZ« = -^ 1 <^(— i,-^ + -^a;J (Za;. r ci 4.1 J. r^siniccZa; tt^ zr^i '. Shew that -— — = — , (Change x into tt - x .) J Q -L "T" COS OG T i. Shew that r (2ax-x')^Yers-'-dx = ?^. Jo a 16 (Change x into 2a — x.) I. Find the limit when n is infinite of 1 . 1 , 1 1 n ^{n'-l) ' VK-2^} "^VK-(n-in- Result ^. 74 MISCELLANEOUS EXAMPLES. 10. Find the limit when n is infinite of i- -i + i- + TT-i + \7^ + 7r-\ +...tow terms Result. i- (x) denote any function of x ; then we have ;en that the integral of cf) (x) is a quantity ii such that - = {cc). The integral may also be regarded as the limit - a certain sum (see Arts. 2..,G), and hence is derived the rmbol / (f) (x) dx by which the integral is denoted. We ow proceed to extend these conceptions of an integral to ises where we have more than one independent variable. 57. Suppose we have to find the value of ^t which satis- es the equation , , = <^ {x, y), where ^ [x, y) is a function [ the independent variables x and y. The equation may e written dy\ dv dy ' v= -j-. Thus V must be a function such that if we differ- dx ntiate it with respect to y, considering x as constant, the 3sult will be (}> (x, y). We may therefore put latis _|=|^(a.^y)f;y. 78 DOUBLE INTEGRATION. Hence u must be sucli a function that if we differentiate it with respect to x, considering 7/ constant, the result will be the function denoted by I (f> {x, y) dy. Hence u =\\j<^{^>y)dy\dx. The method of obtaining u may be described by saying that we first integrate ^ {x, y) with respect to y, and then integrate the result with respect to x. The above expression for u may be more concisely written thus, \\ j> {x, y) dy dx, or \\ cji (x, y) dx dy. ■ On this point of notation writers are not quite uniform ; we shall in the present work adopt the latter form, that is, of the two symbols dx and dy we shall put dy to the right, when we consider the integration with respect to y i3erformed before the integration with respect to a?, and vice versa. 58. We might find u by integi'ating first with respect to X and then with res]3ect to y ; this process would be indicated by the equation ^= jj (1^ (.^> y) dy dx. 59. Since we have thus two methods of finding w from the d\i equation t—t" = (^) y)y it "^^ill be desirable to investigate if more than one result can be obtained. Suppose then that u^ and u^ are two functions either of which when put for u satis- fies the given equation, so that ^ = ^{^,j) and ^ = *(»., 2,). We have, by subtraction, d'ii^ _ cPti^ __ ^ dxdy dxdy that is, -J- ( ^- ) = 0, where v = u, — il. dx \dyj ^ ^ DOUBLE INTEGRATION. 79 Now from an equation -7- = we infer that w must be a constant, that is, must be a comtant so far as relates to x ; in other words, w cannot be a function of x, but ina7/ be a func- tion of any other variable which occurs in the question we are consiJerinu:. o Thus from the equation -r-|-7-l = we infer that -r- dx \ajjj dij cannot be a function of x, but may be any arbitrary function of y. Thus we may put By integration we deduce V = if Of) dy + constant. Here the constant, as we call it, must not contain y, but may contain x\ we may denote it by %(^r). And \f{y)dy we will denote by -v/r (?/) ; thus finally ^ = 'f (.y) + %(^)- Therefore two values of u which satisfy the equation , — T- = {x, y) can only differ by the sum of two arbitrary functions, one of x only and the other of y only, 60. We shall now shew the connexion between double integration and summation. Let <^ [x, y) be a function of x and y, which remains continuous so long as x lies between the fixed values a and h, and y between the fixed values a and /3. Let a, x^, x^, x„_^, 6 be a series of quantities in order of magnitude; also let a, y^, y.^, y,„_^, /3 be another series of quantities in order of magnitude. Let x^-a^h^, x^-x^ = h.^, ^-^„-i = ^^i; also let y,-ci=l\, y._-y,^k, (3- y„,_^ = k„,. 80 DOUBLE INTEGRATION. We propose to find the limit of the sum of a certain series in which every term is of the form hjc^^ {x,_„ ?/^_J, where r takes all integral values between 1 and n inclusive, and s takes all integral values between 1 and m inclusive ; and ultimately m and n are to be supposed infinite ; also x^ and T/o are to be considered equivalent to a and a respectively. Thus we may take hkcf) (sc, y) as the type of the terms we wish to sum, or we may take AccA;/^ {x, y) as a still more expressive symbol. The series then is \[k^4>{a, a) + k,cf){a, y^) + k,(ji {a, y^ + ^-„. («, Vm-d] + h^ {k^(f> {x^, a) + k,^{x„ y,) + k^(f> {x^, y^ + \<^ {x^, ?/„,_J} + h^ {k^cf) {x^_^, a) + k^(f> {x„_„ y,) + + ^'> {x„.„ y „,.,)]. Consider one of the horizontal rows of terms, which we may write h^^^ {k^cji {x,, a) + k^(f> {x^, y,) + kj>{x,, y,) + k„,(f> {x,, O}. The limit of the series within the brackets when k^, k^,.. .k are indefinitely diminished is, by Art. 3, m /' J a ^{«^r>y)dy. Since this is the limit of the series, we may suppose the series itself equal to /•|3 J a where p^^^ ultimately vanishes. Let I ^{x^ ; then add all the horizontal rows and we obtain a result which we may de- note by Sli "^ (x) + %hp. dx. DOUBLE INTEGRATION. 81 Now diminish indefinitely each term of wliicli h Is the type, then S/i/3 vanishes, and we have finally rb I -^^ (x) dx ; J a that is, J U j>{x,7j)dij[ This is more concisely written b f? (f) (x, y) dxdy, dy being placed to the right of dx because the integration is performed first with respect to y. 61. We may again remind the student that writers are not all agreed as to the notation for double integrals. Thus we use I \ ^ {x, y) dxdy to imply the following order of J a J a operations : integrate ^ {x, y) with respect to y between the limits a and ^ ; then integrate the result with respect to x between the limits a and h. Some writers would denote the same order of operations by I I ^ {x, y) dydx. J a •! a G2. We might have found the limit of the sum in Art. GO by first taking all the terms in one vertical column, and then taking all the columns. In this way we should obtain as the r/S fb sum I I (f) (x, y) dydx; and consequently J a J a '•p rb rh ,-^ I (^> y) dydx =\ 1 4>{x,y) dxdy. J a J a J a J a The identity of these two expressions may also be esta- bhshed by the aid of Art. 59, as we will now shew. Let F{x, y) denote the integral of (j) (x, y) with respect to y, supposing x constant ; and let / (x, y) denote the integral of F (x, y) with respect to x supposing y constant. Then T. I. c. G 82 DOUBLE INTEGRATION, h rP rb (f) {w, y) dxdy = \ [F (x, ^) — F{x, a)] dx J a. J a = { F{x, /3)dx- [ F{x, a)dx J a J a =f{^> /3) -/(«, ^) -f{h, 0) +/(a, a) (1). Now let us first integrate {x, y) with respect to x, sup- posing y constant, and then integrate the result with respect to y, supposing x constant; \Qif\{x, y) denote the final result. Then we obtain ' \\[x, y)dydx=f^{b, ^)-f,{h, o)-f,{a, /3) +f,{a, a). ..(2). a. J a But, by Ai-t. 59, f^{^> y) =fix' 2/) +f (y) + %(^) (S), where "^ (y) is some function of y without x, and -^^ (x) is some function of x without y. By making use of (3) we shall find that the right-hand member of (2) reduces to the right-hand member of (1). The function ^(x, y) is assumed to be fijiite through the range of the integration : for that is involved in the notion of continuity : see Arts. 2 and 43. 63. Hitherto we have integrated both with respect to x and y between constant limits ; in applications of double integration, however, the limits in the fust integration are often functions of the other variable. Thus, for example, the rb r^ix) symbol I I <})(x, y) dxdy will denote the following opera- JaJxix) tions : first integrate with respect to y considering x con- stant ; suppose F {x, y) to be the integral ; then by taking the intecrral between the assigned limits we have the result F[x,^{x)]-F[x,x{x)}. We have finally to obtain the integral indicated by \f[x,^{x)]-F[x,x{x)]'\cIx. The only difference which is required in the summatory process of Art. GO is, that the quantities a, y^, y^, ...3/„,_i will DOUBLE INTEGRATION. 83 not have the same meaning in eacli horizontal I'ow. In the {r + 1)^ row, for example, that is, in we must consider a as standing for x{^r)> ^^<^ 2/i> 2/2' ^^ a series of quantities, such that x{^'r)> V^^IU, 2/m-i. '^ (^r), are in order of magnitude, and that the difference between any consecutive two ultimately vanishes. Hence, proceeding as before, we get ^ ix^, y) dy for the limit of the sum of the terms within the brackets in the {r + 1)"* row. 64. It is not necessary to suppose the same number of terms in all the horizontal rows ; for m is ultimately made indefinitely great, so that we obtain the same expression for the limit of the (r + 1)"' row whatever may be the number of terms with which we start. 65. "When the limits in the first integration are functions of the other variable we cannot perform the integrations in a different order, as in Art. Q'2, without special investigation to determine what the limits wdll then be. This question will be considered in Chapter XI. QQ. From the definition of double integration, it follows that when the limits of both integi'ations are constant, \\ (^) f (y) dxdy =j(f){a;)dxx j^lr{y) dy, supposing that the limits in j-^ (y) dy are the same as in the integration with respect to 1/ in the left-hand member, and the limits in I ^ (x) dx the same as in the integration with respect to X in the left-hand member. For the left-hand member is the limit of the sum of a series of terms, such as and the right-hand member is the limit of the product of h,{x,) +kcf>{x;) +h^c}i{x^) + h^c}> (a;„J, and Jc.yjr (t/J + k^f (y^) + k^f (yj + k^yjr (y^ J. 6—2 84 DOUBLE INTEGRATION. G7. The reader will now be able to extend the processes given in this Chapter to triple integrals and to inultiple integrals generally. The symbol I j (p (x, y, z) dxdydz will indicate that the following series of operations must be performed : integrate {x, y, z) with respect to z between the limits ^0 and ^^ considering x and y constant ; next integi-ate the result with respect to y between the limits tj^ and t]^ con- sidering X constant ; lastly integrate this result with respect to cc between the limits ^^ and l^j. Here ^^ ^^^ ?i ^'^^J be functions of both x and y ; and rj^ and rj^ may be functions of w. This triple integral is the limit of a certain series which may be denoted by %(fi (x, y, z) Aa; A^/ As;. MISCELLANEOUS EXAMPLES. Obtain the following eight integrals. / ^lia'-x') dx. (Put y = x".) Result. -X sin"^ — 2. \ J ix — a a- x"^ "^dx {x — a){x — h) {x — c) ' {a—h){a — c) {b—a){b—c) {c—a){c — h) ' I i2cn.xdx -r> 7, log {co5^ X + m^ sm^ x) 3. z r> ^— . Result '•/r + ni^ tan^ x' ' 2 (??r — 1) Result. — ^ log -^ na a + Y (ot -rx ) MISCELLANEOUS EXAMPLES. 85 5. I sec a; sec 2xdx. T. ,, 1 , 1 + V2sina; 1, l + sin^. lies lilt. -r=r lOOr ——. — - log -. . V2 ° 1 - V2 sm a; 2 ° 1 - sin a; ,. ,"tan a — tana; , 0. I ^ ; dec. tan a + tan a; Result, sin 2(X log sin (a +x) — x cos 2a. t7a r dx J x^ + a V" + a -r, , 1 1 x^ + ax+a^ 1 , _i a;a\/3 Result -r-s log -^s — ■ — ^ + ^^ 3 /o t^J^ -^ i • (a — hx^) dx /-p + ^ «■ J^^^f^i- (P-!-^-^) Result, cos ^ -77 — -■■ ■ ,, /v/(c + 4a6) 9. Find the limit when n is infinite of 1 f . TT . 27r . Stt . wtt - tt " „ ,, 1 -^sm- sm — sm — sm —> . Mesult. -^ { n n n n } z 10. Shew that fx (tan-^ xf dx = '^('^-l)+ log ^2. 11. Shew that 8 •' 0-' 0-' / e'''^dxdyd2 = -^--^ + e''- 12. Let ^ = 1 1 ?t- cZa; cZy, B= ijuvdxdy, G=ljv^dxdi/, and suppose the limits of the integrations the same in the three integrals ; then shew that A C is never less than B^. (See Example 21 at the end of Chapter IV.) 86 MISCELLANEOUS EXAMPLES, 13. If I 4> {z) dz is equal to unity, and j> (z) is always positive, shew that (\(j)(z) cos czdzj +( (f) [z) sin cz dzj is less than unity. (See History of...Prohahility, page 564.) 14. If ( (^ (z) dz is equal to unity, and (/> (2) is always positive, shew that j 2'^^(z)d2-l\ z(p {z) dz\ is positive. (See History of... Probability, page 566.) 87 CHAPTER VI. LENGTHS OF CURVES. Plane Curves. Rectangular co-ordinates. 68. Let P be any point on the curve APQ, and let x, y be its co-ordinates ; let s denote the length of the arc AP measured from a fixed point A up to P; O 'jP jtr JV' •» then {Differential Calculus, Art. 307) -(I)}- Hence ds dx s = i+'l)V- dy From the equation to the curve we may express -^^ in terms of x, and thus by integration s becomes known. 69. The process of finding the length of a curve is called the rectification of the curve, because we may suppose the question to be this : find a right line equal in length to any assigned portion of the curve. 88 LENGTHS OF CURVES. In the preceding Article we have shewn that the length of an arc of a curve will be known if a certain integral can be obtained. It may happen in many cases that this integral cannot be obtained. Whenever the 'length of an arc of a curve can be expressed in terms of one or both of the co- ordinates of the variable extremity of the arc, the curve is said to be rectijiahle. 70. Application to the Parabola. The equation to the parabola is ?/ = A/(4ax) ; hence dy _ /a ds _ I Ix + a\ _ dx" y x' 'dx~ y \ X ] ' thus s = \\/\- ) dx (See Example 6, page 19.) = >,/{ax + x^) + a log yx + \/(« + ^)] + 0. Here (7 denotes some constant quantity, that is, some quan- tity which does not depend upon x ; its value will depend upon the position of the fixed point from which the arc s is measured. If we measure from the vertex, then s vanishes with X ; hence to determine C we have a log \fa+ (7=0; and thus s = i\J{ax + x^) + a log [^/x + ^J [a + x)] — a log /y/a ,/ . 2\ , 1 \/x + ^J{a + x) = iJ(ax + ic^) 4- « log -r . \/a If then we require the length of the curve measured from the vertex to the point which has any assigned abscissa, we have only to put that assigned abscissa for x in the last expression. Thus, for example, for an extremity of the latus rectum x = a', hence the length of the arc between the vertex and one extremity of the latus rectum is 71. In tlie preceding Article we have found the value of the constant G, but in applying the formula to ascertain the lengths of assigned portions of curves this is not necessary. LENGTHS OF CURVES. 89 For suppose it Is required to find the length of the arc of a curve measured from the point whose abscissa is ^, up to the point Avhose abscissa is x.^. Let ■xjr {x) denote the integral of a/ \^ + [-7 ) \ > and let s^ and s^ be the lengths of arcs of the curve measured from any fixed point up to the points whose abscissa) are a?j and x^ respectively, so that s^ — s^ is the required length ; then hence s^=-^ (x) + C; s^^^jr (x^) + C ; therefore s.^ ~^i—'^ (■^2) "~ "^ (^i)* Hence to find the required length we have to put x^ and x^ Buccessively for a; in i/r (x) and subtract the first result from the second. Thus we need not take any notice of the constant C; in fact our result may be written -».= rA/n+(t)V^. xi dxj 72. Application to the Ci/cloid. In the cycloid, if the origin be at the vertex and the axis of ?/ the tangent at that point, we have {Differential Calculus, Art. 858) ds^_ //2a\ dx~\/ \^) ' therefore s = »J{8ax) + C. The constant will be zero if we measure the arc s from the vertex. Conversely if 5 = ^/(Sax) + (7 we infer that the curve is a cycloid. And more generally if we have s + A= ^{B + C^x + C„jj), where A, B, C^, and C^ are constants, we infer that the curve 90 LENGTHS OF CURVES. is a cycloid. For by suitable changes in tbe origin and axes the last equation can be put in the form s = s/{8ax) + C. 73. Application to the Catenary. c - -- The equation to the catenary is y = ^ (e" + e ") ; hence thus s = \ Ue^ + e~^) dx = ^ (e' - e~') + C. The constant will be zero if we measure the are s from the point for which a) = 0. 7-i. Application to the Curve given by the equation z x3 +y3 z= aK Here ^ = _^ ^ = Ctulf ^^• dx x^' dx \ x^ / x^' ,1 i [da Sa'\v^ „ thus s = a^ -^ = — ^ h C. J X'i ^ The constant will be zero if we measure the arc from the point for which a; = 0. The curve is an hypocycloid in which the radius of the revolving circle is one-fouith of the radius of the fixed circle. (See Differential Calculus, Art. 362.) 75. In the same way as the result in Art. 68 is obtained we may shew that VI- (l)}^^- LENGTHS OF CURVES, 91 Or we may derive this result from the former thus ; From the equation to the curve we may express -j- in terms of y, and thus by integi-ation s becomes known. In some cases this formula may be more convenient than that in Art. 68. 76. Application to the Logarithmic Curve. X The equation to this curve is ?/ = ha", or i/ = he'' if we y b suppose a=e°; thus a;=clog^, therefore ^==^, ^^'JSlllfl dy y dy y ' and s-i'^-^^^^dv-l-^y +{ y^y s-j ^ rf^_J^^^^,_^+J__^. The latter integral is \/(c^ + y^) ; the former is 77. If sc and y are each functions of a tliird variable t, we have {Differential Calculus, Art. 807) ds _ /ifdxV /dy Jt~V [VdiJ '^\di ^=''/^'¥+©J^'- 92 LENGTHS OF CURVES. 78. Application to the Ellipse. The equation to the ellipse is -^ + t^ = 1. We may there- fore assume a; = a sin ^, y = h cos (/>, so that <^ is the com- plement of the excentric angle [Plane Co-ordinate Geometry, Art. 168). Therefore, by the j)receding Article, ds '_,— = V(a^ cos^^ + 6^ sin^^), and s = I /^/(a^ cos^ (f) + b^ sin^^) d(f) = a j V(l — e^ sin^ ^) d(p. The exact integral cannot be obtained ; we may however expand VCl — e^ sin" <^) in a series, so that s = al[l —\e^ sin^^ — ^-^ e^ sin*^ ~ ^' a' p 6^sin^(/) ]d^, and each term can be integrated separately. To obtain the length of the elliptic qiiadrant we must integrate between the limits and — . Plane Curves. Polar Co-ordinates. 79. Let r, 6 be the polar co-ordinates of any point of a curve, and 5 the length of the arc measured from any fixed point up to this point; then {Differential Calculus, Art. 311) lence s = Hi^V'- 80. Application to the Spiral of Archimedes. In this curve r = ad, thus -v^ = ci : dd hence s= j^/{r'-\- a')de = a j^/{l + 6"-) d0 The constant will be zero if we measure the arc s from the pole, that is, from the point where ^ = 0. LENGTHS OF CURVES. . 93 81. Application to the Cardioid. The equation to this curve is r = a (1 + cos 6) ; thus = [V [a? (1 + cos Of- + a' sia' 6] dd = a f V(2 + 2 cos 6) dO = 2./ cos ^ dO = 4a siu ^- + C. The constant will be zero if we measure the arc s from the point for which ^ = 0, that is, from the point where the curve crosses the initial line. The length of that part of the cuive which is comprised between the initial line and a line through the pole at right angles to the initial line is 4 a sin ^ • The length of half the perimeter of the curve is 4a sin - , that is, 4a. 82. Suppose we require the length of the complete peri- meter of the cardioid ; we might at first suppose that it riiT would be equal to 2a I cos - dO; but this would give zero as Jo ^ the result, which is obviously inadmissible. The reason of this may be easily seen ; we have in fact shewn that ^| = aV(2 + 2cos^), a a and this ought not to be put equal to 2a cos ^ but to + 2a cos - , and the proper sign should be determined in any application of the formula. Now by s we understand a positive quantity, and we may measure s so that it increases with 0, and thus ~ is positive. Therefore when cos - is positive, we take the dU ^ upper sign and put -^^ = 2a cos ^ ; when cos ^ is negative, we take the lower sign and put -^=- 2a cos 2. Hence the 9-i , LENGTHS OF CURVES. r2^ Q length of the complete perimeter is not 2a I cos - d9, bi 2a cos ^d9- 2a I cos ^ cZ^, that is, 8a. This result migl have been anticipated, for it will be obvious from the sjtc metry of the figure that the length of the complete perimet is double the length of the part which is situated on one sic of the initial line, and this was sheAvn to be 4a in the prece( ing Article. 83. It may sometimes be more convenient to find tl length of a curve from the formula which follows immediately from that in Art. 79. 84. AjjplicatioJi to the Logarithmic Spiral. The equation to this curve is r — ha^, or r = he'' if we su - r dd e pose a = 6"; thus ^ = c log ^ ; therefore j- = ~ and s =1 V(l + c') dr = V(l + c') r + C. Thus the length of the portion of the curve which has and r^ for the radii vectores of its extreme points is r V(l + c') dr, that is, V(l + C) (r, - r J. The angle between the radius vector and the correspondii tangent at any point of this curve is constant {Differenti Calcidus, Art. 354) ; and if that angle be denoted by ds we have c = tan a ; thus /^(l + c^) = sec a ; therefore -v; = sec and s = r sec a + C. Hence {r^ — rj sec a is the length of tl portion mentioned above. LENGTHS OF CURVES. 95 Formulce involving the radius vector and perpendicular. 85. Let (f) be tho angle between the radius vector r of any point of a curve and tho tangent at that point ; then cos^ = -T- (Differential Calculus, Art. 310). Let p be the perpendicular from the j)ole on the same tangent ; then sin (p = - , therefore cos = — — ; dr^ ^{r'-f) ^ ds r ' thus ds __ r J — f '"'^^ therefore -t- = —rrr, w . and 86. Application to the Epicycloid. With the notation and figure in the Differential Calculus, Art. 360, it may be shewn that the equation to the tangent to the epicycloid at P is COS a — cos — ^ — o sm d — sm ■ — 5 — t7 where x and y are the co-ordinates of P, and x and y the variable co-ordinates. Hence it will be found that the per- pendicular p from the origin on the tangent at P is given by p = [a + 2h) sm ^ ; also r = a^ + 45 (a + h) sin" ^7 ; thus p' = ^-A Y^ > where c =- a + 2J. ^ c — a Hence, by Art. 85, 9G LENGTHS OF CUIIVES. At a cusp 7' = a, and at a vertex r = c ; thus the length of the portion of the curve between a cusp and the adjacent vertex is -^ -Tr^> 2T ■, ttia-t IS , that IS — ^ . Hence the length of the portion between two consecutive cusps . 8h(a + h) a 87. A remark may be made here similar to that in Art. 82. If we apply the formula a ^ ^ to find the length between two consecutive cusps, we arrive at the result zero, since r = ft at both limits. The reason is that we have used the formula ds _ V(c^ — (f) r dr a \J{c^ — r") while the true formula is ds \/(c^ — <^") ^ dr ~ a ^/{c^ — r') ' Since s may be taken to increase continually, it follows that ds . . . ■J- is positive when r is increasing, and negative when r is diminishing. Now in passing along the curve from a cusp to the adjacent vertex r increases, thus -j- is positive, and wc should take the upper sign in the formula for y- ; then in passing from the vertex to the next cusp r diminishes, thus ds . . ~r- is negative, and the lower sign must be taken. Hence the dr ^ ' *= length from one cusp to the next cusp LENGTHS OF CURVES. 97 _ V(c'-a') p rdr ^/(c^-a') f^ rdr _ 2 V(c^ - g") p rdr ^ 86 (a + h) 88. From what is stated in the preceding Article, it ap- pears that if the arc s begin at a vertex the proper formula is ds \/{c^ — a^) r therefore s = -^-^l^ \ ^^^^Jl^^^[o^ -r^). No constant is required since we begin to measure at the point for which r = c; the formula holds for values of s less a It may be observed that thus a 89. Similarly for the hypocycloid we may shew that p = — 4 2-^ > where c = a — 26. ^ a^ — c^ Suppose (? less than a' ; then we may shew that ds _ \/(a^ — c^) r dr a '^{r^ — c")' and thus s may be found. The length of the curve between two adjacent cusps is «''("-*) a Next suppose c" greater than a^ ; then we should -vviite (is the value of -7- thus, dr ds _ a/(c^ — <^^) ^ dr ^ a s/{c'-r') T. I. C. ,»\ » 98 LENGTHS OF CURVES. in this case h is greater than a, and we shall find the length of the curve between two adjacent cusps to be — ^^ . When a = 2b we have c = and p = 0; in this case the hypocycloid becomes a straight line coinciding with a dia- meter of the fixed circle. If a = J we have c^ = a" ; in this case the denominator in the value of p^ vanishes ; it will be found that the hypocycloid is then reduced to a point, and r = a. It may be shewn as in Art. 88, that if s be measured from a vertex to a point not beyond the adjacent cusp, we have s = + a' J{r^-f), the upper or lower sign being taken according as c is greater or less than a. Formulce involving the Perpendicular and its Inclination. 90. Another method of expressing the length of a curve is worthy of notice. Let P be a point in a curve ; x, y its co-ordinates. Let s be the length of the arc measured from a fixed point A up to P. Draw F a perpendicular fi'om the origin on the tangent at P, suppose OY=p, PY = u, Y0x = 6; then LENGTHS OF CURVES. 99 p = x COS ^ + 2/ sin 0, u = x&inO — y cos 0, dy . a ^s a ~= —coiO, -T-= — cosec ; ax ax therefore dp . yy „ ^dx . ^dy j^ = — a: sm ^ + 7 cos ^ + cos p -jt; + sin -y^ = — M, do da da d^p du „ . ^ . ^dx -, Jy -f^ = — -Tn = — xcos0—ysm0 — sin0-j^ + cos9-f^ d(f d0 ^ d0 d0 ndx ds therefore, by integration, ^ = -lpd0 + s. d0 this may also be written s + u= I pd0. Suppose 5j and Mj the values of s and u when 9 has the value 6^ , and s^ and u^ their values when ^ has the value 0^, then therefore « = -th + 5,^ -s^ + u. re, — u^=\ pd0. We have measured u in the direction of revolution from P and have taken it as positive in this case ; when u is negative it will indicate that Y is on the other side of P. The preceding results may be used for different purposes, among wdiich two may be noticed. (1) To determine the length of any portion of a curve when the equation to the curve is given ; for from that equa- tion together with -^ = — cot ^ we can find x and y in terms of 0, and therefore j) which is equal to x cos -{y sin ; then s may be found from the equation = |4./.... 7—2 100 LENGTHS OF CURVES. (2) To find a curve such that by means of its arc a pro- posed integral may be represented ; for if the proposed inte- gral bo jpdd, where p is a function of 9, the required curve is found by eliminating 6 between the equations X =]) cos 6 -~sm0, y = psin6 + -^ cos 6 and then the integral may be represented ^J ^-^q- This Article has been derived from Hymers's Integral Calculus, Art. 136. 91. The results of the preceding Article may be obtained in another way. Let p denote the radius of curvature of the curve at F ; let OP = r, and let s, u, and have the same meaning as before, then from the Differential Calculus we have , = r|, therefore f^=r^' . dr ds J d9 Also Pr=r cos OP F=-r^; LENGTHS OF CURVES. 101 therefore ~:. = — PY= — u. do Let PC be the radius of curvature at P; draw OQ perpen- dicular to FC. The locus of C is the evolute of the curve AP; and QG is with respect to this locus what PF is with respect to the locus of P. Let 6', p be the polar co-ordi- nates of Q, and let QC= u; then 0' = 6 — -^ and p = u. Ana OC-u'--^- dp'__du_d:'p . Also p=PQ + QG=p+u'=p + j^,', but P~W *^®^6fore s = -L + jpdd. From the value of PY we can obtain an easy proof of a theorem of some interest in the Differential Calculus (Differ- ential Calcidus, Art. 329). Let p^ denote the perpendicular from on the tangent at Y to the locus of Y; then [Differ- ential Calculus, Art. 284) 1 1 1 prp^'^p' (dpV \dd)' since ^ is the radius vector of Y. Thus 1 P.' _1 ^r_2)' f P' therefore • A particular case ; of the formula ^2 -s^ + u^-u^ = pdd should be noticed. Suppose we take a complete oval curve without singular points; then 6^=6^ + 27r, and u^=u^', thus the complete perimeter of the curve is I ^;rf^. J 01 102 LENGTHS OF CURVES. 92. Application to the Ellipse. • ' c A Let APB be a quadrant of an ellipse, CFthe perpendicu- lar on the tangent at P; let ACY=e. Then {Plane Co- ordinate Geometry, Art. 196) CY=^ a ^J{1 - e' sin' 6) ; therefore AP + FY =^ aL{l -e" sin= 6) dO, the constant to be added to the integral is supposed to be so taken that the integral may vanish with 6. If 7i be a point such that its excentric angle is ^ — ^, we have, by Ai't. 78, thus And PY BP = aj >^{l-e' sin' 6) d9; AP + PY=BP _ dp _ ae^ sin 6 cos 6 .(!)• dd V(l - e' sin' 6) ' Let X be the abscissa of P ; then by Art. 90, a: = p cos — -, sin 6 dd ,,-, s • 2/1N n ae'sin^^cos^ = a \/{l — e' sm'^) cos 6 + a cos V(l-e'sin'^) V(l-e'sin'^)' Thus PY= e'x sin ; and if x' be the abscissa of P we have -^ — ^ j so that PY= . Thus (1) may be written BB-AP = -xx ... a this result is called Faemani's Theorem. .(2); LENGTHS OF CURVES. 103 From the ascertained values of x and x we have 2 a' — a' sin* "^ 1-e'^sin-^^ a' - x'^ 1- (TX'^ therefore eV^-'* — a* {x^ + x'^) + a* =0. Thus the equation which connects x and x involves these quantities symmetrically ; hence from (2) we can infer that BP—AR = — XX . This is also obvious from the figure. a The length of FY is also equal to the length of the corresponding straight line at R. We may observe that the value of FY may be obtained more simply by means of a known property of the ellipse. For suppose the normal at F to be drawn meeting GA at O ; and through F di-aw a straight line parallel to GA meeting GY at Q. Then FQ, = GG — e^x, by the nature of the ellipse ; and FY=FQ^me = e^xdnd. 93. Application to the Hyperbola. Let G be the centre and A the vertex of an hyperbola, CFthe perpendicular on the tangent at P. Let AGY=6, and CY = p ; then it may be proved that PY-AF= a [ V(l - e' shi' 6) dO. 104 LENGTHS OF CURVES. This may be proved in the same manner as the corresponding result of the preceding Article ; we may either make the requisite changes of sign in the formulae of Art. 90, which are produced by difference of figure ; or we may begin from the becfinninof asfain in the manner of that Article. The constant to be added to the integral is supposed to be so taken that the intes^ral may vanish with 6. Suppose a the greatest value which 6 can have, then PF has its least inclination to the axis GA, and {Plane Co- ordinate Geometry, Art. 257) cot a — V(e^ — !)• When P moves off to an infinite distance PY— AP becomes the excess of the length of the infinite asymptote from C over the length of the infinite hyperbolic arc from A. Thus this excess ', [V(l - e' sin^ 6) de. IS a Inverse questions on the lengths of Curves. 94. In the preceding Articles we have shewn how the length of an arc of a known curve is to be found in terms of the abscissa of its variable extremity; we will now briefly notice the inverse problem, to find a curve such that the arc shall be a given function of the abscissa of its variable ex- tremity. Suppose <^ {x) the given function ; then s = ^{x); therefore ^ (^) =£= V {l+ (^ thus f^= [{' {x)Y-l]\ and 2/=/[(^'(^)r-l]^^^- 95. As an example of the preceding method, suppose (f){x) = ^/{'icx) ; thus 'ix) = jJ^; therefore LENGTHS OF CURVES. 105 Ih - — x]dx . , ax J iJicx — x') 2 J I sjicx — X') 2 J /^{cx — x^) c _, 2x = \/{cx —x^)+^ vers ^ — • + C. We may ^vrite y for y — C and thus we find that the curve is a cycloid. {Differential Calculus, Art. 858.) f(^) 9G. For another example suppose (f)(x) = a log x ; thus A" ' (a"^ — x^) dx Here 2/=/y(^- l) ^^ = /l^(^._^.^ Involutes and Evolutes. 97. We may express the length of an arc of a curve with- out integration when we know the equation to the involute of the curve. Suppose s' to represent the length of an arc of a curve, p the radius of curvature at that point of the involute which corresponds to the variable extremity of s, then {Dif- ferential Calculus, Art. 331) s ±p = l, where I is a constant. If the equation to the involute is known, p can be found in terms of the co-ordinates of the point in the involute ; then these co-ordinates can be expressed in terms of the co-ordi- nates of the corresponding point of the evolute, and thus s is known. By this method we have to perform the pro- cesses of differentiation and algebraical reduction instead of integration. 106 LENGTHS OF CURVES. 98. Application to the Evolute of the Parabola. Take for the involute the parabola which has for its equa- tion y^ = ^ax\ let x, y be the co-ordinates of the point of the evolute which corresponds to the point [x, y) on the parabola. Then by the ordinary methods {Differential Cal- culus, Art. 330} we have «' = 2a + ^x, y' = -^2, and p = 1ai j . Thus we shall obtain for the equation to the evolute and /, = 2a(^); therefore s + 2a \—~-^ ] = I. 3a J ~ Suppose we measure s from the point for which x = 2a, that is, from the point which corresponds to the vertex of the parabola; then we see that s increases with x, so that we must take the lower sign in the last equation ; also by sup- posing X = 2a and s' = we find l= — 2a; thus This value of s may also be obtained by the application of the ordinary method of integration. 99. When the length of the arc of a curve is known in terms of the co-ordinates of its variable extremity, the equa- tion to the involute can be found by the ordinary processes of elimination. For we have (Differential Calculus, Art. 331) dx' dx _ Ids x —X ~ p dx' LENGTHS OF CURVES. 107 where the accented letters refer to a point in a curve, and the unaccented letters to the corresponding point in the in- volute. Thus , — ax y- « ^ = ^"+^^' ^^)- Similarly y = y'T p^ (2). If then s is known in terms of x, or of y', or of both, by means of this relation and the known equation to the curve (IX civ we may find -tt and -p ; and p is known from the equation s + p = I. It only remains then to eliminate x and y from (1) and (2) and the known equation to the curve; we obtain thus an equation between x and y, which is the required equation to the involute. 100. Application to the Catenary. The equation to the catenary is y =1^ + e XT s' =l(^ — e xf and supposing s' measured from the point for which a;' = and y' =c', we shall now find the equation to that involute to the catenary which begins at the point of the curve just specified. We have then dy _ s ds _y\ dx c ' dx c ' , dy _ s' dx' _ c ^^""^ di~y" dl~^' and p = s , no constant being required, because by supposition p vanishes with s\ 108 LENGTHS OF CURVES. Hence equations (1) and (2) of the preceding Article become _ ' ^'^ . CO — vC ■"" ■ J J y y=y - — = ■ — ; — = -0 y y y And s = V(3/" - c^) = ^/(p -0^) = '- V(c' - y) ; therefore i^^^lil^lf), y c thus x = x' — \J{c^ — y^) ; therefore x = \/{c^ — y^) + x. We have then to substitute these values of x' and y in the equation to the catenary, and thus obtain the required relation between x and y. The substitution may be con- veniently performed in the following manner : c - --' y=-^{^'+e')', therefore ^J[y'^ _ c^) = £ (/ _ e" ^) ; therefore y + \/{y'^ — c^) = ce°, thereiore x = c lo? ^-^ . c Thus finally, x + V(c' -y') = c log "^ "*" ^^''""''^'^ . This curve is called the tractor^y ; on account of the ra- dical, there are two values of x for every value of y less than c, these two values being numerically equal, but of opposite signs. There is a cusp at the point for which x = and y = c\ and the axis of x is an asymptote. 101. The polar formulae may also be used in like manner to determine the involute when the length of an arc of the evolute can be expressed in terms of the polar co-ordinates of its variable extremity. We have {Diferoitial Calculus, Art. 832) r" = p' + r'-2pp (1), p'' = r'-p'' (2). LENGTHS OF CURVES. 109 Here, as before, the accented letters belong to the known curve, that is, to the cvolute, and the unaccented letters to the required involute ; hence since the evolute is known, there is a kno\\'n relation between 'p and r. And s' + p = /, so that if s can be expressed in terms of j'j' and r we may eliminate I) and r by means of (1), (2), and the kuo-vvn relation between 'p and ?•'. Thus we obtain an equation connecting 'p and r, which serves to determine the involute. 102. Application to the Logarithmic Spiral. In this curve p' = r sin a, where a is the constant angle of the' spiral. If we suppose the involute to begin from the pole of the spiral, and s to be measured from that point, we have /3 = s' = ?•' sec a (Art, 84). Thus (1) of the preceding Article becomes r'^ = r'^ sec' a + 7"' — ^r'p sec a = r"^ sec^ a + r'* sin* a +p^ — 2r'p sec a, by (2). From this quadratic for p we obtain p^r sec a = + r' cos a. V (1 ~f" cos'' Ot) If we take the upper sign we find » = — ^^ , and ^^ ° ^ cos a then from (2) we find r' = '—^ r'*. But this solution ^ ' cos a must be rejected, because from it we should find p or r^r — ■ — ■ — 7^ 2-T f', which is inconsistent with the dp cos a (1 + cos a) equation p = r sec a. / • 9 r SNi ct If we take the lower sign we find « = , and then ^ ^ cosa '2 „• 2 from (2) we find r" = ^ — : thus » = r sin a. Hence the ^ ' cos a -^ involute is an equiangular spiral with the same constant angle as the evolute has. , 110 LENGTHS OP CURVES. Intrinsic Equation to a Curve. 103. Let s denote the length of an arc of a curve measured from some fixed point, (f> the inclination of the tangent at the variable extremity to the tangent at some fixed point of the curve; then the equation which determines the relation between s and is called the intrinsic equation to the curve. In some investigations, especially those relating to involutes and evolutes, this method of determining a curve is simpler than the ordinary method of referring the curve to rectangular axes which are extrinsic lines. 104. We will first shew how the intrinsic equation may be obtained from the ordinary equation. Suppose y =f{x) the equation to a curve, the origin being a point on the curve, and the axis of y a tangent at that point; from the given equation we have S = -^'(^)=tiZ^^y^yP"^^''^^' thus X is known in terms of tan (f), say x = F (tan 0) ; then -j-i=F' (tan 0) sec'^ j> ; also J- = cosec <^ ; ds therefore 'Tl~^' ^^^ ^ ^^^ ^ cosec ^ ; from this equation s may be found in terms of (f) by integra- tion. A similar result will be obtained if at the origin the axis of X be the axis which we suppose to coincide with a tangent. 105. Application to the Cycloid. By the Differential Calculus, Art. 358, we have ^= //2a -^N 1 . dx V \ tc / tan ^ ' LENGTHS OF CUEVES. Ill therefore — = . „ , , cc = 2a sin* 6, X sin'' . Similarly y = Ids cos (p. Now s is by supposition known in terms of ; thus by integration we may find cc and 7/ in terms of <^, and then by eliminating (f> we obtain the ordinary equation to the curve in terms of a;*"and y. 107. Application to the Cycloid. Here s = 4tt sin <^ ; thus a; = I c?s sin ^ = 4a I sm we can obtain the ordinary equa- tion ; if the origin of the rectangular axes is the vertex of the curve, we shall have C = a and C" = 0. 112 LENGTHS OF CURVES. 108. "We shall now give some miscellaneous examples of intrinsic equations. The intrinsic equation to the circle is obviously s = a(p. 109. The equation to the catenary is the origin being on the curve. Hence thus if ^ be the angle which the tangent at any point makes with the tangent at the origin, s = c tan (j). 110. We have seen in Art. 86, that for the epicycloid , cos 6 — cos — J— a -^ = =; = tan 6 suppose, ax . a + ^ . ^ sm — -J — £^ — sm c/ o thus ■ — j; ^ EXAMPLES, 1. For what values of m and n are the curves a"'?/" = a-"'"^" rectifiable? (See Art. 15.) Result. If TT- or ^^ — 1- ^ is an integer. '2.111 2m 2 ° 2. Shew that the length of the arc of a tractory measured from the cusp is determined by s = c log - . 3. Shew that the cissoid is rectifiable. 4. Shew that the whole length of the curve whose equation is 4 (a;' + ?/") - a* = Sa^y^ is equal to fia. It may be shewn that i-r] = — k — i i^ 120 EXAMPLES. 5. The length of the arc of the curve "between the limits {x^, y^ and [x, y) is 2^ {(^ + yf +{x- yff - 2^ {(x, + y^^ + {x, - y^f. 6. If s = ae% find the relation between x and y. 7. Shew that the intrinsic equation to the parabola is ds 2a a , 1 + sin , a sin rf> -TT = TT or S = 75 log 3 -. 2 + ^1 ^2-r • d<^ cos' (j) 2°l-sin(/) 1- sm' ^ 8. The intrinsic equation to the curve y^ = aa^ is 9. Shew that the length of the arc of the evolute of a parabola from the cusp to the point at which the evolute meets the parabola is 2a{S/^3 — 1) ; where 4a is the latus rectum of the parabola. 10. The evolute of an epicycloid is an epicycloid, the radius 2 of the fixed circle being - — ^ and the radius of the ° a + zo generating circle t^. (Arts. 110 and 114.) 11. Shew that if the equation to a curve be found by eliminating 6 between the equations and y = cos dy^r' (6) — sin 6^|/' {6), then s = yir{e)-\-ylr"{e). 12. Shew that the length of the curve SaJ'y = x* + Ga'x'^ measured from the origin is ^-3 (x^ + 4a") . 121 CHAPTER VII. AREAS OF PLANE CURVES AND OF SURFACES. Plane Areas. Rectangular Formulce. Single Integration. 128, Let DPE be a curve, of which the equation is 7/ = (x), and suppose x, y to be the co-ordinates of a point P, Let A denote the area inchided between the cui-ve, the axis of X, the ordinate PM, and some fixed ordinate DB, such that OB is algebraically less than x ; then {Differential Cal- culus, Art. 48) dA bence -^ = I <^ (*) dsc. Let '^{x)-\-C be the integral of ^ (cc) ; thus A=^\r{x) + a Let J. J denote the area when the variable ordinate is at a iistance x^ from the axis of y, and let A^ denote the area when 122 AREAS OF PLANE CURVES AND OF SURFACES. the variable ordinate is at a distance x^ from the axis of y\ then therefore A^ — A^ = '>Jr (x^) — yjr (x^) = (^{x) dx. J Xi 129. Application to the Circle. The equation to the circle referred to its centre as origin is i/^ = a^ — x^ ; here (f> (x) = ^{a^ — x^) ; thus A = U (x) dx = L{a'- x') dx = '^^^^^ + ^ sin' - + a The constant G vanishes if we suppose the jixed ordinate to coincide with the axis of y. It will be seen by draAving a figure, that the area comprised between the axis of x, the axis of y, the circle, and the ordinate at the distance x from the axis of y, may be divided into a triangle and a sector, the values of which are given by the first and second terms in the above expression for A. This remark may serve to assist the student in remembering the important integral JV(«' - ^') dx = - ^^^^ ~ ^'^ 4- 1' sin-^ ^ . 130. Application to the Ellipse. Suppose it required to find the whole area of the ellij)se. The equation to the ellij)se may be written y^ = —^{a'^—x'). Co Hence the area of one quadrant of the ellipse = / J v(a' - .V- :-/„ ^/(«= -') ^- 1 ^ = hence the area of the ellipse is irah. 131. Application to the Parabola. The equation to the parabola is y"^ = 4 \

ositive. If y be really negative the area will be I (— y) dx. Thus in the present example the area will not be c I sin - dx but c\ %\n- dx + c\ ( — sin - dx. AREAS OF PLANE CURVES AND OF SURFACES. 125 X 2aTT X that Is, c I sin - dx — c \ sin - dx ; Jo a JaT « this will give 2ca + 2ca, that is, 4ca. Plane Areas. Rectangular Formula. Double Integration. 137. In Art. 128 we have sui>plied a formula for finding the area of a curve ; that formula supposes the area to be the limit of a number of elemental areas, each element beinc;' a quantity of which T/Aa; is the type. We shall now proceed to explain another mode of decomposing the required area into elemental areas. ^ y^ / ^ tJ r _j^ 1 ,i' ■^ ■^ 6 u ? i y 1 i J r -B Suppose we require the area included between the curves BPQE and hpqe, and the straight lines Bh and Ee. Let a series of straight lines be drawn i^arallel to the axis of y, and another series parallel to the axis of x. Let st represent one of the rectangles thus formed, and suppose x and y to be the co-ordinates of s, and x + Aa? and y -\- Ay the co-ordinates of t; then the area of the rectangle st is Aa:;A_y. Hence the required area may be found by summing up all the values of AxAy, and then proceeding to the limit obtained by sup- posing Ax and Ay to diminish indefinitely. We effect the required summation of such terms as AxAy in the following way : we first collect all the rectangles 12G AREAS OF PLANE CURVES AND OF SURFACES. similar to st which are contained in the strip PQqp, and we thus obtain the area of this strip ; then we sum up ail the strips similar to this strip which lie between Bh and Ee. The error we may make by neglecting the element of area which lies at the top and bottom of each strip, and which is not a complete rectangle, will disappear in the limit when Ace and Ay are indefinitely diminished. Let y = 4> {x) be the equation to the upper curve, and y—-y\r{x) the equation to the lower curve; let OC'=cand 0H= h, then if A denote the required area, we have rh r4>{x) A= I I dxdy, J cJ >l'{x) for the symbolical expression here given denotes the process which we have just stated in words. r r4> (X) Now \dy = y, therefore dy = (j:) have rh A= {(f){x)-ylr (x)] dx. J c In this form we can at once see the truth of the expression, for (f>{x)—^lr (x) =PL—j)L = Pp\ thus [(^{x)—'^ {x) ] Aa? may be taken for the area of the strip PQ^p, and the formula asserts that A is equal to the limit of the sum of such strips. The straight lines in the figure are not necessarily equi- distant : that is, the elements of which AxAy is the type are not necessarily all of the same area. 138. The result of the preceding Article is, that the area A is found from the equation A = r{(P(x)-ylr{x)]dx. ■J c This result may be obtained in a very simple manner as shewn in the latter part of the preceding Article, so that it was not absolutely necessary to introduce the formula of double integration. We have however drawn attention to the formula rhr{x) A =^ I i dxdy J cJ

{y), and straight lines parallel to the axis of x at distances respectively equal to c and /(, we have in a similar manner dydx=\ {<^i:y)-f{y)]chj. Some examples of the formulas of Arts. 137 and 139 will now be considered ; we shall see that either of these formuhe may be used in an example, though generally one will be more simple than the other. 140. Kequired the area included between the parabola if=ax and the circle y" = 2ax — x^. The curves pass through the origin and meet at the point for which x = a; thus if we take only that ai'ea which lies Dn the positive side of the axis of x, we have A = j {'^ {2ax — x^) — aJ (ax)] dx = -J ^. The whole area will therefore be 2 (— ^^j . Suppose that we wish in this example to integrate with •espect to X first. From the equation y'^=2ax — x^ we deduce : = a ± /\/ (a" — 1/') , and it will appear at once from a figure hat we must take the lower sign in the jsresent question. {6)Y cW 2 • Hence A=h [\4>(0)Yd0, =hj[4>m Let fid) be the integral of i^^ , then A = 'f{e) + c. Let ^j denote the area when the variable radius vector is at an angular distance 6^ from the initial straight line, and let A^ denote the area when the variable radius vector is at an angular distance 6„ from the initial straight line ; then A = ^(^J + a A,=^{r{e,) + C, therefore A^- A^= f[d^) - f (^J = h l^'^W)]' dd. J Oi 145. Ajoplication to the Logarithmic Spiral. In this curve r = he'' ; thus 2« 6^c ^' A = \\l/e^dd = ^-fe''-{-G, 9—2 132 AREAS OF PLANE 'CURVES AND OF SURFACES. and A,-A,= ^jye"^ cZ^ = '^^ (e « - e ^ ) = | (r/ - r/), where r, and r„ are the extreme radii vectores of the area considered. 146. Application to the Parabola. Let the focus be the pole, then COS 2 J COS - = — f 1 + tan^ ^ J sec^ ^ "^ = « tan - + -^ tan^ 2 + ^• Hence ^,- ^, = «'/tan | - tan |) + | (tan^ | - tan' |) . TT Suppose that 6^ = and ^2 = 9' ''^^'^'^ "^^ obtain for the . 4a' area a^ -^ -^ , that is, ^; this agrees with Art. 131. o o For another example we will suppose the parabola refer- red to the intersection of the directrix and the axis as pole, the axis being the initial straight line. Here ^ cos^-V(cos26') ^ = -^ sh^^^ ' . c 2 fcos'^ + COS 2^-2 cos 6 V(cos 26) , . thus A = 2a I r-^-fl "t/ J sm t/ „ , r2 cos"^ - sin'^ , . . , [cos d V(cos 20) ,. = 2a — — ^-Ta- d6- 4a- t-^^ "^• J siyrd J sm 6 AREAS OF PLANE CURVES AND OF SURFACES. 133 Now [ ^cos'^^-sm"^ ^^ ^ f ^^^,^ _ ^^^^^, ^^^^ J sm d J ^ = -§ cot'^ + cot^. , /• COS ^ V(cos 2e)dd _ fJ(l-2 sin' ^)Jsm^ assume sin 6 = ~ , then the intesral becomes t' ° - h{t- - 2) tdt, that is, - 1 (f - 2)1 Hence, adding the constant, we have ^ = ^ (cosec' ^ - 2) t - ^' cot^ ^ + 2a' cot 6' + C 4a' (cos2g)^-c os^ ^ 3 sin' d = 2a cot ^ + -5- -'^ J 3 ^ 1- C. The constant will be zero if A commences from the initial straight line ; for it will be found on investigation that „ , . 4 (cos 2^)^ - cos'' 6 . , , ^ ^ 2 cot o + 7i f^-TTi vanishes when £7 = 0. 3 sm 147. Application to the curve r = a (^ + sin 6). Here A = I" [(^ + sin ey dd = ^j(e^' + 20 sin e + sin' ^) dd ; and I ^ sin 6d9 = — 6 cos ^ + sin 0, [sin' ^ cZ^ = 1 f (1 _ cos 2^) c^^ = I - ^v ^ «M^' ozi n ^ ■ n ^ sin 2^] ^ thus ^= 2]3-2^cos^+ 2sm(9 + 2 ^f+^- Suppose we require the area of the smallest portion which is bounded by the curve and by a radius vector which is si n 2^ 4 ' 134 AREAS OF PLANE CURVES AND OF SURFACES. inclined to the initial straight line at a right angle ; have and hrrr as the limits of the integration, required area is then Ave Thus the a' 2 1 1-2 24 ^ 4 ^ Plane Curves. Polar Formulce. Double Integration. 148. In Art. 144 we have obtained a fonnula for finding the area of a curve ; that formula supposes the area to be the limit of a number of elemental areas, each element being a quantity of which ^r"^ A^ is the type. We shall now proceed to explain another mode of decomposing the required area into elemental areas. x/ Suppose we require the area included between the curves BPQE and hpqe, and the straight lines Bh and Ee. Let a series of radii vectores be drawn from 0, and a series of circles Avith as centre ; thus the plane area is divided into a series of curvilinear quadrilaterals. Let st represent one of these elements, and suppose r and 6 to be the polar co-ordinates of s, and r + Ar and 6 -f A^ the polar co-ordinates of t ; then the area of the element st will be ultimatelv rA6^Ar. Hence the required ai'ea is to be found by summing up all the values of rA^A?-, and then proceeding to the limit obtained by sup- posing A^ and Ar to diminish indefinitely. AREAS OF PLANE CURVES AND OF SURFACES. 135 We effect the required summation of such terms as ?-A^Ar in the following way: we first collect all the elements similar to St which are contained in the strip PQqp, and thus obtain the area of the strip ; then we sum up all the strips similar to this strip which lie between Bh and Ee. Let r = (^ (6) be the equation to the curve BPQE and r = -^{6) the equation to the curve hpqe, let a and ^ be the angles which OB and OE make respectively with Ox ; and let A denote the required area, then A= rcld dr ; J a J ^(0) for the symbolical expression here given denotes the process which we have just stated in words. Now lrc?r = -^, therefore I' J li thus we have J a In this form we can see at once the truth of the expres- sion, for 0P= (6) and Oj) = -^ {&), and thus may be taken for the area of the strip PQqp, and the formula asserts that the area A is equal to the limit of the sum of such strips. 149. The remark made in Art. 138 may be repeated here ; we have introduced the process in the former part of the preceding Article, not because double integration is absolutely necessary for finding the area of a curve, but because the process of finding the area of a curve illustrates double integration. 150. If the area which is to be evaluated is bounded by the curves whose equations are d=(p{r), 6=-^{r) respectively, 13G AREAS OF PLANE CURVES AND OF SURFACES. and by the circles whose equations are r — a and r = h re- spectively, it will be convenient to integrate with respect to 6 first. In this case, instead of first summing up all the elements like st, which form the strip PQqp, we first sum up all the elements similar to st which are included between the two circles which bound st and the curves determined by 6= <^ (it') and 6 = ylr[r). Thus we have rb r^{r) A = \ \ rdr dO. J aJ 'dr wo nave A = ^\ —-f-r, ttt . 156. Application to the Epicycloid. Here «^ = — -, r— : thus ^ c'-a' ' A - i[^ \/(?'" — «^) rdr _ c f \/{r'^ — a^ rdr 'J a^ic'-r') ~2^Jv{c'-a'''-(/''-a-')} = TT -jT^i -i T\ y where r = r - a . Now & — (T . ^^ z z \j{(? — a" — 2;^) c' - a^ . _, V(r' - a') V(^-' - a') V(c' - r') = — ;; — sm / _ V N / V V(c'-a') Taking this between the limits r = a and r = c, we get 2 2 ) — a TT ■, that is, b (a + h) ir. Hence the area is ^-h {a+ h) tt, that is, " ^ ^ — . By doubling this result we obtain the area between the curve and the radii vectores drawn to , ,. -u- 1 • ^-u f (a + 2?;) i (a + &) TT two consecutive cusps, which is thereiore ^ a The area of the circular sector which forms part of this area is irah ; subtract the latter and we obtain the area between an arc of the epicycloid extending from one cusp to the next 142 AREAS OF PLANE CURVES AND OF SURFACES, cusp and the fixed circle on which the generating circle rolls ; the result is — (3a +26). Similarly in the hypocycloid the area between the fixed circle and the part of the curve which extends between two consecutive cusps may be found. If a is greater than h the result is — (3a -26). Area between a Curve and its Evolute. 157. In the figures to Art. 114, if we suppose the string or straight line PQ to move through a small angle A0, the figure between the two positions of the straight line and the curve AP may be considered ultimately as a sector of a circle ;- its area will therefore be -|p^A0, where p = PQ. Thus if ^ denote the whole area bounded by the curve, its evolute, and two radii of curvature corresponding to the values 0^ and cf)^ of = tan (f) + I t-An^ be the angle between these perpendiculars and the fixed initial line. Then, as in Art. 157, A = ^jpUl - 6') cZc/) (1). Let a; = r cos 9, y = r sin 6 ; then A'=A- {hx + hj) + M + 2ma;?/ + nf (2), where h, Jc, I, m, n are certain quantities which remain con- stant for every position of 0'. Now (2) shews that the locus of [x, y) for a given yahie of A is a conic section ; and that the conic sections obtained by assijrnino: different values to A' are concentric. The conic section is in general an ellipse. For, by putting for I, m, n their values, we have 4 (m' - In) = ] [sin cf) cos -e)dcl); as the second term on the right-hand side is positive. A' is necessarily greater than A, so that the origin is that which makes the pedal area least. In the particular case in which the primitive curve is a closed curve the conic section becomes a circle. For the limits of (/) may then be supposed to be and 27r ; and thus we have l = 7i and m = 0. AREAS OF PLANE CURVES AND OF SURFACES. 145 We may just advert to the effect of the existence of singular points on the primitive curve. In this case it may liappen that (f) does not always increase from tlie lower limit jf the integrations to the upper limit, but sometimes increases md sometimes decreases. Suppose now, for example, that irst increases from Q to -it, then diminishes from -tt to - tt o 3 4' md then increases from - tt to - tt. The values of h, k, I, m, I Avill then be the same as if ^ had always increased from ) to - TT. The area of that part of the pedal curve traced out -s ^ decreases from -^-ir to - tt will count as a negative [uantity. A memoir by Professor Hirst on the Volumes of Pedal hirfaces will be found in the Philosophical Transactions for 8G3. Area of Surfaces of Pevolution. Pectangidar Formidce. 161. Let ^ be a fixed point in the curve APQ ; let x, y e the co-ordinates of any point P, and s the length of the re ylP. Suppose the curve to revolve round the axis of cc, ad let S denote the area of the surface formed by the revolu- on of ^P; then {Differential Calculus, Art. 315) dS . T. I. C. 10 14G AREAS OF PLANE CURVES AND OF SURFACES, therefore >S' = I I'lryds (1) ; thus S = I 27ri/ -J- dx (2), f ds and S = \ 2^-7/ -J- dy (3). Of these three forms we can choose in any particular ex- ample that which is most convenient. If y can be easily ds expressed in terms of s we may use (1) ; if -7- can be easily expressed in terms of 7/ we may use (3) ; generally however ds it will be most convenient to express y and -j- in terms of x and use (2). In each case the area of the surface generated by the arc of the curve which lies between assigned points will be found by integrating between appropriate limits. 162. Application to the Cylinder. Suppose a straight line parallel to the axis of x to revolve round the axis oi x, thus generating a right circular cylinder: let a be the distance of the revolving straight line from the axis of X ; then 2/ = ^. ^^^ ^ ~^> thus by equation (2) of Art. 161, /S = 27r I adx = 27rax + C. Suppose the abscissae of the extreme jDoints of the portion of the straight line which revolves to be x, and x^: then the surface generated = 2-!ra I dx — 2'Tra {x^ — x^. J Xi AHEAS OF TLANE CURVES AND OF SURFACES. 147 1G3. Application to the Gone. Let a straight line whicli passes tlirougli tlie origin and is inclined to the axis of x at an angle a revolve round the axis of X, and thus generate a conical surface. Then y = x tan a, and -r-; = sec a ; thus by equation (2) of Art. 161, S=27r I tan a sec a xdx = 'ir tan a sec ax^+ C, Hence the surface of the frustum of a cone cut off by planes perpendicular to its axis at distances x^, x^ respectively from the vertex is TT tan a sec a (x^ — x^). Suppose x^ = 0, and let r be the radius of the section made by the plane at the distance x^., then r = x^isoa.a, and the area is TT cosec ar*. 1G4. Application to the Sphere, Let the circle given by the equation y^=a^ — x^ revolve round the axis of x ; here dy _ X dx y ' a?\ a 1+-J = y^ y Hence by equation (2) of Art. IGl, dx = lirax + G. *S = 27r (y- dx = 2TTa r y Thus the surface included between the planes determined by x — x^ and x = x^ is 27ra (^2 — rrj. Hence the area of a zone of a sphere depends only on the height of the zone and the radius of the sphere, and is equal 10—2 148 AREAS OF PLANE CURVES AND OF SURFACES. to the area wliicli the planes that bound it would cut off from a cylinder having its axis perpendicular to the planes and circumscribing the sphere; and thus the surface of the whole sphere is 4!7rd\ These results are very important. 165. Application to the Prolate Spheroid. Let the ellipse given by a^ + Z'V= a'h'^ revolve round the axis of X which is supposed to coincide with the major axis of the ellipse ; here dif _ Ifx dx (ly ' dx V \ a^if) ay Hence by equation (2) of Art. 161, The surface generated by the revolution of a quadrant of the ellipse will be obtained by taking and a as the limits of X in the integration. This gives 166. For another exami3le suppose the catenary to revolve round the axis of x. Here s = ^[6" - e " j , by Art. 73, if we measure from the point for which x = 0. Thus we see that y- = s^+c'^. In this case we shall find that we can use any of the three formulae in Art. 161; but (2) will be the most convenient. 167. Suppose one curve to have for its equation y = (^), and another curve to have for its equation y = -^{^), ^nd let AREAS OF PLANE CURVES AND OF SURFACES. 149 both curves revolve round the axis of x. Let s^ and s^ denote the lengths of arcs measured from fixed points in the two curves up to tlie point whose abscissa is x. Let & denote the sum of the areas of both surfaces intercepted between two planes perpendicular to the axis of x at the distances x^ and x„ respectively from the origin. Then, by Art. 161, ^=277 £{^wg + f(.)grf.. For a simple case suppose that there is a curve which is bisected by the straight line y = a, so that we may put ■y = C5 -f ^ (a;) for the upper branch and y = a — xi-^) ^^r the lower branch. Hence ds^ _ ds^ dx dx' p2 ds f and S = 47ra I -j-^dx = iira I ds^ , the limits for s^ being taken so as to correspond with the assigned limits of x. Hence, if there be any complete curve which is bisected by a straight line and made to revolve round an axis which is parallel to this straight line at a distance a from it and which does not cut the curve, the area of the whole surface gene- rated is equal to the length of the curve multiplied by '"lira. For example, take the circle given by the equation [x-hy+{y-kY-c''=0. Here the area of the whole surface generated by the revolu- tion of the circle round the axis of x will be ^irk x lire. There is no difficulty in this example in obtaining sepa- rately the two portions of the surface. For the part above the straight line y = k, -we have 27r I yds, that is, 27rjllc + ^{c'-(x-hf}]ds, that is, 2'rrjkds + lir f V [c' -{x- lif] ds. 150 AREAS OF PLANE CURVES AND OF SURFACES. The former of these integrals is ^-nhs ; the latter is equal to which will reduce to 27r I cdx, that is, ^ircx. Hence the sur- face required is found by taking the expression '^itlis + ^ircx between proper limits. Area of Surfaces of Revolution. Polar Formulw. 168. It may be sometimes convenient to use polar co- ordinates ; thus from Art. IGl we deduce S = hiryds =kiry %de = J2'7rr sin ^ dO, ds where ^ 169. Application to the Cardioid. Here r = a (1 + cos ^) ; thus ^ = a V[(l + cos 6')^+ sin^6'} = a V(2 + 2 cos 6) = 2a cos ^ ; do therefore S = 47ra' (1 + cos 0) cos - sin 0d9= IGTra'Jcos* ^ ^^^^ ^^ = 5- cos 2+ C. The surface formed by the revolution of the complete curve about the initial straight line will be obtained by takmg „, . . 327ra and TT as the limits of in the integral. This gives —^ . AREAS OF PLANE CURVES AND OF SURFACES. 151 Any Surface. Double Integration. 170. Let X, y, z be the co-ordinates of any point 2> of a surface ; x + A.r, y + A^, ^ + A^ the co-ordinates of an ad- x*^ /y jacent point q. Through -p draw a plane parallel to that of ix, z), and a plane parallel to that of (y, z) ; also through q draw a plane parallel to that of {x, z) and a plane parallel to that of (7/, z). These planes will intercept an element ^5- of the curved surface, and the projection of this element on the plane of {x, y) will be the rectangle PQ. Suppose the tangent plane to the surface at p to be inclined to the plane of {x, y) at an angle 7, then it is known from solid geometry that sec 7 ^ + '1, + chV dy) where ^- and -^ must be found from the known equation to ax dy the surface. Now the area of PQ is AxAy, hence by solid geometry the area of the element of the tangent plane at p of which FQ is the projection is AxAysecy. We shall assume that the limit of the sum of such terms as Ax Ay sec 7 for all 152 , AREAS OF PLANE CURVES AND OF SURFACES. values of x and y comprised between assigned limits is the area of the surface corresponding to those limits. Let then >S^ denote this surface ; thus ^-(£hm>'y dy) the limits of the integrations being dependent upon the portion of the surface considered. 171. With respect to the assumption in the preceding Article, the reader is referred to the remarks on a similar point in the Differential Calculus, Art. 308 ; he may also here- after consult De Morgan's Differential and Integral Calculus, page 444, and Homersham Cox's Integral Calculus, page 96. 172. Application to the Sphere. Let it be required to find the area of the eighth part of the surface of the sphere giveu by the equation a^' + y + s' = a^ ,T clz X dz V Here -r- = — . -i- = — -: a' thus 8 = dx s ' dy ^^{d^-x^-f) Now in the figure we suppose OL = x\ put ?/j for 12, then 3/j = VC^* — ^0' ^^^ ^^ value of y^ is obtained from the equation to the surface by supposing 2 = 0. If we integrate with respect to y between the limits and y^, we sum up all the elements comprised in a strip of which LMml is the pro- jection on the plane of (x, y). Now pi dy^ _ r V(a^-^^-y) U^[y{-f) 2 y^- dy _ ir _ thus S= ~^idx. If we integrate with respect to x from to a, we sum up all the strips comprised in the surface of which OAB is the AREAS OF PLANE CURVES AND OF SURFACES. 153 ira projection. Thus --^ is the required result ; and therefore the whole surface of the sphere is 4>7ra^. If we integrate with respect to x first, we shall have r fi. adr/dx ioio ^{d'-x'-f)' where x^ = \l{a^ — if). As another example let it be required to find the area of that part of the surface given by the equation z^ + (x cos a + 3/ sin a)""* — a^ = 0, which is situated in the positive compartment of co-ordinates. This surface is a right circular cylinder, having for its axis the straight line determined by z = 0, xcosa +?/ sin a = 0, and a is the radius of a circular section of it. Here dz _ cos a {x cos a + ;/ sin a) dx z dz _ sin a [x cos a + y sin a) dy~ z thus &= [[^^^ = (I , . , "^"^^ ■ ,., . J J z jj Vl«— (^' COS a + 3/ sm a)} The co-ordinate plane of [x, y) cuts the surface in the straight lines a = + (a; cos a + ?/ sin a), and if the upper sign be taken, we have a straight line lying in the positive quad- rant of the plane of [x, y). To obtain the value of >S' we integrate first with respect to y between the limits y = and y={a — x cos a) cosec a ; now r dy 1 . _.x cos a + V sin a J i^[d^ — [x cos a -1- y sin a)"} sin a a ' take this between the assigned limits, and we obtain TT . _, rr cos a -X — sm sma V2 a ' ' 154 AREAS OF PLANE CURVES AND OF SURFACES. Gj C I tt . 00 cos ol) therefore >S'= -. — \ \^ — sin"^ -[ dx, sm a j ( ti a ) and the limits of the integration are and . Hence we cos a shall find sin a cos a 173. It is worthy of notice that two different surfaces may have their corresponding elements of area equal. Take ■ for example the surfaces determined by 2a2 = ac^ + y^, and by az=-xij\ in each case dz^ (dz_^ _ x^-^y- \dxj \dy) a' Euler has discussed this matter in a Memoir entitled Evolutio insignis paradoxi circa cequalitatem super Jicier am. Novi Comm. Acad. Petrop. Tom. xvi. Pars prior. He calls two such surfaces superficies congruentes. The following surfaces are congruent : the cone {z - c)'' = {(a; - aY +(y- hf] tan' 7, and the plane x cos ci + y cos ^ + z cos y= p. Again, the surfaces determined by the following equations are congruent: 2az = «" + 1/^, 2az = {x' -7f)c + 2xy V(l - c'), 2a2 = [{x' + yj - Uxy + 2c {x' - y') + J' + c^}^, 2az = {x^ - 2/') cos e+2xysmd-[ (6) dd, where ^ (6) is any function of 6, and ^ is a function of x aud y determined by 2xy cos 6 — {x' — y') suxO = ^ {6). 174. Instead of taking the element of the tangent plane at any point of a surface, so that its projection shall be the AREAS OF PLANE CURVES AND OF SURFACES. 155 rectangle Aa;A//, it may be in some cases more convenient to take it so that its projectiou shall be the jjolar element rAOAr. Thus we shall have S= 1 1 secy rdO dr. For example, suppose we require the area of the surface cvij = az, which is cut off by the surface x~ + if = c" ; here sec7 = y^ (1 +-3 + ^.j = ^-^ smce x- + f = r\ Thus S=rr^^^^^^rdedr = '^{{c^+a^f-a^]. J Jo ct. ou 175. Suppose a; = r sin ^ cos (j),y = rsmO sin (f),z = r cos 9, so that r, 6, ^ are the usual polar co-ordinates of a point in space; then Ave shall shew hereafter that the equation may be transformed into An independent geometrical proof will be found in the Cambridge and Dublin Mathematical Journal, Vol. IX., and also in CarmichaeFs Treatise on the Calculus of Operations. It will be remembered that in this formula 7' = V(d:;^-l- 3/^ + 2''), while in Art. 174 we denote V(-^^ + 2/'^) ^7 *'• Approximate Values of Integrals. 176. Suppose y a function of x, and that we require ydx. If the indefinite integral \ydx is known we can at once ascertain the required definite integral. If the inde- finite integral is unknown, we may still determine approxi- mately the value of the definite integral. This process of /, 15G AREAS OF PLANE CURVES AND OF SURFACES, approximation is best illustrated by supposing y to be an ordinate of a curve so that 1 ydx represents a certain area. Divide c— a into n parts each equal to li and draw n — \ ordinates at equal distances between the initial and final ordinates; then the ordinates may be denoted by y^, y^, Vn^ Vn^v Hence we may take ^^(2/1 + 2/2 + + :!/n) as an approximate value of the required area. Or we may take as an approximate value. We may obtain another approximation thus ; suppose the extremities of the r"" and r + lj"" ordinates joined; thus we have a trapezoid, the area of which is (yr + ^/r+Jo* ^^^ sum of all such trapezoids gives as an approximate value of the area ^{l+y.+ys +y.+'^}- This result is in fact half the sum of the two former results. It is obvious we may make the approximation as close as we please by sufficiently increasing n. The following is another method of approximation. Let a parabola be drawn having its axis parallel to that of y ; let Vv V"-' 2/3 represent three equidistant ordinates, h the distance between y^ and 3/^, and therefore also between y^ and y^. Then it may be proved that the area contained between the parabola, the axis of x, and the two extreme ordinates is 3 (2/1 + %2 + 2/3)- This will be easily shewn by a figure, as the area consists of a trapezoid and a parabolic segment, and the area of the latter is known by Art. 143. AEEAS OF PLANE CURVES AND OF SURFACES. 157 Let us now suppose that n is even, so tbat the whole area we have to estimate is divided into an even number of pieces. Then assume that the area of the first two pieces is 3 {y, + 4y, + 2/3). that the area of the third and fourth j)ieces is . 3(^3 + 43/4 +yJ> and so on. Thus we shall have finally as an approximate result 0(^1 + 2(2/3 + ^5+ 2/«-i)+Z/«+i + 4(y.. + 2/4 +yj]- 3 Hence we have the following rule : add together the first ordinate, the last ordinate, twice the sum of all the other odd ordinates, and four times the sum of all the even ordinates ; then multiply the result by one-third the common distance of the ordinates. This rule is called Simpson's Rule: see Simpson's Mathematical Dissertations 1743, page 109. Simpson however merely made the obvious extension of supposing n to be any even number ; the case of « = 2 really involves the whole principle, and this had been given before : see Cotes De Methodo Differentiali, page 32. As an example of Simpson's rule let it be required to find n dx the value of :j 5 . Suppose n = 10; then we have -1 1 __]_ ^ ' 1 + -01' '^^ 1 + -04'*" '^^ 1 + 1" If the calculation be carried to six places of decimals it will be found that the approximate value of the definite integTal is equal to •785398. In this case the exact value is known, namely - ; and this agrees to six places of decimals with the approximate value. 158 AREAS OF PLANE CURVES AND OF SURFACES. 177. Instead of referring to Art. 143 in the preceding investigation we might have used the following method. Assume for the equation to the curve y =A + Bx + Cx^, where A, B, and G are constants; and let y^, y.^, y^ denote the values of y corresponding to the values 0, h, 2h of x respectively. Then y^=A, y^ = A+Bh+Ch\ y^ = A + 2Bh+4.Ch'; and from these equations we can express A, Bh, and Cli' in terms of y^, y^, and y^ The area contained between the curve, the axis of x, and the two extreme ordinates = ydx = 2Ah + 2Bh^ + JO substitute the values of u4, Bh, and Ch"^, and this expression becomes If the first of the three equidistant ordinates had been drawn at any point x = a, instead of the point x=0, we should have obtained the same result. For put x = a + x in the equation to the curve ; the equation will become y = P + Qx + Bx'y where P, Q, and R are constants; and y^, y.^, y^ will now denote the values of y corresponding to the values 0, h, 2h of x\ so that the process and result will be as before. If we take y = A-\-Bx-\- Cx" + Dx^ for the equation to the curve, then as we have only three equations connecting the four quantities A, Bh, Ch'', and Dli? with ;/, , y^, and y^ we cannot determine these four quantities ; it is however worthy of notice that the area will still be expressed by the formula just given. For we have o o and this is equal to /, 2h {A + Bx+ Cx"" + Dx') dx. AREAS OF PLANE CURVES AND OF SURFACES. 159 Let us now investigate an analogous expression f(^r tho case in which four equidistant ordinates are known. Assume foi: the equation to the curve y = A+Bx+ Cx" + Dx^, and let Vv 2/2' Vv y* denote the values of y corresponding to the Values 0, h, 2/i, 3A of x respectively. Then y^ = A + Bh + Ch' + Dh\ y^ = A + 2Bh + 4C/i' + SDh\ y, = A + 2Bh + 9 Ch' + 27 Dh' ; and from these equations we can obtain A, Bh, Ch^, and D?i^ ill terms of y^, y^, y^ and y^. The area contained between the curve, the axis of x, and the two extreme ordinates r3'» , ^^^ dBh' ^^,3 , 8Wh* = ydx = SAh+—^ + 9Ch^+~-^ ; substitute the values of J., Bh, (7/l^ and DA", and this expres- sion becomes -r;-''(y,+ 8^^+ Si/s + ^/J- This result Avas given by Newton ; see the end of his Methodus Differentialis. Then proceeding as in the latter part of Art. 176 we ob- tain the following approximate rule, the whole area being supposed divided into a number of pieces which is some multiple of three : add together the first ordinate, the last ordinate, twice the sum of every third ordinate, excluding the first and the last, and three times the sum of all the other ordinates ; then multiply the result by three-eighths of the common distance of the ordinates. In the methods of finding approximate values of areas of curves which we have explained, we have supposed the successive ordinates to be drawn at equal distances. Another method of approximation has been proposed by Gauss in which the successive ordinates are drawn, not at equal dis- tances, but at intervals which the method shews will ensure the most advantageous results. For an account of this method the student may consult the tenth Chapter of the Elementary Treatise on Lajilace's Functions, Lamp's Functions and Bessel's Functions. IGO EXAMPLES. EXAMPLES. 1. If A denote the area contained between the catenary, the axis of x, the axis of y, and an ordinate at the extremity of the arc s, shew that A = cs. The arc s begins at the lowest point of the curve. 2. The whole area of the curve {-) +[f)"^'^ ^^ i'^^^- (The integration may be effected by assuming a; = a cos^ <^.) 3. The area of the curve y {x^ + a^) =c^{a- x) from x = to a; = a is c'^ (1^ — 2 log 2 J . 4. Find the whole area between the curve y'^x = 4a' (2a — x) and its asymptote. Result. 47ral 5. Find the whole area between the curve y^ (x^-\r a^) = aV and its asymptotes. Result. 4a^ /jQ [ft _1_ n(y\ G. Find the area of the loop of the curve y"^ = . Result. 2a' (l - ^) • x^ (a, -\- x') 7. Find the area bounded by the curve y'^ = — ^ and the asymptote x = a, excluding the loop 4 Result. 2aM 1 + "^ 8. Find the whole area between the curve y' (2a — x)= x^ and its asymptote. Result. oTra". 9. Find the whole area of the curve (y - xf = a" — x". Result. 7^a^ 10. Fiad the area included between the curves 2/'-4aa; = 0, x'-4ay = 0. Result. -^^ 11. Find the whole area of the cur^^e a*y^ + tV = a'b'xi Result, ^ab. EXAMPLES, IGl 12. Find the area of a loop of the curve a'y* = x*(a' — x^). Result, —zr- . 5 .13. The area between the tractory, the axis of y, and the asymptote is -t~ . (See Ai't. 100, and Ai't. 134.) 1-i. Find the area of a loop of the curve i/~ (a* + cc") = a;- (a^ — x^). Result -^ (tt - 2). 15. Find the area of the loop of the curve 16aV = Z>V (a^ — 2ax). • Result. ^ ^ ^ SO IG. Find the area of the loop of the curve 2f {a' + x') = {a' - xy-. Result, a" [S V2 log (1 + V2) - 2}. 17. Find the whole area of the curve 2if {a? + x") - ^ay (a' - x") + (a' - xj = 0. Result. aV •] 4 ^— Y . 18. Find the area of the curve y = c sin - . lofj sin - from ic — to a; = air. Result. 2ac (1 — log 2). 19. Find the area of the curve -= (-) between x = a and c V«/ X = /3, and /ro??i tJi5 result deduce the area of the hyperbola xy = a* betvyeen the same limits. 20. Find the area of the ellipse whose equation is ax~ + 2hxy + cy" = 1. Result. TT »J{ac — y^) ' 21. Find the area of a loop of the curve r^ = a^ cos 29. Result. -^. T. I. C. 11 1G2 EXAMPLES. 22. Find the area contained by all the loops of the curve r = a sin nd. 2 2 7ra Tra Result. -4- or —r according as n is odd or even. 4 2 23. Find the area between the curves r = a cos nO and r = a. 24. Find the area of a loop of the curve r^ cos 9 = oj' sin 3^. O 2 2 Result -r- — IT log 2. 4 2 25. Find the whole area of the curve r=a (cos 2(9 + sin 2^). ^ ■ Result. Tra'. 26. Find the area of a loop of the curve (x^ + y'^f = 4aV7/^ Result. -5- . o 27. Find the whole area of the curve (af + yy = 4!a'x^ + Wf. Result. 277 (a" + b^. 28. Find the whole area of the curve -^ + Tz = ^2\—2 + i^] ' Result. ^riV^+^J- 29. Find the area of the loop of the curve 2/' - oaary + x^ = 0. Result. — . 30. Find the area of the loop of the curve r cos 6 =a cos 20. Result. [ 2 — ^ j a\ 31. Supposing a greater than h find the area of the curve V(a— 6cos"d^) \l(a—b) 2 32. In a logarithmic spiral find the area between the curve and two radii vectores drawn from the pole. EXAMPLES. 103 33. Find the area between the conchoid r = a + h cosec and two radii vectores drawn from the polo. 3-i. In an ellipse find the area between the curve and two radii vectores drawn from the centre. 35. In a parabola find the area between the curve and two radii vectores drawn from the vertex. 36. Find the area between the curve r = a. (sec ^ + tan ^) and its asymptote r cos = 2a. Result. ( ^ + 2 ) a\ 37. Tlie whole area of the curve r = a (2 cos ^ + 1) is / 3 /3\ a- f 2-77 H — ^— j , and the area of the inner loop is a' ( 3 \/o\ 38. Find the whole area of the curve r = a cos ^ + J, where a is greater than h. Also find the area of the inner loop. 39. If X and y be the co-ordinates of any point of an equi- lateral h}T)erbola x''-y-=a^, and u the area inter- cepted between the curve, the central radius vector drawn to the point {x, y), and the axis, shew that 40. Find the whole area of the curve which is the locus of the intersection of two normals to an ellipse at right angles. Besult. ir [a -%)~. It may be shewn that the equation to the curve is , ^ ( a''-&-)'(a''sin^^-6^cos^^ )'' ^' (a' -F ¥) {a' sin^ O+h- cos''' Bf ' (See Plane Co-ordinate Geometry, Example 53, Chap- ter XIV.) 11—2 164 EXAMPLES. 41. Find the area included within any arc traced by the extremity of the radius vector of a spiral in a com- plete revolution, and the straight line joining the ex- tremities of the arc. If, for example, the equation to ^}> prove that the area corre- sponding to any value of 6 greater than 27r is ira' 2 f ' fi \ '^"■^'^ / ^ -"'''^ .)^ ) 19^-^ 2w + l [V^tt/ XLtt 42. Find the area contained between a parabola, its evolute, and two radii of curvature of the parabola. (Art. 157.) 43. Find the area contained between a cycloid, its evolute, and two radii of curvature of the cycloid. 44. Find the area of the surface generated by the revolution round the axis of x of the curve xy — k^. X 45. Also of the curve y = ae''. 46. Find the area of the surface generated by the revolution c - -- of the catenary y = ^ (e° + e ") round the axis oi y. 47. Shew that the whole surface of an oblate spheroid is 48. A cycloid revolves round the tangent at the vertex : shew that the whole surface generated is -^ ira'. 49. A cycloid revolves round its base : shew that the whole G4 surface generated is -7,- ira^. 50. A cycloid revolves round its axis : shew that the whole surface generated is 87ra" (tt — f ). EXAMPLES, 1G5 51. The whole surface generated by the revolution of the tractory round the axis of x is iirc". 62. A sphere is pierced perpendicularly to the plane of one of its gi-eat circles by two right cylinders, of which the diameters are equal to the radius of the sphere and the axes pass through the middle points of two radii that compose a diameter of this great circle. Find the surface of that portion of the sphere not included within the cylinders. Result. Twice the square of the diameter of the sphere. 53. Find the surface generated by the portion of the curve cc 7/ = a 4- a log - between the limits x = a and x = ae. o - o ^ Result, ^-na- |l + V(l + O - V2 + log .^ _^Y(/+,-J " 54. Find I — , where d^ represents an element of surface, and 'p the perpendicular from the origin upon the tangent plane of the element, the integral being ex- tended over the whole of the ellipsoid -7,-\-^,-\- —, = \. ^ a b c Result s^(«'^' + ^'c'^ + cV). 166 CHAPTER VIII. VOLUMES OF SOLIDS. FormulcG involving Single Integration. Solid of Revolution. 178. Let ^ be a fixed point on a curve APQ, and P any other point on the curve whose co-ordinates are x and y ; and suppose X algebraically greater than the abscissa of A. Let the curve revolve round the axis of x, and let V denote the volume of the solid bounded by the surface generated by the curve and by two planes perpendicular to the axis of x, one through A and the other through Pj then {Differential Calculus, Art. 314) dV Tx=''y therefore V = liry'dx. From the equation to the curve 3/ is a knoAvn function of X \ suppose i/r {x) to be the integral of iry"^ ; then v=^{x)^-a VOLUMES OF SOLIDS. 1G7 Let Fj denote the volume when the point P has x^ for its abscissa, and V^ the vokime when the point P has x^ for its abscissa; thus therefore V^— T"i = "^/^ (-^'o) — "^ (^i) = tt I y^dx. 179. Application to the Bight Circular Cone. Let a straight line pass through the origin and make an angle a with the axis of x ; then this straight line will gene- rate a right circular cone by revolving round the axis of x. Here y = x tan a ; thus V= jirtan^ax'^dx = — ~ — x^ + C, ■TT- TT- "^ tan Ct , g g. Suppose iCj = 0, and let r = x^ tan a ; thus the volume becomes — — ^ — - , that is, — ^ . Hence the volume of 3 3 a right circular cone is one-third the product of the area of the base into the altitude. 180. Application to the Sphere. Here taking the origin at the centre of the sphere we have y^= c^ — x^ ; thus \'iry^dx = 'JTi/r {x)Y dx -'^\\:['i>{x)Y-{{x)Y-{ir{x)Y = 4ax{x), and F= tt j-iiax (x) dx. Suppose the abscissge of the extreme points of the curve are x^ and x^, then the volume generated by the revolution of the closed curve round the axis of x is 4(Z7r 1 ^ (^) ^^• And 2 1 % (a;) dx is the area of the closed curve, so that the volume is equal to the product of 2a7r into the area. This demonstration supposes that the generating curve lies en- tirely on one side of the axis of x. If the generating curve be the circle given by {x-hr+{y-hy = c\ we have ttc^ for its area, and therefore 2/ircV^ for the volume generated by the revolution of it round the axis of x. 186. In a similar way if the curves x= q> (y), x = -^ (y), revolve round the axis of y we obtain for the volume bounded by these surfaces and by planes jDerpendicular to the axis of y V='jrj[[cf>iy)Y-{f{y)Y]dy. 187. The method given in Art. 178 for finding the volume of a solid of revolution may be adapted to any solid. The method may be described thus : conceive the solid cut up into thin slices by a series of parallel planes, estimate approxi- mately the volume of each slice and add these volumes ; the limit of this sum when each slice becomes indefinitely thin is the volume of the solid required. Suppose that a solid is cut VOLUMES OF SOLIDS. 171 up into slices by planes perpendicular to the axis of cc; let (j) (.^') be the area of a section of the solid made by a plane which is at a distance x from the origin, and let £c + Ax be the distance of the next plane from the origin; thus these two planes intercept a slice of which the thickness is Ax, and of which the volume may be represented by <^ (x) Ax. The volume of the solid Avill therefore be the limit of ^0 {x) Ax, that is, it will be (f> (x) dx; the limits of the integration will depend upon the particular solid or portion of a solid under consideration. For example take a prism as defined in Euclid, Book XL Cut up the prism into slices by planes which are parallel to the two equal and similar ends; take the axis of x perpen- dicular to the two ends. Thus ^{x) is a constant, say A ; the volume of the i^v'ism = A dx ^ Ah, where h is the perpen- dicular distance between the two equal and similar ends. 188. Application to an Ellipsoid. The equation to the ellipsoicj is if a section be made by a plane perpendicular to the axis of x at a distance x from the origin, the boundary of the section is an ellipse, of which the semiaxes are h ,/ (1 A and c / f 1 jj ; hence the area of this ellipse is irhc ( 1 — — ^ ) ; this is therefore the value of [x). Hence the volume of the eUipsoid r 7 ft ^\ 7 ^-rrdbc = I iroc 1 7,\dx= — ^ — . ■/>(i-?) 189. Application to a Pyramid. Let there be a pyramid, the base of which is any recti- linear figure ; let A be the area of the base and h the height. 172 VOLUMES OF SOLIDS. Take the origin of co-ordinates at the vertex of the pyramid, and the axis of x perpendicular to the base of the pyramid, then the volume of the pyramid rh = (f> {x) dx. J Now the section of the pyramid made by any plane pa- rallel to the base is a rectilinear figure similar to the base, and the areas of similar figures are as the squares of their homologous sides; and x and h are proportional to homo- logous sides ; hence we infer that x^ 4> {x) = p A. Thus the volume of the pyramid _A [^ 27 _^ fl J o This investigation also holds for a cone, the base of which is any closed curve. 190. For another example we will find the volume lying between an hyperboloid of one sheet, its asymptotic cone, and two planes perpendicular to their common axis. Let the equation to the hyperboloid be „2 , .2 _2 and that to the cone 2 2 2 a' b' c' If a section of the former surface be made by a plane perpendicular to the axis of x and at a distance x from the origin, the boundary is an ellipse of which the area is irhc ( — 2 + 1 ) ; the section of the second surface made by the same plane also has an ellipse for its boundary, and its VOLUMES OF SOLIDS. 173 •nhcx^ area is — s— . Therefore the difference of the areas is irhc. a' Hence the required volume, supposing it bounded by the planes x = x^ and x = x„, is I rrhcdx, that is, ':rlc{x^ — x^. 101. Sometimes it may be convenient to make sections by parallel jDlanes not perpendicular to the axis of x. If a be the inchnation of the axis of x to the parallel planes, then 4> {x) sin a^x may be taken as the volume of a slice and the integration performed as before. 192. The remarks made in Arts. 176 and 177 have an apjjlication to the subject of the j^resent Chapter. Let there be a solid such that the area of a section made by a plane jDarallel to a fixed plane and at a distance x from it is always equal to P + Qx + Rx^ + Sx^, where P, Q, B, S are constants. Let three equidistant sections of the solid be made by planes parallel to the fixed plane, 2/i being the dis- tance between the two extreme sections. Let the area of the sections, taken in order, be denoted by A^, A^, A^. Then the volume of the portion of the solid contained between the two extreme sections is equal to {A^ + 4^A^ + A^). 3 lifour equidistant sections be made, oh being the distance between the extreme sections, and the area of the sections taken in order be denoted hy A^,A^, A^,A^, then the volume of the portion of the solid contained between the two extreme sections is equal to ^(a^ + sa^+sa^+a;). Hence we may obtain rules for estimating approximately the volume of any solid. Make equidistant parallel sections of the solid ; the areas of these sections must then take the place of the ordinates which occur in tlie Eules given in Arts. 176 and 177. 174 VOLUMES OF solids; Formulce involving Double Integration. 193. We will first give a formula for the volume of a solid of revolution. In the figure, let x, y be the co-ordinates of s, and X + Ax, ;/ + A^/ those of t Suppose the whole figure to revolve round the axis of x, then the element st will generate a ring, the volume of which will be ultimately 2713/ A^ A?/: this folloAvs from the consideration that Aa;A?/ is the area of st and 27r_?/ the perimeter of the circle described by s. Hence the volume generated by the figure BEeh, or by any portion of it, will be the limit of the svmi of such terms as Stt?/ Aa; Ay. Let F denote the required volume, then V— 27r \\ydxdy \ the limits of the integration being so taken as to include all the elements of the required volume. 194. Suppose that the volume required is that which is obtained by the revolution of all the figure BEeh; let y=j){x) be the equation to the upper curve, y=-y\r{x) that to the lower curve, and let OG=x^, 011= x^. We should then integrate first with respect to y between the limits y = -\^{x) and y — (f)(x); we thus sum up all the elements like 27ryAxAy which are contained in the solid formed by the revolution of VOLUMES OF SOLIDS. I7u the strip PQqp ; then we integrate with respect to x be- tween the limits x^ and x^. Tims to express the operation symbolically rx., r^(x) F= 27r y dx dij J x^J ^ (x) = 'rr\^\[j>{x)Y-[f{x)Y]dx. The second expression is obtained by effecting the inte- gration with respect to y between the assigned limits, and it coincides with that already obtained in Art. 185. 195. Thus in the preceding Article we divide the solid into elementary rings, of which liryb^xl^y is the type; in the first integration we collect a number of these rings, so as to form a figure which is the difference of two concentric circular slices ; in the second integration we collect all these figures and thus obtain the volume of the required solid. The truth of the formulee of the preceding Article is obvious as soon as the notation of the Integral Calculus is under- stood. 196. Suppose the figure which revolves round the axis of X to be bounded by the curves x = (f) (y) and x = ylr (?/), and by the straight lines y =y^ and y = y^ ', then in applying the formula for V it will be convenient to integrate first with respect to x ; thus rvi ri'iy) V=27r ydydx. ■J Vi J ^ {y) In this case in the integration with respect to x we collect all the elements like 27ry^yAx which have the same radius y, so that the sum of the elements is a thin cylindrical shell, of which Ay is the thickness, y is the radius, and (p{y) — "^{y) the height. Thus o F=27r [c^{y)-'^{y)]ydy. 197. As an example of the preceding formulte, let it be required to find the volume of the solid generated by the re- 17G VOLUMES OF SOLIDS. volution of the area ALB round the axis of x in the figure already given in Art. 141. This volume is the excess of the hemisphere generated by the revolution of SLB over the pa- raboloid generated by the revolution of ABL ; the result is therefore known, and we propose the example, not for the sake of the result, but for illustration of the formulae of double integration. Let S be the origin. Suppose the positive direction of the axis of X to the left, then the equation to AL is ?/*= 4a (a — a;) and that to BL is if= 4(1^— x". Let V be the required volume, then /•2a rV(4a--2/^ V= I ^iry dij dx. Jo J jgl-p-i ia, If we wish to integrate with respect to ;/ first, we must, as in Art. 141, suppose the figure ALB divided into two parts; thus V=\ I 27rydxdi/+j lirydxdy. •! J ^{ia'-iax) J a J Again, let it be required to find the volume generated by the revolution of LDG about the axis of x. Let the positive direction of the axis of x be now to the right, then the equa- tion to LC is 2/^ = 4a (a + x-) and that to LD is i/^ = 4 O t> -^ + r2 + ^ = 1- a c Here we have to find First integrate with respect to y, then the limits of y are and LI, that is, and b ./ (l — -rA ; we thus obtain the sirna of all the columns which form the slice between the planes Zpl and 3Icpn. Now between the assigned limits 1-" ,2 .fi^ „J d' l)^^ = ?(-J)^ thus V=\j^lc(l 7,)dx. The limits of x are and a ; we thus obtain the sum of VOLUMES OF SOLIDS. 170 all the slices which are comprised in the solid OABG. Hence ■,,_7rt/6c 201. Suppose the given surface to be determined by xy = az, and we require the volume bounded by the jjlaue of (x, y), by the given surface, and by the four planes x = x^, so = x^, y = y^, y = y„. Here the volume is given by = 4 - (^2 - ^'i) {1/2 - 2/1) l-^iyi + ^,y, + ^i3/2 + ^,i/.x] where z^, z^, z^, z^ are the ordinates of the four corner points of the selected j)ortion. 202. Find the volume comprised between the plane 2; = and the surfaces xy = az and {x — hy + {y — kj = c\ Here we have to integrate li—dx dy between limits de- termined by {x — Kf +(y— k)" = cl Now lydy = ^, and the limits of y are ^' - Vic' - {x - h'f] and /.• + ^{c' -{x- JiY]. Thus we obtain 2k Vic' ~{x- hy]. Hence finally the required volume = ~ja^^{c'-(x-hr}dx, where the limits of x are h — c and h-]-c. 12—2 180 VOLUMES OF SOLIDS. And jx v[c' -(x- ny] dx =j{x - h) v[c' -(x- hy] dx + hj>^[c'-{x-hy]dx. Put x — h = t; thus we obtain jt V(c' - f) dt + a[v(c' - f) dt The limits of t are — c and + c ; therefore the result is AcV , ,, • J 1 • ^^'cV — ^r— ; and the required volume is . ^ Co This result however assumes that ^;/ is positive throughout the limits of the integration ; that is, the circle determined by (^x — hy+{y—ky = c^ is supposed to lie entirely in the first quadrant or entirely in the third quadrant. If this condition be not fulfilled our result does not give the arithmetical value of the volume, but the balance arising from estimating some part of the volume as positive and some part as negative; for example, if h and k vanish our result vanishes. Similarly in the result of the preceding Article, it is assumed that xy is ^jositive throughout the limits of the in- tegration. 203. Instead of dividing a solid into columns standing on rectangular bases, so that zl^xi^y is the vokime of the column, we may divide it into columns standing on the polar element of area; hence 2;?'A^Ar is the volume of the column. Therefore for the volume F of a solid we have the formula =// zrdO dr. From the equation to the surface z must be expressed as a function of r and 6. For example, required the volume comprised between the plane z = 0, and the surfaces x^ -^if= iaz and y"^ = 2cx - x\ Here z = z-\ and the limits of r and 6 must be such as to 4a' VOLUMES OF SOLIDS. 181 extend the inteQfration over the whole area of the circle y' = 2cx — x^. Let )\ = 2c cos 6 ; then the required volume \~dedv = ^\ cos*ede==— cos*dd0 204, Required the volume of the solid comprised between the plane of {x, y) and the surface whose equation is z = ae ~c^ . Here, since x^ + if = r"^, we have V= a \\e ''' rdO dr. The surface extends to an infinite distance from the origin in every direction ; thus the limits of 6 are and Itt, and those of r are and oo . > Now je '^^ rdr = ^ c^\ thus Jo rdr = 2' And r2rr dO Jo = 27r. Hence the required volume is Trad'. There is a point involved in this Example which deserves notice ; it relates to the limits of the integral. It is plain that in general corresponding to the limits ± c for ^ and y it would not be sutficient to integrate between the limits and 27r for 0, combined with the limits and c for r ; the integration in the latter case instead of extending over a certain square would extend only over the inscribed circle. In like manner the limits + go for x and y do not certainly con-espond to the limits and 27r for 6, combined with and CO for r. But in the present Example it is easy to see that 182 VOLUMES OF SOLIDS. no error arises; tlie part of the integral which depends as it were on the difference between the square and the circle vanishes in comparison with the rest of the integral. The subject has been noticed by mathematicians: see the Melanges Math4- onatiques et Astronomiques, St Petersbourg, 1859, Vol. 2, page Go, and a paper by Professor Cayley in the Messenger of Mathematics, 1874. Formulce involving Triple Integration. 205". In the figure to Art. 199, suppose we draw a series of planes perpendicular to the axis of z; \et z be the distance of one plane from the origin and s+As the distance of the next. These planes intercept from the column jjqPQ an elementary rectangular parallelepiped, the volume of which is A^A_?/A2. The whole solid may be considered as the Hmit of the sum. of such, elements. Hence if V denote its volume, V = \\\ dxdydz. 206. Eequired the volume of a portion of the cylinder determined by the equation x^ + y""- 2ax = 0, wdiich is intercepted between the planes z = x tan a and z = x tan /3. Here if y, stand for ^{2ax-x-), we have Y—jj dxdydz Jo J -y J X tun a (I fill \ (tan/3 — tana) iTcZicay ^ -2/1 = 2 (tan /3 - tan a) x »J{2ax - x"") dx Jo = 2 (tan /3 — tan a) -^ . VOLUMES OF SOLIDS. 183 207. The polar element of plane area is, as we have seen in previous Articles, rA0/\r. Suppose this were to revolve round the initial line through an angle 27r, then a solid ring would be generated, of which the volume is 27rrsin ^rA^A?-, since 27r?- sin 6 is the circumference of the circle described by the point whose polar co-ordinates are r and 0. Let (f) denote the angle which the plane of the element in any position makes with the initial position of the plane, (f>+A(f) the angle which the plane in a consecutive position makes with the initial plane ; then the part of the solid ring which is inter- cepted between the revolving plane in these two positions is to the whole ring in the same proportion as A^ is to 27r. Hence the volume of this intercepted part is r^sm0A(f)MAr. This is therefore an expression in polar co-ordinates for an element of any solid. Hence the volume of the whole solid may be found by taking the limit of the sum of such ele- ments ; that is, if V denote the required volume, V=jjlr'sm0d(f>d0ch The limits of the integration must be so taken as to in- elude in the integration all the elements of the proposed solid. The student will remember that r denotes the distance of any point from the origin, the angle which this distance makes with some fixed straight line through the origin, and (f> the angle which the plane passing through this distance and the fixed straight line makes with some fixed plane passing through the fixed straight line. 208. Suppose, for example, that we apply the formula ta find the volume of the eighth part of a sphere. Integrate with respect to ?' first ; we have / r^ dr = - . Suppose a the radius of the sphere, then the limits of r are and a ; thus V = jj'^s,med + /\(p. TT Lastly, integrate with respect to (f) from to ;^ ; thus In this example the integrations may be performed in any order, and the student should examine and illustrate them. 209. A right cone has its vertex on the surface of a sphere, and its axis coincident with the diameter of the sphere passing through that point : find the volume com- mon to the cone and the sphere. Let a be the radius of the sphere ; a the semi-vertical angle of the cone, V the required volume, then the polar equation to the sphere with the vertex of the cone as origin is r = 2a cos 6. Therefore '2w fa r2acoa r2w fa f2aco3 9 F= r'' s'm d d

/ dz to find the volume of a closed surface of the second order whose equation is ax^ + bi/ + cz' + a'yz + h'xz + c'xy = 1. 25, State between what limits the integrations in dx dy dz must be performed, in order to obtain the volume contained between the conical surface whose equa- tion is z = a — i\J(x^ -{-y"^), and the planes whose equa- tions are x = z and x = Q; and find the volume by 2a^ this or by any other method. Residt. -^ . 26. State between what limits the integrations must be taken in order to find the volume of the solid con- tained between the two surfaces cz = mx^ + ny^ and z= ax+ hii : and shew that the volume is - — when m = 11 = a = h = 1. 27. A cavity is just large enough to allow of the complete revolution of a circular disc of radius c, whose centre describes a circle of the same radius c, while the plane of the disc is constantly parallel to a fixed plane, and perpendicular to that of the circle in which its centre moves. Shew that the volume of the cavity is f (.Stt + 8). 28. Find the volume of the cono-cuneus determined by 2 2 X which is contained between the planes x = nnd X = a. Result. -^—^ . 190 EXxiMPLES. 29. The axis of a right cone coincides with a generating line of a cylinder ; the diameter of both cone and cylinder is equal to the common altitude : find the surface and volume of each part into which the cone is divided by the cylinder. Mesidts. „ „ 47r Vo - 3 V15 2 J 27r -v/5 + S^/lo ^ Surfaces, ^ a and — — a ; 87r + 27V3-64 3 , 64-27V3-27r 3 Volumes, ^ a and ^, ■ — a^ ; where a is the radius of the base of the cone or cylinder. 30. A conoid is generated by a straight line which passes through the axis of z and is perpendicular to it. Two sections are made by parallel planes, both planes being parallel to the axis of z. Shew that the volume of the conoid included between the planes is equal to the product of the distance of the planes into half the sum of the areas of the sections made by the planes. 191 CHAPTER IX. DIFFERENTIATION OF AN INTEGRAL WITH RESPECT TO ANT QUANTITY WHICH IT MAY INVOLVE. 211. It is sometimes necessary to differentiate an inte- gral with respect to some quantity wliicli it involves ; this question we shall now consider. Required the differential coefficient of I (f> {x) dx with J a respect to h, supposing (^ {x) not to contain h, and a to be indej^endent of b. Let 10= \ (f) (cr) dx ; J a suppose b changed into b + Ab, in consequence of which w becomes u + Au ; thus rb + Ab u + A?( =1 ^ (•^) dx 5 J a rb + Ab rb therefore Au= j (x) dx — I (j) {x) dx J a J a b + Ab b (f>{x) dx. Now, by Art. 40, rb + Ab ' j {x)dx = Abc}){b + eAb), where 6 is some proper fraction ; thus ^^ = c^ (6 + dAb). 192 DIFFERENTIATION OF AN INTEGRAL Let Ab and Au diminish without limit ; thus 212. Similarly, if we differentiate u with respect to a, supposing (f) {x) not to contain a, and b to be inde23endent of a, we obtain 213. Suppose (p (x) to contain a quantity c, and let it be required to find the differential coefiicient of I cf) (x) dx J a with respect to c, supposing a and h independent of c. Instead of ^ (x) it will be convenient to write ^ {x, c), so that the presence of the quantity c may be more clearly indicated ; denote the integral by u, thus •b u= i (f> (x, c) dx. J a Suppose c changed into c + Ac, in consequence of which u becomes u + Au ; thus ti+ Au= i 4) {x, c + Ac) dx ; J a ' therefore Aw = (f){x,c + Ac) dx — j (f) {x, c) dx J a •'a = 1 { WITH RESPECT TO AXY QUANTITY. 193 where p is a quantity which diminishes without limit when Ac does so. Thus we have Ac J a etc J a When Ac is diminished indefinitely, the second integi'al vanishes ; for it is not gx'oater than {b — a) p, where p' is the greatest value p can have, and p ultimately vanishes. Hence proceeding to the limit, Ave have du f^dd) (x, c) , -J- = — ^-1 dx. dc J a dc 214. It should be noticed that the preceding Article sup- poses that neither a nor b is infinite ; if, for exam2:)le, b were infinite, we could not assert that (6 — a) p' would necessarily vanish in the limit. 215. We have shewn then in Art. 213 that We will point out a useful application of this equation. Suppose that -v/r [x, c) is the function of which

,c) (2), let us suppose that b does not occur in (f) (x, c), and that a is also independent of b ; then (2) may be written ^?^+C = %(i.o) (S). dc where C denotes terms which are independent of b, that is, are constant with respect to b. Hence as b may have T. I. c. 13 lO-i DIFFERENTIATION OF AN INTEGRAL any value we jolease in (3), we may replace h by x, and write %(-.c) = '^%^ + c w. dc This equation may be applied to find ;^ {x, c) ; as the constant may be introduced if required, we may dispense with writing it, and put (4) in the form I -?-^-^—^ dx = -j-\(p{x, c) dx. For example, let ^ {x, c) = , ., .^ 5 then dx 1 , _i = - tan ex, I 6 (x, c) dx = \ ^ .-, 2 — thus -1- ( - tan"^ ex] = I -^ ( -, r-j dx ac\c J J dc \l + c'x J -I 2cx'' dx. (1+cV/ dx Thus from knowing the value of = s— „ we are able to ^ J 1 + ex deduce by differentiation the value of the more complex intes^ral ( r-- — ^-^r-„ dx. * j (1 + cV)" 216. Eequired the differential coefficient cf 1 (^{x, c) dx J a Avitli respect to c when both b and a are functions of c. Denote the integral by u ; then -j- consists of three terms, one arising from the fact that ^ (x, c) contains c, one from the fact that h contains c, and one from the fact that a contains c. WITH KESPECT TO ANY QUANTITY. 195 Hence by the preceding Ai-ticles, du _ /"* d^ (x, c) , du db du da dc J a dc db do da dc J a dc C^d(b (x, c) J , ,, .dh . , . 217. With the suppositions of the preceding Article we may proceed to find -^ . By differentiating with respect to c the term ( -^-^-^ dx we obtain J a dc r^ c?'^ {x, c) 7 ,d(f) {h, c) db d(f) (a, c) da J a dc^ dc dc dc dc From the other terms in -r- we obtain by differentiation ^^ ' ^ dc' db \dcj dc dc ,, s d'^a d(}> (a, c) /daV d(f> (a, c) da Thus ^!=r^(^'^^. dc J a dc ^ ^ ^ dc db \dcj dc dc __, f ^ d^a dl«,cj^^, d^[d^) ~^ dc Tc- dhi Similarly -j-^ may be found and higher differential co- efficients of u if required. 13—2 19G DIFFERENTIATION OF AN INTEGRAL 218. The following geometrical illustration may be given of Art. 216. M M' uV N' Let y = (f)(x, c) be the equation to the curve APQ, and y = cf>(x, c + Ac) the equation to the curve A'P'Q'. Let 0M= a, MM' = Aa, NN' = M. Then u denotes the area FMNQ, and u + Au denotes the aveaP'iM'N'Q'. Hence and Au = P'pq Q + qNN'q - PMM'p, Au _ P'pqQ' QNN'q PMM'p Ac Ac ' Ac Ac It may easily be seen that the limit of the first term is the limit of f ^ ^(^>g + ^c)-(^'»g) ^^ that the limit of the L Ac second term is the limit of ^ (h, c) -r- , and that the limit of the third term ia the limit of ^ {a, c) -^ . This gives the result of Art. 216. WITH RESPECT TO ANY QUANTITY. 107 219. Example. Find a curve such that the area between the curve, the axis of x, and any ordinate, sliall bear a con- stant ratio to the rectangle contained by that ordinate and the corresponding abscissa. Suppose ^ {x) the ordinate of the curve to the abscissa x; then I <^ (x) dx expresses the area between the curve, the J axis of X, and the ordinate (c) : hence by sujoposition we must have Jo n where n is some constant. This is to hold for all values of c; hence we may ditferentiate with respect to c ; thus ^ n n therefore c0' (c) = {n— 1) cf) (c), , 6' (c) n — 1 (c) = {n — 1) log c + constant ; thus (}) (c) = ^c""S where A is some constant ; thus we have finally (f> {x) = Ax''~\ which determines the required curve. 220. Find the form of ^ {x), so that for all values of c x[(^ {x)Y dx I' Jo c n {(}i{x)Ydx By the supposition fx{cf>ix)Ydx = ir{cf^{x)Ydx. JO itJo 198 DIFFERENTIATION OF AN INTEGRAL Differentiate with respect to c ; thus thus o{l-^{i>{c)r = lj\{x)Yda,. Differentiate again with respect to c ; thus (l - i) [c/, (c)r + 2c (l - ^^) c/> (c) f (0) = ^-^^ hence (l " ^) ^ (c) + 2c (l - ^^) ' (c) = ; . ^ <^'(C) 2-71 1 therefore , , . = zr? iT ~ • (p (c) 2 (n — 1) c Integrate ; thus 2—7?. log (/) (c) = 2-(,7:riy log c + constant ; 2-n therefore (x) + 2x(f)' (x) must be zero. Therefore £^ = -1-. 9 {x) zx therefore log (^ (a:) = — ^ log x + constant, therefore (h (x) =—- , \/x where A is some constant. This is the solution of a problem in Dynamics, which may be enunciated thus. Find a curve, such that the time of falling down an arc of the curve from any point to the lowest point may be the same. If s denote the arc of the curve measured from the lowest point, x the vertical abscissa of the extremity of s, then we have ds -V- = (a;) and s=2A >Jx; 30 that the curve is a cycloid (Art. 72). 200 MISCELLANEOUS EXAMPLES. MISCELLANEOUS EXAMPLES. 1. If the straight line SP^P^P^ meet three successive revo- lutions of an equiangular spiral, whose equation is r= a^, at the points P^, P^, P^, find the area included between P^P^, P^Pz^ ^"^ ^^^^ *wo curve lines P^P^,P^P^. Result. ~rj^ — (P,-PJ'. 2. Find the area of the curve y'^ — axy^ + x^ = 0. 7ra"\/2 Result. 16 3. Find the area of the curve a'" + 2/^" = a" {xyY ^, where n is a positive integer. 2 Result. If n is an even integer -^r- ; if n is an odd . , aV integer . 4. A string the length of which is equal to the perimeter of an oval is wound completel}'' round the oval, and an involute is formed by unwinding the string, begin- ning at any point : shew that when the length of the involute is a maximum or a minimum the length of the string is equal to the perimeter of the circle of curvature at the jooint from which the unwinding beojins. 5. Find the portion of the cylinder oc^ + y^ — rx= inter- cepted between the planes ax + by + cz = and ax + h2/ + cz — 0. TT (a' — a) ?-' Result. 8c 6. Find the volume of the solid bounded by the para- boloid y'^+ z^= 4a(a;+a) and the sphere x^ + y^-{- z'^ = c\ supposing c greater than a. Result. 2'jra ( c^ — -^j- 1 . 201 CHAPTER X. ELLIPTIC INTEGRALS. 222. The integrals [ „, ^f . .^, , fv(l - c' siu'^) cW, \ (IB and /-, . ., ,,, — -r-, o . „ -, , are called elliptic fanc- j(l + asm-^) \/(l -c'sm^^; ' ^ -^ tions or elliptic integrals of the first, second, and third order respectively; the first is denoted by F{c, 6), the second by E (c, 6), and the third by II (c, a, 6). The integi'als are all supposed to be taken between the limits and 6, so that they vanish when 6 vanishes. 9 is called the amjilitude of the function. The constant c is supposed less than unity; it is called the modulus of the function. The constant a, which occurs in the function of the third order, is called the para- meter. When the integrals are taken between the limits and — , they are called coinplete functions; that is, the ampli- tude of a comjDlete function is ^ . 223. The second elliptic integral expresses the length of a portion of the arc of an ellipse measured from the end of the minor axis, the excentricity of the ellij)se being the modulus of the function. JFrom this circumstance, and from the fact that the three integrals are connected by remark- able properties, the name elliptic integrals has been de- rived. 22-i. The theory of elliptic integrals and the investiga- tions to which it has led constitute a part of the Integral Calculus of great extent and importance, to which much attention has been recently devoted. We shall merely give a few of the simpler results. For further information the student is referred to the elementary treatise on the subject by Professor Cayley. 202 ELLIPTIC INTEGRALS, 225. If 6 and cf) are connected by the equation F(c,e) + F{c,c}>) = Fic,f^), where /m is a constant ; then will cos 6 cos ^ — sin ^ sin ^ ^(1 — c^ sin^ fi) = cos /x. Consider $ and ^{l- & sin' ; therefore df "" ~ 2 ^^^"^ - ^^^ '^^' Let ^ + ^ = -v//- and ^ — ^ = ^j^ ; thus -^ = -c-sm'»/rcosx, -^ = -c sm^cosi/r. therefore , , , = cot y, ■ , , , = cot ^Ir ; d±dx d^dx at dt dt dt ELLIPTIC IXTEGRALS. 203 therefore therefore log -^ = log sin x + constant, _ clylr , . therefore W^ ^^^"^ ^ ■(2), and similarly -jj; = B sin ^p' where A and B are constants. Hence -^ sin % -^'^ = jK sin -v/r — , therefore A cos % = -B cos -v/r + C (3). Now from the original given equation we see that if ^ = F{c,6) = F{c,,.); therefore then 6 = (x and % = ■^/^ = /^ ; thus from (3) {A- B)cos[Jb= C; thus A cos {e-(p)=B cos (^ + <^) + {A - B) cos ^l ; therefore (A - B) cos ^ cos (^ + (^ + -B) sin ^ sin <^ = (^ - B) cos /a.. .(4). In (2) put for -^ its value ^{1 - c^ sin^ 0) - ^(1 - c" siir 0), and for ^ its value V(l - c' sin' 6) + V(l - c' sin'' ), and then suppose ^ = ; thus V(l — c" sin' /x) — 1 = ^ sin //,, and V(l - c' sin' fi) + 1 = B sin fx. Substitute for ^ - ^ and ^ + J5 in (4) ; thus cos ^ cos ^ - sin ^ sin ^ \f(l - c' sin' jjl) = cos fi. 204 ELLIPTIC INTEGRALS. 226. The relation just found may be put in a different form. Clear the equation of radicals ; thus (cos 6 cos (j) — cos /a)" = (1 — c" sin" /x) sin" 6 sin'^ + cos^ ^ — 2 cos d cos (f) cos /i = 1 — c^ sin^ fjb sin^ 6 sin^ . Add cos* (f) COS" yu. to both sides and transpose ; thus (cos 6 — cos (f) cos nY = 1 - cos" ^ — cos" fj, + cos* ^ cos* /x — c* sin* yu. sin* ^ sin* ^ = sin* (f) sin* /z (1 — c* sin* 6) ; therefore cos ^ = cos cf) cos /i + sin (j) sin /x. a/(1 — c* sin* ^). The positive sign of the radical is taken, because when ^ = 0, we must have ^ = /i. 227. We shall now shew how an elliptic function of the first order may be connected with another having a different modulus. Let F[c,6) denote the function; assume c + cos 2(^ therefore 1 de 2 (1 + c cos 2(^) cos' dd(f3 {c + cos'2cf)y ' therefore dd _ 2 (1 + c cos 2(f)) dcf) 1 + 2ccos2) V(l-|-2ccos2(^+c'-') , J J{l-c^sm^0) il + 2ccos20 + c«* l + ccos2 c' c (1 + c)' and since c is less than unity, 4 is gi-eater than c (1 + c)^ IT If (f)= - , then 6 = 7r ; thus jf^l'(.„|)=f(c,.) = 2i.(o,|). 228. We will give one more proposition in this subject, by establishing a relation among Elliptic Functions of the second order, analogous to that proved in Art. 225 for func- tions of the first order. If cos Q cos — sin ^ sin ^^/(l -~ c^ sin*^/*) = cos /x, then will E (C; 6) +E (,: and the surface of each intercepted by the 3 sm/3 other IS sin/3' MISCELLANEOUS EXAMPLES. 209 8. The centre of a variable circle moves along the arc of a fixed circle ; its })lane is normal to the fixed circle, and its radius equal to the distance of its centre from a fixed diameter : find the volume generated ; and if the solid so formed revolve round the fixed diameter, shew that the volume swept through is to the volume of the solid as 5 is to 2. 9. The centre of a regular hexagon moves along a diameter of a given circle of radius a, the plane of the hexagon being perpendicular to this diameter, and its magni- tude varying in such a manner that one of its diago- nals always coincides with a chord of the circle : shew that the volume of the solid generated is 2V3a\ Shew also that the surface of the solid is a^ (27r + 3 v/3). 10. Prove that f ^ dx ^ / ^\ where c = ^. 11. Shew that the length of an arc of the lemniscate T^ = a^ cos 26 measured from the vertex can be ex- pressed as an elliptic integral of the first kind. 12. P and Q are any two points on a lemniscate of which A is the vertex, and is the pole. Find the relation between the vectorial angles of P and Q in order that the arcs ^Pand QO may be equal. nesult. Cos xi OP cos AOQ = ^,. V2 T. L C. 14 210 CHAPTER XI. CHANGE OF THE VARIABLES IN A MULTIPLE INTEGRAL. 229. We have seen in Art. G2 that the double integral rb rp ^ rp rb I \ (}> {x, y) dx dtf is equal to I I <^ {x, y) dy dx when the J a J a. J a J a limits are constant, that is, a change in the order of integra- tion produces no change in the hmits for the two integrations. But when the limits of the first integration are functions of the other variable, this statement no longer holds, as we have seen in several examples in the seventh and eighth Chapters. We' give here a few additional examples. 230. Change the order of integration in ••a r-^/(a^-x!>) <^{x, y)dxdy. The limits of the integration with respect to y here are ,y = and y = ^f{a^-x''); that is, we may consider the 'integral extending from the axis of x to the boundary of a CHANGE OF THE VARIABLES. 211 circle, having its centre at the origin, and radius equal to a. Then the integration with resjDect to cc extends from the axis of 1/ to the extreme point A of the quadrant. Thus if we consider z = (f) (.r, y) as the equation to a surface, the above double integral represents the volume of that solid which is contained between the surface, the plane of {x, y), and a straight line moving perpendicularly to this plane round the boundary OAPBO. It is then obvious from the figure that if the integration with respect to x is performed first, the limits Avill be ic = and x = \/(a^ — y"), and then the limits for y will he y=0 and y = a. Thus the transformed integral is a r^J(a*-y-) CO {^y y) dy dx. 231. Change the order of integration in 2 rSacosfl /7' JO ^0 (/)(r, e)rd9dr. Let OA = 2a, and describe a semicircle on OA as dia- meter. Let POX= e, then 0P= 2a cos 6. Thus the double integral may be considered as the limit of a summation of values of ^(r, d)rM^r over all the area of the semicircle. Hence when the order of integration is chancjed we must integrate for 6 from to cos"' ~ , and for r from to 2a. 14--2 212 CHAXGE OF THE VARIABLES Thus the transformed integral is r2a rcos-'gr (f>{r,e)rdrd9. JO ^0 Jo 232. Change the order of integration in "2a rZa-x (j) (x, y) dx dy. X The integration for y is taken from 3/ = t- to ?/ = 3a — ar. The equation y = j- belongs to a parabola OLD, and the equation y=2>a~x to a straight line BLC, which passes through L, the extremity of the latus rectum of the parabola. Thus the integration may be considered as extending over the area OLBSO. Now let the order of integi'ation be changed ; we shall have to consider separately the spaces OLS and BLS. For the sj^ace OLS we must integrate from x = to x=2 \/{ay), and then from y = to y = a\ and for the space BLii we must integrate from a; = to x = 3a — y, and then from y = a to y = 3a. Thus the trans- formed integral is ra r2^(,ay) rSa rZa-ij cj) {x, y) dy dx + \ {x, y) dy dx, J J J a J IN A MULTIPLE INTEGRAL. 213 233. Change the order of integration in /:/ 1 /•x(2-ar) (f> (x, y) dx dy. Here the integration with respect to y is taken from y = x to y = x {^— x). The equation y = x represents a straight line, and the equation y = x(^ —x) represents a parabola. The reader will find on exainin>ng a figure, that the trans- formed integral is •1 ry /:/ (^ {x, y) dy dx. QJ i-Vd-y) 234. Change the order of integration in a rx+2a <^{x,y)dxdy. Here the integration with respect to y is taken from y = /^{a^ — x") to y = x-i-2a. The equation y = sj{a? — x^) represents a circle, and the equation 2/ = a? + 2a represents a straight line. The reader will find on examining a figure, that when the integration with respect to x is performed first, the integral must be separated into three portions; the transformed integral is ra ro r2a ra (f){x,y)dydx+ j>{x,y)dydx J J -Jia^-y-) J a J rZa ra {x,y)dydx. J 2a J y-2a 235. Change the order of integration in + _b 'a rb+x ra rb+x (f>(x,y)dxdy. J J Here the integration with respect to y is taken from y = to y = r • The equation y = j represents an hyper- bola ; let BDE be this hyperbola, and let OA = a. Then the integration may be considered as extending over the 2U CHANGE OF THE VARIABLES space OBDA. Let the order of the integration be changed; we shall then have to consider separately the spaces OADG and CDB. For the space OADG we must integrate from h a; = to x = a, and then from -?/ = to y = j—_ — . For the ' '^ ^ 6 + a space CDB we must integrate from a; = to x — -., and then from ?/ = t to y = 1. Thus the transformed in- tegral is fjO-y) rb+a fa f^ f ^ I ^{x,y)dydx+ 4> (x, y) dy dx. JQ Jo J b Jo b+a 236. Chancce the order of integration in where h = y rh cc-ixx <^{x,y)dxdy, J •/ Ax . The transformed integral is c-y rxh TA re r iJ. 4>{x,y)dydx+ <}> (x,y) dy dx. J J J Kh J IN A MULTIPLE INTEGRAL. 215 237. Change the order of integration in I I j>{x,y,z)dxdydz. J J J The integration here may be considered to be extended throughout a pyramid, the bounding planes of which are given by the equations z = 0, z = 1/, y = x, x = a. The integral may be transformed in different ways, and thus we obtain or or or or fa ra ry (^{x,y, z) dy dx dz, Jo J y J ra ry ra {x, y, z) dy dz dx, J J J y ra ra ra <^{x,y,z)dzdydx, Jo J s J y fa rx fx \ <^{x,y, z) dx dz dy, J J J z fa fa fx I j (f) {x,y, z) dz dx dy. J J S J 2 These transformations may be verified by putting for (j) (x, y, z) some simple function, so that the integi'als can be actually obtained ; for example, if we replace

^) ^^> where V^ is what V becomes when we put for y its value in V. Hence the original double integral becomes 1 1 ^i^' C-^' ^) ^^ ^^• Thus we have removed y and taken v instead. As the limiting values of y between which we had originally to IN A MULTIPLE INTEGRAL. 217 integrate are kno^v^l, we shall from (2) know the limiting values of v, between which we ought to integrate. It will be observed, that in finding -— from (2), we supposed x constant ; this we do because, as already remarked, when we integrate the proposed expression with respect to y wo must consider x constant. The next step is to change the ordei^ of the above integra- tions with respect to x and v, that is, to perform the integra- tion with respect to x first. This is a subject which we have already examined ; all we have to do is to determine the new limits properly. Thus, supposing this point settled, we have changed the original expression into I / ^li^' {^> ^) ^^ ^^• It remains to remove x from this expression and replace it by w. We proceed precisely as before. From equations (1) eliminate y, and obtain a; as a function of v and u, say a- = %(v, u) (3), from which we get dx=')l [v, ti) du, where ^ (v, u) means the differential coefficient of ;!^ (y, m) with respect to u. Substitute then for x and dx, and the double integral be- comes \\ V'yjr' (x, v) X {v, w) dv du, where F' is what F, becomes when we put for x its value in Fj. Thus the double integral now contains only u and v, since for the x which occurs in yjr' (x, v) we suppose its value substituted, namely, ^ (y> '^)- Moreover since the limits between which the integration with respect to x was to be taken have been already settled, we know the limits between which the integration with respect to u must be taken. 218 CHANGE OF THE VAEIABLES We have thus given the complete theoretical solution of the problem ; it only remains to add a j^^ctd^cci^ method for determining ^^r' (x, v) and ^ (^'; ^0 • ^^ this we proceed. We observe that -v|r' {x, v) or ■— is to be found from equa- tions (1) by eliminating u, considering x constant ; the fol- lowing is exactly equivalent : from (1) we have dy dv du dv dv ' dij dv da dv dv d^dy^d^^ d^dy^d^ _,.. . , du ,, dy dv dv dy dv dv Llimmate -y- : thus , , = -^ n > dv d^ tt9j du du therefore d<^^ cZ(^2 d<^i d'^2 dy _ dv du du dv dv d(f)^ d(f)^ c/(/)j d(f>,^ ' du dy dy du This then is an equivalent for ^jr' (x, v), supposing that after the differentiations are performed we put for y and u their values in terms of x and v from (1). Again, x {^y ^0 or -r- is to be found from equations (1) by cttt eliminating y, regarding v as constant ; the following is exactly equivalent : from (1) we have d^ dx d^ dy_ d^^ _ d(f>„ dx d^ dy d^ _ „ dx du dy du du ' dx du dy du du . . • du From these equations by eliminating -.- we find d4id(f>^_d^d^ dx _ du dy dy du du d(f)^ d(f>^ d(f)^ d(f),^ ' dy dx dx dy IN A MULTIPLE INTEGRAL. 219 This then is an equivalent for ^ {y, xi). , . - , , . dv dii. da dv Thus y {x, V) X [v, u) = . , , , , , .. . dy dx dx dy Hence the conckision is that //K.W,.//K ^_gJg^ ...,. (*), dy dx dx dy where after the differentiations have been performed, we must substitute for x and y their values in terms of w and v to be found from (1); also the values of x and y mvist be substituted in V. An important particular case is that in which x and y are given explicitli/ as functions of u and v; the equations (1) then take the form oc-f,{u,v) = 0, y-fAu,v)=0 (5). Here ^^ = 1 ^^ = ^^ = ^^=1 dx ' dy ' dx ' dy ' and the transformed integral becomes J J \dudv dvduJ where we must substitute for x and y their values from (5) in F. Thus we may write \\^'^^HKtt-r/l>^'^ («)• Again ; suppose that u and v are given explicitly as func- tions of X and y ; the equations (1) then take the form u-F,{x,y) = Q>, v-F^{x,y) = (7). . 220 CHANGE OF THE VARIABLES Hence we obtain Vdv du j!vdxd>/ = \dF^ dh\ dF^ dF^ ' jjvdxdy dx dy dx dy wliere we must substitute for x and y their values to be obtained from (7). Thus we may write V dv du \du dv du dv ^ '' ' dx dy dy dx The formula in (4), (6), and (8) are those which are usually given ; they contain a simjile solution of the proposed problem in tliose cases where the limits of the new integra- tions are obvious. But in some examples the difficulty of determining the limits of the new integrations would be very great, and to ensure a correct result it would be necessary instead of using these formulse, to carry on the process pre- cisely in the manner indicated in the theory, by removing one of the old variables at a time. 240. The following is an example. Kequired to transform Vdx dy, having given Jo ^0 y + X = u, y = 2iv. From the given equations we have x = ic (1 —v), y = uv; - dx _ dx dy dy thus '^ = l—v, -j- = — u, -T-=v, -r- = u; da dv du dv - , dx dy dx dy ._ , therefore -^ -f- —-j- ~ = u (I —v) + uv = u. du dv dv au Hence by equation (6) of Art. 239, we have I 1 Vdx dy= jl Vu dv du ; but we have not determined the limits of the integrations with respect to u and v, so that the result is of little value. We IN A MULTIPLE INTEGRAL. 221 will now solve this example by following the steps indicated in the theory given above. From the given equations connecting the old and new variables we eliminate u ; thus we have y = ~ — ; therefore - — X 1-v' dv {l-vf to the limits y = Q and y = h, correspond respectively v = and V = 1 ; thus b + x' h ra fb fa rb+x Vdxdy=\ J\x{l -vpdxdv. J J J J We have now to change the order of integration in nb+x i\x{i -vrdxdv. This question has been solved in Art. 235 ; hence we obtain b fa fb fa rb+x b b{\-v) fb+afa fl f V = j^ j^V^x{l-vrdvd.x+j j V^x{l-v)-'dvdx. b+a We have now to change x for ii where 6 a b fb + a fl-v f\ fv thus we obtain 1 V'udvdu+ 1 V'udvdu, Jo Jo J b Jo a + b since to the limits and a for x correspond respectively and Y^^ for u, and to the limits and — ^^ ^ for x correspond respectively and - for w. 222 CHANGE OF THE VARIABLES 1{ a = h the transformed integral becomes 1 a a_ V'udvdu+ V'udvdu. D Jl Jo a If a is made infinite, these two terms combine into the single expression nV'u do du. 241. Second Example. Required to transform nc-x Vdxdy, having given y -\-x = u, y = uv. Perform the whole operation as before ; so that we put vx , dy X y = ~ and -~ = 1 — v dv {1 —v)^' When y = we have v = 0, and when y = c — x we have V = . Thus the integral is transformed into c r [ " V^x{l-v)-^d^dv. Jo Jo Now change the order of integration ; thus we obtain /•I rc{i-v) J\x{l-v)-'dvdx. Jo J dx Now put x = u(l — v) and -7- =1 — v; the limits of u will be and c. Hence we have finally for the transformed integral n J J ( 1 re V'u dv du. QJO IN A MULTIPLE INTEGRAL. 223 2-t2. Third Example. Transform \\V dxdy to a double integral with the variables r and 6, siq^posing X = r cos 6, y = r sin 6. "We may put 6 for v and r for u in the general formulae ; thus dx dij dx dif J ^ , . 2 - du av dv du and the transformed intec^ral is V'rdedr. This is a transformation with which the student is pro- bably already familiar ; the limits must of course be so taken that every element which enters into the original integral shall also occur in the transformed integral. A particular case of this example may be noticed. Sup- pose the integral to be 1 1 <}) (cix + hy) dx dy ; by the present transformation this becomes ^ [hr cos {6 — a)] rdO dr, ■11^ where Jc cos a = a and k sin a = h. Now put — a = 6', so that the integral becomes I (f> {kr cos 6') rdd'dr ; then suppose r cos d' = x and r sin 6' = y and the integral may be again changed to 1 1 ^ (kx) dx dy. 224 CHANGE OF THE VARIABLES Thus suppressing the accents we may write 1 1 (j) (ax +hy) dxdi/= jj(f> [hx) dx dy, where k = '^{a^+h^). The limits will generally be different in the two integrals ; those on the right-hand side must be determined by special examination, corresponding to given limits on the left-hand side. 243. Fourth Example. Transform I 1 Vdx dy, having JO io given x=au-\-hv, y = hu + av, a being greater than h. Eliminate u, thus ay —lx= [a? — ¥) v, and the first trans- formation gives X «2 _ 52 re r^^ I V.dxdv, a JoJ__^_ ^ hx a' — ¥ where V, is what V becomes when we put [ v for ^ ^ a a y. Next change the order of integration ; this gives c V^dvdx + - '-\ \ V^dvdx. ^ Jo J {a + b)v ^ J _ be J jC'-h"- a-^-b-i b '" We have now to change from x to u by means of the equation x = au -\-hv, which gives -^ = a ; the limits of ^^, corresponding to the known limits of x are easily ascer- tained. Thus we have finally for the transformed integral e c-bv c—bv ra+b r~~a fO f a~ (tt^-60 V'dvdu-\-{d'-V)\ V'dvdu. J J V J be J av a- — b'^ b The correctness of the transformation may be verified by supposing V to be some simple function of x and y ; for \ IN A MULTIPLE INTEGRAL. 225 example, if V be unity, the value of the original or of the transformed integral is - . 244. Fifth Example. The area of a surface is given by the integral //. 1 Kjiii » / v> \j Kj ^ r' and the denominator is dx dy ded

'^ cos 10 dx f ■ dr\ r sin Q ( r cos 6 + sin 6 -,^ Similar dz ■ly r cos d) ^-r dxp + r sin Q cos , ^ sm (/> ^ - r sm e sin dy A* on v» fi \ fv* n, net fl _l_ cin fl IN A MULTIPLE INTEGRAL. 2 1^7 therefore /^Y fdz\^ \d4)) V \dx) \du/ 1 ■ 'ia( /J . • a^^^'\ ^ ^^ 7-^sm^ (rcos^ + sm^-^j and finally the transformed integral is • \ '•* ' 245. There will be no difficulty now in the transformation of a triple integral. Suppose that F is a function of x, y, z, and that \\\V dxdy dz is to be transformed into a triple integral with respect to three new variables u, v, w, which are connected with .-r, ;/, z by three equations. From the investi- gation of Art. 239, we may anticipate that the result will take its simplest form when the old variables are given ex- plicitly in terms of the new. Suppose then ^ =/i ("» ^'' ^^')y y =/2 ("^ ■^^ '^^)' ^ =fz i^'' '^'' '^^) (!)• AYe first transform the integral with respect to z into an integral with respect to ?«. During the integration for z we regard x and y as constants; theoretically then we should from (1) express ^ as a function of x, y, and w, by eliminating XI and V ; we should then find the differential coefficient of z with respect to w regarding x and y as constants. But we may obtain the required result by differentiating equations (1) as they stand; ji df. du df. dv df, ail aw dv aw aw df^ fZw .df^dv^df^^ die dw dv dw dw ' df^ du dfg dv df^_ dz du div ' dv dw dw diu' 15—2 228 CHANGE OF THE VARIABLES Eliminate -y- and -^— ; thus we find dw aw dz N dit dv du dv where N = ^-^^ (if,iL Jf^if^ j.'^L (¥Af, _if^¥^^ dw \du dv du dv ' dw \du dv du dv df,fdf,df,_df,df, dw \du dv die dv Hence the integral is transformed into lll'^^wwr^m'"'''''' du dv du dv where V^ indicates what V becomes when for z its value in terms of x, y and lo is substituted. We must also determine the limits of w from the known limits of 2. Next we may- change the order of integration for y and w, and then pro- ceed as before to remove y and introduce v. Then again we should change the order of integration for tv and x and then for V and x, and finally remove x and introduce w. And in ex- amples it might be advisable to go through the process step by step, in order to obtain the limits of the transformed integral. We may however more simply ascertain the final formula thus. Transform the integral with respect to 2 into an inte- gral with respect to lo as alaove; then twice change the order of integration, so that we have du dv du dv Now we have to transform the double integral with respect to X and y into a double integral with respect to u and v by means of the first two of equations (1). Hence we know by Art. 239 that the symbol dxdy will be replaced by (' du dv du dvj IN A MULTIPLE INTEGRAL. 229 and the integral is finally transformed into V'N divdvdi', where V is what V becomes when for x, i/, and z, their values in terms of n, v, and w are substituted. The student will now have no difficulty in investigating the more complex case, in which the old and new variables are connected by equations of the form <^i (^, y, ^, "■, V, w) - (}).^{x, y,z, u, V, iv) = \ (2). 3 ('^. y> ^> "> V, w) = Here it will be found that dz_^N^ dl_N^ dx ^ N^ dw ~J),' dv~l)^' da ~ I)^ ' also that N^ = D^, and N^ = D,^. Thus 1 1 1 Vdx dy dz = 1 1 1 V -j^ du dv dw, where ^ ^ dj>^ ^d^ d^ _ d^^ d^A ^ f% /d^ d^ _ dc^^ d^\ * dw \du dv du dv J dw \du dv du dv J + ^^3 /^ #2 _ % #^^ dw \d2i dv du dv J ' and — i)j is equal to a similar expression with x, y, z instead of u, V, w respectively. It may happen that equations (2) will impose some restric- tion as to the way in which the transformations arc to be effected. For example suppose we have x + y-\-z — u = 0, x + y — uv=0, y — uviu = 0. From these equations we cannot express z in terms of w and X and y, and therefore we cannot begin by transforming from z to IV. We may however begin by transforming from ;; to ti or from ^ to v ; or we may begin by transforming from x or y to M or V or IV. 280 CHANGE OF THE VARIABLES 246. It may be instructive to illustrate these transforma- tions geometrically. We begin with the double integral. -bB' Let 1 1 Vdx d)/ be a double integral, which is to be taken for all the vahies of x and ?/ comprised witliin the boundary A BCD. Suppose the variables x and y connected with two new variables u and v by the equations 3/=/.(«,^) (!)• u and V be found in terms of X ■■ft ("> ^). From these equations let sc and y, so that we may write u = F^ (x, y), v=F,{x,y) (2). Now by ascribing any constant value to u the first equa- tion of (2) may be considered as representing a curve, and by giving in succession different constant values to w, we have a series of such curves. Let then APQGho. a curve, at every point of which F^ {x, y) has a certain constant value u; and let A' SRC be a curve, at every point of which F^ (.r, y) has a certain constant value u + hu. Similarly let BPSD be a curve, at every point of which F.^ {x, y) has a certain constant value V \ and let B'QRD' be a curve, at every point of which IN A MULTIPLE INTEGRAL. 231 F„ (.r, y) has a certain constant value v + Sv. Let x, y nov/ denote tlic co-ordinates of P ; we shall proceed to express the co-ordinates of Q, 8, and li. The co-ordinates of Q are found from those of P, by chang- ing V into v + hv; hence by (1) they arc ultimately, when Zv is indefinitely small, x -t- -,- Si' and y + -r ^v. •' dv ^ dv Similarly the co-ordinates of 8 are found from those of P by changing u into u + hw, hence by (1) when hic is indefinitely dx dii small they are ultimately a; -f -y- hit, and y + -f- ^u- The co-ordinates of P are found from those of P by changing both u into u + Bu and v into v + Bv; hence by (1) they are ultimately x + -j- Bu -{■ -j- Bv, and y + -f^ Bti + ctu civ au av These results shew that P, Q, P, 8 are ultimately situated at the angular points of a parallelogram. The area of this parallelogram may be taken without error in the limit for the area of the curviliuear figure PQB8. The expression for the area of the triangle PQB in terms of the co-ordinates of its angular points is known (see Plane Co-ordinate Geometry, Art. 11), and the area of the parallelogram is double that of the triangle. Hence we have ultimately for the area of PQRS the expression fdx dy dx dy\ ^ ^ ~ \du dv dv duj Thus it is obvious that the integral 1 1 Vdx dy may be replaced by ± JjF'g | - g |) du dv ; the ambiguity of sign would disappear in an example in which the limits of integration were known. In finding the value of the transformed integral, we may suppose that we first integrate with respect to v, so that u is kept constant ; this amounts to taking all the elements such as PQRS, which 232 CHANGE OF THE VARIABLES form a strip such as A A' CO. Then the integration with respect to u amounts to taking all such strips as AA'C'C which are contained within the assigned boundary ABCD. 247. We proceed to illustrate geometrically the trans- formation of a triple integi-al. ::>£ ,->^ Let 1 1 1 Vdxdy dz be a triple integral, which is to be taken for all values of x, y, and z comprised between certain as- signed limits. Suppose the variables x, y, and z connected with three new variables u, v, w by the equations ^ =/i (w, V, to), 2/ =/, {u, v,w), z =/3 {ii, v,'vc) (1). From these equations let xi, v, and iv be found in terms of X, y, and z, so that we may write u = F^ (x, y, z), '0 = F^ [x, y, z), w = F^ (x, y, z) (2). Now by ascribing any constant value to u, the first equa- tion of (2) may be considered as representing a surface, and by giving in succession different constant values to u we IN A MULTIPLE INTEGRAL, 233 have a series of such surfaces. Suppose there to be a surface at every point of which F^ {x, ?/, z) has the constant vahie u, and let the four points P, B, D, be in that surtice ; also suppose there to be a surface at every point of which •^i.(^. y> ^) has the constant value u + Zu, and let the four points A, F, G, E be in that surface. Similarly suppose P, A, E, G to be in a surface at every point of Avhich -^2 {^' y. ^) li3,s the constant value v, and B, 1), G, F to be in a surface at every point of which F^ {ic, y, z) has the constant value v + Sy. Lastly suppose P, A, F, B to be in a surface at every point of which F^ {x, y, z) has the constant value w, and C, D, G, E to be in a surface at every point of which F^ {x, y, z) has the constant value w + hw. Let X, y, z now denote the co-ordinates of P; we shall proceed to express the co-ordinates of the other points. The co-ordinates of A are found from those of P by chano-ino- u into iL + hu; hence by (1) they are ultimately when SiT is indefinitely small, , dx ^ di/ dz ^ du •" da du The co-ordinates of B are found from those of P by chang- ing V into V -\-hv\ hence by (1) they are ultimately dv ^ do dv Similarly the co-ordinates of are ultimately dx ^ dy ^ dz ^ diu ^ dio dw The co-ordinates of D are found from those of P by chang- ing V into v + 8v, and w into w + Stu; hence by (1) they are ultimately Similarly the co-ordinates of ^, P and G may be found. These results shew that P, A, B, G, D, E, F, G are ulti- mately situated at the angular points of a parallelepiped ; and the volume of this parallelepiped may bo taken without error 234 CHANGE OF THE VARIABLES ia the limit for the volume of the solid bounded by the six surfaces which we have referred to. Now b)'^ a known theo- rem the volume of a tetrahedron can be expressed in terms of the co-ordinates of its angular points, and the volume of the parallelei^iped PQ is six times that of the tetrahedron ABPG. Hence finally we have for the volume of the paral- lelepiped + {dx fchi dz dy dz\ dy fdz dx_ _dz^ dx\ \diL \dv dw dw dvj du \dv dw diu dvj dz dx dii dx dy\\ ^ ^ ^ , at 5. ^ 5. du\dv dw dwdvj) Hence the triple integral is transformed into ±{{[v'Ndudvdw\ the ambiguity in sign would disappear in an example where the limits of intecfration were kno\vn. 248. We have now given the theory of the transforma- tion of double and triple integrals ; the essential point in our investigation is, that we have shewn how to remove the old variables and rejalace them by the new variables one at a time. We recommend the student to pay attention to this 2)oint, as we conceive that the theory of the subject is thus made clear and simple, and at the same time the limits of the transformed integral can be more easily ascertained. We do not lay any stress on the geometrical illustrations in the two preceding Articles ; they require much more development before they can be accepted as rigid demonstrations. 249. Before leaving the subject we will briefly indi- cate the method formerly used in solving the problem. This method w^e have not brought prominently forward, partly because it gives no assistance in determining the new limits, and partly on account of its obscurity ; the latter defect has been frequently noticed by writers on the subject. Suppose 1 1 Vdxdy is to be transformed into an integral with respect to two new variables u and v of which the old variables are known functions. IX A MULTIPLE INTEGRAL. 235 Let the variables undergo infinitesimal changes : tlius dx = -, du + -j- dv (1), du dv du = -/du + -rdv (2). '^ du do ^ ^ Now in the original expression Vdx dy in forming dx we suppose y constant, that is, dy = ; hence (2) becomes = ^du + '$dv (3), du dv find dv from this and substitute it in (1) ; therefore dx dy dx dy , du dv dv du ^ Tbj ^''••- ('^)' dv Again, in forming dy in Vdxdy we suppose x constant, that is, dx = 0; hence by (4) we must suppose du = 0; there- fore from (2) ^y=£^^ (^)- From (4) and (5) , , [dx dy dx dy\ , , dx dy = — - -/ — - / du dv : ■^ \du du dv du) ' and 1 1 Ydx dy becomes [[r[^^^-^^'f\dudv. J J \au dv dv an) With respect to the limits of integration we can only- give the general direction, that the new limits must be so taken as to include every element which was included by the old limits. 23G CHANGE OF THE VARIABLES. 250. Similarly in transforming a triple integral Vdx dy dz ///' the process was as follows. Let tlie new variables be ?i, v, w ; in forming dz we must suppose x and y constant ; thus we have 7 dz , dz , dz ^ dz = -y- du + ^r dv + -j— dw, du dv dw ^ dx , dx , dx J = -7- ait + -7- rfy + -,— aw, du dv dw = -/ du + -^ dv+ -~ dw, du dv dw therefore dz= , , , , (l), dxcly dxdy ^ ^ du dv dv du where N has the same value as in Art. 247. Next in forming dy we have to regard x and z as constant; hence by (1) we must regard w as constant ; thus we have = f, du du + '^ dv; dv ' fdy dx \dv du dy dx\ du dv) dx du therefore dy — 1 (2) And lastly in forming dx we suppose y and z constant, that is, by (1) and (2) we suppose w and v constant; therefore 7 dx , , , dx = -T- du (3). From (1), (2), and (3) dx dy dz = Ndu dv dw. EXAMPLES. 237 251. The student who wishes to investigate the history of the subject of the present Cliapter may be assisted by the following references. Lacroix, Calcid Dif. et Integral, Vol. ii. p. 208 ; also the references to the older authorities will be found in page XI. of the table prefixetl to this volume. De Morgan, J) If. and Integral Calculus ^ p. 392. Moigno, Calcul lyif. et Integral, Vol. Ii. p. 214; Ostrogradsky, Memoires de V Academie de St Petershourg, Sixieme S(^rie, 1838, p. 401. Catalan, Memoii'es Couronnes par I' Academie... de Bruxelles, Vol. XIV. p. 1. A memoir by Haedenkamp in Crelle's Journal, Vol. XXII. 1841. Boole, Cambridge Mathematical Journal, Vol. IV. p. 20. Cauchy, Exercices d' Analyse et de Physique Mathematique, Vol. IV. p. 128. Svauberg, Nova Acta Regice Societatis Scientiarum, Upsaliensis, Vol. xiii. 1847, p. 1. De Morgan, Transactions of the Camhridg.e Phil. Society, Vol. ix. p. [133]. Winckler, Denhschriften der Kaiserlichen Akad. Math....Classe, Vol. xx. Vienna 1862, p. 97. A memoir by Holmgren was communicated to the Stockholm Academy in 1864, and published in Vol. V. of the Transactions. EXAMPLES. 1. Shew that if ic = asin^sin^ and y~h cos 6 sin (}), the double integral Ijdxdy is transformed into + 1 1 a5 sin ^ cos ^ d^ dO. If X = u sin a. -\- V cos a and y — u cos a — v sin a, prove that //■^(^' s-) jiM^^:f) =///■ ("' ") dudv ^^{l-io'-v')' 3. In the problem of Art. 239, supposing the limits of x and y are both constants, shew how the limits of u and V are to be found, in each of the three parts of which the transformed integral will in general be composed, 238 EXAMPLES. 4. Prove that (•OO -00 »x> I ^ (aV + 6^7/') c?a; f/y = -"^y \ 6 (x) dx. J J 'iUDJ 5, Transform ilVdx di/, where y = xu and x= — ~ . If the limits of y be and x and the h'mits of x be and a, find the limits in the transformed integral. ri ra(\+u) Besult V'v[l+u)-\Jadv. J oJ 6. Transform lje~^^'''^-'^^'^''^'''^^J'^dxdi/ from rectangular to polar co-ordinates, and thence shew that if the limits both of x and y be zero and infinity, the value of the integral will be *= 2 sin a ■ 7. Transform 1 i (f) {x, y) dx dy to polar co-ordinates, and J J ^ indicate the limits for each order in the transformed integi'al. Shew that p p dx dy 1 ^^^_, ab Jo Jo (c' + x' + y'f c c V(a' 4- i"" + c') ' 8. Apply the transformation from rectangular to polar co- ordinates in double inteofrals to shew that + 00 r +00 iZI a dx dy 27r -«> ^a;' + y- + a'f {x' -\- y" + a'^ « + « 1). Transform the double integral \jf(x, y) dxdy into one EXiVJtfPLES. 239 in -svliich r and 6 shall be the independent variables, having given x = r cos ^ + a sin ^, y = r sin 6 + a cos 6. Result. Uf{r cosd + a sin 6, r sin ^ + a cos 6) (a sin 29 - r) dO dr. 10. Transform U e' ''''-'■'' dxdy into a double integral where r and t are the independent variables, where - = t and r^ = x"^ + if ; and if the limits of x and y be each and CO , find the limits of r and t Result. I -, — ^2 — . Jo h !+« 11. If X and y are given as functions of r and 9, transform the integral \\\dxdydz into another where r, 9 and z are the variables ; and if a? = r cos 9 and y = r sin 9, find the volume included by the four surfaces whose equations are r = a, s = 0, ^ = 0, and z = mr cos 9. Result. The volume = | \ r^m cos 9d9dr = -^ . J 12. If ax = yz, fiy = zx, ^z = xy, shew that jjjf{ru,^,ry)doid^dy = 4>jjjf(^^, J, ^)dxdydz. 13. Transform jjjjvdx^dx.^dx^dx^ to r, ^, (^ and -v/r where iTj = r sin 9 cos ^, ^3 = '^ cos 9 cos A/r, cc^ = r sin ^ sin ^, x^ = r cos ^ sin ilr. i?eswZ^ jjjjy'r' sin ^ cos ^ cfr cZ^ # cZ^/^. 240 EXAMPLES. 14. Find the elementary area included between the curves (f> {x, y)=u, yjr (x, y) = v, and the curves obtained by giving to the parameters u and v indefinitely small increments. Find the area included between a parabola and the tangents at the extremities of the iatus rectum by dividing the area by a series of parabolas which touch these tangents and by a series of straight lines drawn from the intersection of the tangents. 15. Transform the triple integral I i j f{.v, y, z) dx dy dz into one in which r, y, z are the independent variables, having given -^ {x, y, z, r) = ; and change the vari- ables in the above integral from x, y, z to r, 6, (f), having given f (^, y, ^, r) = 0, -f 1 (?/, ^, r, 6) = 0, f^ (z, r, 0, <^) = 0. df ^, djr^ dx dy dz 16. Transform the double integral in which x, y, z are connected by the equation a;^ + ?/^ + 2" = 1, to an integral in terms of Q and <^, having these relations, X = sin ^ /^(l — m" sin^ 0), y = cos 6 cos (f), z = sind a/CI — n' sin'^ (f)), m^ + n^ = 1. Hence prove that ff JT '^'■2 on^ cos~ 6 + n^ cos^ (f) -,„.. ir 11' •' • V(l - m' &m' 6) V(l - n' sin'^ ) ^^^'^ 2 EXAMPLES. 241 17. Transform the integral Indxdijdz to r, 6, (J), where a; = r sin ^ ^/(l — n"^ cos" 6), y = r cos (p sin 0, z = r cos 6 \/(cos^ ^ + n^ sin' <^). p , [/•/ >•' {(?^' - 1) cos'-' (/) - n'^ sin' 6] dr d6 d<^ 18. Transform the expression 1 1 - sin ^ J^ d(f) for a volume, to rectansfular co-ordinates. Besult. ^ jj(z—px — qy)dxdi/; this should be in- terpreted geometrically. 19. lfx+7/ + z = u, x-i-7/ = uv, 1/ = uvw, -pvove ihat Vdxdydz=\ Vu^vdudvdio. J J oJ CO ;. 00 /• 00 J ^ 20. If x^ = rcose^, x^ = r sin 6^ cos ^j, iTg = r sin 6^ sin ^^ cos ^j^, ^„-i = ?' sin ^j^ sin $^. . .sin ^^^ cos 0„_^, x^ = r sin ^^ sin 0^. . .sin ^^.^ sin 6'„_j', shew that \\\ Vdx^ dx^. . . dx^ = ±jlj y V'-'Edr d9^ dd^ dd^_„ where V is any function oi x^, x^,...x^, and V what this function becomes when the variables are changed, and // stands for (sin ^J"-» (sin ^,)"-' sin^„_,. T. I. c. 16 242 CHAPTER XII. DEFINITE INTEGRALS, 252. When the indefinite integral of a function is known, we can immediately obtain the value of the definite integral corresponding to any assigned limits of the variable. Some- times however we are able by special methods to assign the value of a definite integral when we cannot express the indefinite integral in a finite form; sometimes without actually findino- the value of a definite integral we can shew that it possesses important properties. In some cases in which the indefinite integral of a function can be found, the definite integral between certain limits may have a value which is worthy of notice, on account of the simple form in which it may be expressed. "We shall in the present Chapter give examples of these general statements. We may observe that a collection of the known results with respect to Definite Integrals has been published in a quarto volume at Amsterdam, by D. Bierens de Haan, under the title of Tables d'Integrales Definies. 253. Suppose f{x) and F{x) rational algebraical functions of cc, and f (x) of 'lower dimensions than F (x), and suppose the equation F{x) = to have no real roots ; it is required to find the value of It will be seen that under the above suppositions, the expression to be integrated never becomes infinite for real values of x. Let a + /3 v'(- 1) and oc - ^ V(- I) represent a pair of the imaginary roots o^F{x) = ; then the corresponding quadratic DEFINITE INTEGRALS. 243 fix) fraction of the series into which "v, can be decomposed, may be represented by the constants A and B being found from the equation A-B ^/{- 1) - ^r^^q:^,3-j^j (Art. -1). Now f '^^./"... = 2i?tan--^-" therefore I ,— ^^' — r^i — '—^, = 'UBir, and hence it mioht be said in a certain sense that if the integ-ral be taken between the limits — cc and + co the result will be zero. This however is not satisfactory, for the positive part of the integral and the negative part are both aumerically infinite, so that it is not safe to assume that they balance. But \if{x) is at least two dimensions lower than F(.r), we shall find that the sum of the terms of the type which we are considering is finite for each part of the integral, and then the positive part may be safely taken to balance the negative part. For suppose we require the integral between the limits and h. Let A^, A^,...A^^ denote the constants of which we have taken A as the type ; and let 1 similar notation hold with respect to a and ^. Then we bave for the integral the expression ...+ J„log 16—2 244 DEFINITE INTEGRALS. This may be put in the form 2[A^ + A^+...+A,]logh M\A^ A «.V,/^./ , 4 ^ \ h) ]i* ■ , \^ hj ' /r , ••• + Alog ^ . ... Now since /(a;) is at least two dimensions lower than F{x) we have A^ + A„^ ... + ^^ = 0. Thus the above expression reduces to the second part, which is iinite when h is infinite. Hence when the limits are — cc and + co the sum of the terms we are considering: vanishes. If then we suppose F{x) to be of 2;i dimensions, and Jj^,B^, B^ to be the n constants of which we have taken B as the type, we have when / {x) is at least two dimensions lower than F(x) f_^-l^^-^dx = 2^{B, + B, + +B„]. 254. As an example of the preceding Article we take a?-"' dx i 1+x"" where m and n are positive integers, and m less than 7i. Here 1 A-B^{-1) = ^^^ ^^^^^^_ -^^|.„-.in^, and it is known that the values of a 4-/3 \/(— 1) are obtained from the expression (2r + l)7r^ ., ^- . C2r + l)7r cos • ^-^- + V(- 1) sm ^ , by giving to r successively the values 0, 1, 2, up to w — 1 : see Plane Trigonometry, Chapter xxiii. Thus, by De Moivre's theorem, DEFINITE INTEGRALS. 24'.') (a + /? V(- l)!^"-^-"-^ = cos (/. + V(- 1) sin 4>, where so that cos cf) + v'(- 1) sin (^ = - cos (2r + J)d + V(- 1) sin (2r + 1) ^, where = — -; tt. 2n Hence 2n - cos (2r + 1) ^ + V(- 1) sin (2r + 1) ^ cos(2r + l)^ + \/(-l)sin(2r+l)(9 2?i ' therefore i? = ^^— -— . zn Hence ^-^'^^ = '^Jsin^ + sin3^ + sin5^+... + sin(2/i-l)^l . j _ oc 1 + iC Jl ( ' ) The sum of the series of sines may be shewn to be sin'^ nO — — — ; see Plane Trigonometry, Chapter xxil. ; and in the present case nO = — - — tt, so that sin^ n6 = 1. Therefore -^l + x^" . 2m +1 n Sm t: TT 2n It is obvious that that is, •" x^"" dx ^ ^ is half of the above result 1+a;' r x'"" dx _ Jo l + x'"' TT „ . 2m +1 • zn sin — ^ ■ TT 246 DEFINITE INTEGRALS. 255. In the last formula of the preceding Article put 2m + 1 a;^" = y, and suppose —^ = h ; thus we obtain /. 1 + 2/ ^i^ ^'^'^ .(1). This result holds when k has any value comprised between and 1. For the only restriction on the positive integers m and n is that m must be less than n, and therefore by pro- 2/?i + 1 perly choosing m and n we may make — ^ — — equal to any assigned proper fraction which has an even denominator when 2m + 1 in its lowest terms. And althousjh we cannot make ^^- — — * 2'7J exactly equal to any fraction which has an odd denominator when in its lowest terms, yet we can make it differ from such a fraction by as small a quantity as we please, and thus deduce the required result. In the last result put x" for y, where r is any positive quantity ; thus ' rx'^" X"^' dx IT , . rx'^-'dx TT that IS, 1 + x'' sin /cTT ' ' J 1 + ^''" ^' sm Ictt Let Jcr = s ; thus I J I X ax 77 ___ ___ r sm - TT r The only restriction on the positive quantities r and s is that s must be less than r. The student will probably find no serious difficulty in the method we have indicated for proving the truth of equation (1) when A; is a fraction which has an odd denominator when in its lowest terms ; nevertheless a few remarks may be made which will establish the proposition decisively, and which will also serve as useful exercises in the subject of the jDre- sent Chapter. Let «=rp'^; then«=rC^+r^^; h 1 + y .'o 1+y Ji i+y ' DEFINITE INTEGRALS. 247 and by putting - for y we find that ii l+y Joi + 2 Jo 1 + y ^'--f- t/=2u = sjir. Jo 264. We shall now give an expression for F (n) that will afford another proof of the result in Art. 262. We know that the limit of — r — when h is indefinitely diminished is log x ; hence (logl) = limit of (^-") '; so we may write where ?/ is a quantity that diminishes without limit when h does so. Put h = -; , then, by Art. 258, r (w) = r"-' [ (1 - x'-y-' dx+{ y dx. J •'0 In the first integi'al put x = z' ) thus r (n) -fydx = 7-™ [' z"--' (1 - zy-' dz. 252 DEFINITE INTEGRALS. Wc have it in our power to suppose r an integer; tlicn the integral on the right-hand side, by Art, 33, is 1.2.3 r „_, ?i(?i+l) (w + r — 1) Let r increase indefinitely, then y vanishes and we have 1 2 3 ... r r (n) = limit of —. -^-^ — '^^ ^ , r" \ ^ ' n{n+\) {n + r-V) 2G5. From the result of the preceding Article we have A particular case of this is obtained by suj)posing n = l; thus V{l-m)T{l + m)^\} ~ rj (^ ~ F; V ~ 3V ' the expression on the right-hand side is known to be equal to ; see Plane Trigonometry, Chapter xxili. : thus rtnr r (1 -^ m) r (1 + m) = sm viir therefore T im) Til- m) = . '^ (Art. 259). 266. We shall now establish the following equation, n being an integer, 11- V^ then reversing the order of the factors we have x = r(i-?)rfi--) rfl \ nj \ nj \ii DEFINITE INTEGRALS. 253 Multiply, and use Art. 2G2 : thus —n-l . IT . 'lir . (?J — ijTT sm - siu — sm n n n n The denominator is equal to ^7^1 : see Plane Trirjonometry, Chapter XXIII. Thus the result is established. 267. A still more general formula is r(.)r(. + l)r(.+ 5) t{.+'^) n-l = r {nx) {27r) ''n^^'', ■which we shall now prove. Let {x + 2) = (j) {x + 1) = (J3 (x) ; and by proceeding thus we have (f) (x) = (f> (x + m), where on may be as great as we please. Hence {x) is equal to the limit of (J3 (/x) Avhen fj, is infinite ; thus ^ (x) must he independent of x, that is, must have the same value whatever x may be ; hence (f) (x) must have the same value as it has when x = -: thus the n theorem follows by the preceding Article. This theorem is 254 DEFINITE INTEGRALS. ascribed to Gauss ; a more rigid proof is given in Legend re's Exercices de Calcid Integral, Vol. ii. p. 23 ; see also the Journal de VEcole Poly ted inique, Vol. xvi. p. 212. 268. Take the logarithms of both sides of the formula established in the preceding Article, and differentiate with respect to x; thus we obtain rv ^ TV ^ r'fa. + -) r{x + '^—^] nV {nx) ^ r_Or) V nj \ n ) r(„.)-r(.) J,/ l^ + ^'TlI^^ n ("+^) + n\ogn (1), where r'(^) stands for , . Differentiate again; then, putting z for nx, we obtain |.iogr(.) d' log r {x + ^) d' log r (x + "^"j ' ^ 1 fZ^grjx) _^ ^^ if [ dx^ dx' I • • • 1 ^^2 If n be made infinite the right-hand side vanishes, for it becomes ultimately 1 p-+icZMogr(,r) nJr dx thati, iF'ogr(.+2)_.nogr(aj) n ( a.c ax j Hence we see that if z be infinite ? ,— ^- vanishes. ^^ x x{x+l) a-(ic+l)(a; + 2) ' take the logarithms and differentiate twice with respect to x ; thus ^'^°g^^^^ = i + -J:^- + ^ 4- ad inf (^) DEFINITE INTEGRALS. 255 The series just given is convergent for every positive value of X. Integrate between the limits 1 and x ; thus d lo "when n is tnade infinite, is zero. In (3) suppose x infinite : hence, with the aid of the result just obtained, we see that G is equal to the limit when n is infinite of -, 111 1 1 1 + 2 + 3 + 4+ +,-l°g^^- It is easy to shew by elementary considerations that this limit is finite. See Algebra, Chapter LV, Examj^le 12. The value of C to 10 places of decimals is '5772156G49 ; the calculation has been carried to 263 places of decimals: see a paper by Professor J. C. Adams in the Proceedings of the Royal Society, Vol. xxvii. page 88. 269. In equation (2) of the preceding Article change x into a; + 1 ; thus <^''logr(l+a-) _ 1 1 1 dx"" ~(aj+l)' + (^+2)'' + (a;+8)^+"'*' differentiate w — 2 times ; thus d"l0gr(l + ^-)_, _l(_l)nf 1 . 1 dx'' 1 ^ ' \{x + iy (a; + 2)" + (a; + 3)"+"-J • Let S^ denote the infinite series 1 + .^ + -57, + . . . ; then, if n be not less than 2, the value of ° , „ , when ax a: = 0, is |w-l(-l)">S^„. DEFINITE INTEGRALS. 257 Also the value of — ^^^— y^ -, when x = 0, is — C\ ax and log r (1 + .r) = when ic = 0. Hence, by Maclaurin's Theorem, hgT{l + x) = -Cx+^ 3' + "t~~-" The series is convergent as long as x is numerically less than unity. Now by the property of Art. 2G2, combined with that contained in equation (1) of Art. 259, it follows that F (x) is known for all positive values of x if it be known for all values of x between and - , or for all values between ^ and 1, or for all values between 1 and 1^, and so on. And the series just given will enable ns to determine the value of log r (x), and thence of F (x), for all values of x between 1 and H ; so that we may consider that F (x) can be calcu- lated for any positive value of x. Legendre has constructed a table of the values of log F {x) ; and an abbreviation of this table is given in De Morgan's Differential and Integral Calculus, pages 587... 590. We may also refer to an article by H. M. Jeffery on the Derivatives of the Gamma-Function in the sixth volume of the Quarterly Journal of Mathematics. 270. A higher degree of convergence may be given to the series obtained for log T {1 ■\- x) thus : ^ x^ S v^ logF(l+^)=-Ca. + '-|---^- + ..., logF(l-^)= (7^ + 3^+*-- + ...; now F (1 + a;) . F (1 - a;) = xF {x) T {I - x) = -.^^,byArt. 2G2; sin CCTT "^ T. I. c. 17 258 DEFINITE INTEGRALS. (fjir 1 1 therefore log -. = Sjc' + -r ^.x^ + ;^ S>a^ + • • • , and logr(l+a;) = ^log-^ Cx--f i--.... The result may also be written thus : 1 XTT \ 1 1+x iogr(i + ^0 = ^iog^-ii^-2ios-i3^ the series in the last line converges rapidly when x is numeri- cally less than - . 271. From equation (2) of Art. 26S we see that ^- is always positive, and is finite if x be positive : hence —j — ^ increases algebraically as x increases from to infinity, and therefore cannot vanish more than once. Thus r (x) cannot have any maximum Avithin this range of values of x, nor can it have more than one minimum. It is easy to see that F (x) has one minimum, between x = l and ic = 2 ; for F (2) = F (1). To determine the minimum of F (1 + x) we differentiate one of the series found for log F (1 +x), and equate the result to zero. This gives an equation from which it is found by trial that l+x = 1-4G16:321.... 272. Many definite integrals may be expressed in terms of the Gamma-function; we shall give some examples. The integral I e~"'^^ dx becomes by putting y for a~x^ Jo I ^ — .-^ , that IS, jr- F (i), or -:r- . 2a Vy 2a ^-^ 2a DEFINITE INTEGRALS. 259 Again, in — - — '—^ — put — — = .r-^ ; thus we 7o ix + ar"" ^ x + a 1 + a i-^ + a) obtain I f V- (1 - V)-- d>, that is 1 r (0 r jm) «"• (1 + ay Jo ^ ^' ^^ '^^' ^^'""^ ''' a'" (1 + ay r{l + m) ' Again, in I x'~^ (1 — ic')""' tZiz; put x' = y; thus we obtain ,r i-. r(|)r(„o ij »" (1 - y)- rfy, that is, — i^^ .. 2r(|+,„) IT Thus [^ sin" 6 cos' ddd=\ x^{l- x^y' dx •' .'o r r^ + ^^ r /^^ + ^ . . . r x'-'il-xr-\Ix ^ ly we obtain -1- [ V-i fi _ y\-^-^du that is ^ ^^) ^ (^^) a'6"'Jo^ ^-^ 2/} rfy, thatis,^,^„.j,^^_^^^^. 273. In 1 ic'"^ (« — a:;)"*"^ c?x put x = ay; thus we obtain a--- [ ' 3,'- (1 - yy~^ dy, that is, a^-- ^J:^^'^ ' Jo i {(> + VI) ST-t. It is required to find the value of the multii^le in • teoral {\L . .a;'-^ y""-^ «"-^ ...dxdy dz. 17—2 2G0 DEFINITE INTEGRALS. the integral being so taken as to give to the variables all positive values consistent with the condition that x + y + z-^... is not greater than unity. We will suppose that there are three variables, and conse- quently that the integral is a triple integTal ; the method adopted will be seen to be applicable for any number of variables. We must first integrate for one of the variables, suppose z; the limits then will be and 1-x-y; thus between these limits r.-i,, a-x-yY _ v{n) Next integrate with respect to one of the remaining varia- bles, suppose y ; the limits will be and \ — x\ and between these limits, by Art. 273, r , , , (1 - xT^'^ V (m) r (n + 1) Lastly integrate with respect to x between the limits and 1 ; thus between these limits J ^ ^ r (Z4-W-1-W + 1) Hence the final result is r Qn) r (m) r (n + 1) r(or(>/i + n + i) r (n + 1) r (to + ?i + 1) r (^ + m + ?i + 1) ' V{1) V (TO) r (n) "'''^'^' r(^-i- TO 4-71 + 1) 275. It is required to find the value of the multiple the integral being so taken as to give to the variables all integral DEFINITE INTEGRALS. 261 positive values consistent with the condition that is not greater than unity. Assume a. = ^ , ^=Q, ^=(^^ Then the integral becomes with the condition that x + i/ + z+ ... is not greater than unity. The value of the integral is, therefore, by the pre- ceding Article, , fff i_i «-i :?-! , , , - j...x^ y"^ z'' ...dxdydz a'yS^v".. \p) \qj \rj pqr ... T-, fl m n ^\ V - + - + - + ... +1 \p q r J This theorem is due to Lejeune Dirichlet ; we shall give Liouville's extension of it in Arts. 277 and 278. 276. As a simple case of the preceding Article we may suppose p, q, r, ... to be each unity, and a, /3, 7, ... each equal to a const-ant h; thus the condition is that ^+t] + ^+... is not to be greater than h. Therefore the value of the integral jjj...rv"'-'r'-d^dvdc... ig ;j.»........ r (r)r(m)r(n)... r{i+Qu + ti+ ... + 1) ' which we may denote by Similarly if the integral is to be taken so that the sum of the variables shall not exceed h + Ah, we obtain for the result Hence we conclude that the value of the integral extended over all such positive values of the variables as make tlie 2G2 DEFINITE INTEGRALS. sum of the variables lie between h and h + Ah is and when Ah is indefinitely diminished, this becomes N{l + m + n+..:)h'^'''^''^--'Ah, rfflr(m)r(n). tiiatis, Til + m + n + ...) 277. It is required to transform to a single integral the multiple integral the integral being so taken as to give to the variables all positive values consistent with the condition that a; + 3/ + ^ +. . . is not greater than c. We will suppose for simplicity that there are three variables. By the preceding Article if / (.x- + 3/ + 2) were replaced by unity that part of the integral which arises from supposing the sum of the variables to lie between h and h + All would be ultimately r (i -1- VI + n) And if the sum of the variables lies between h and h^- Ah the value of f{x^y-¥z) can only differ from /(A) by a small quantity of the same order as Ah. Hence, neglecting the square of Ah, that part of the integral_ which arises from supposing the sum of the variables to lie between h and h + Ah is ultimately r(?)r(m)r(» ) „.^.-,^^^^ r (/ + m + ?o -^ ^ ^ Hence the whole integral is r(/ + m + w) Jo^ ^ This process may be applied to the case of any number of variables. DEFINITE INTEGRALS. 2G3 278. Similarly the triple integral ///r.-'ry{(^)'+(|)V©]'^f<'^''? for all positive values of the variables, such that is not greater than c, is equal to Pi"- yI- + - + -] -' \p q rj This process may be applied to the case of any number of variables. 279. It is required to transform to a single integral the double integral 11 {u + ax + hyT" ' where the integral is to be taken for all positive values of X and y such that ^ + y is not greater than k ; the quantities p, q, u, a, and h being all positive constants. Suppose that a is not less than h. We have 11 + ax + hy = u + a {x + y) - {a - h) y = U - 7), where U stands for u-^a{x-\- y), and 77 for (a - h) y. Thus (w + ax + hyy^^ the series here given being convergent. The proposed double integral may now be transformed by applying the method of Art. 277 to every term. Thus the double integral 264 DEFINITE INTEGRALS. ^ [' [ r(j.)r(g ) jF^ T(p)T{q + i) ( a - h) r^ h\V{p + q) {u + atr'''^ Fip + q + l) ^^^^^ (u + at)'^'^' T{p) T(q + 2)(p + q)(p + g + l)(a-hrr^'' ) ■^ r{p + q + 2) 1.2 {u + aty^'^^''^'--r^ = r(v) (' '^"^ f ^^^^^ {p + q)Tiq + l)ia-h)t ^^' Jo {ii + city' [r {p + (/) "^ r{2) + q + 1) u + a^ , {p + q){p + q^l)V{q+^^) {a-hYf \ r(p + 2 + 2) 1.2(M + aO'' J _ r ( y) r (y) p r^-^ f, , q{a-h)t r(i> + ?) io ((i + a^/'-^^l ^ u + at q{q + l) {a-hfe \ "^ 1.2 ' {u + atj +"-|^^ r(i) + ?) Jo (^i + aO^H ^* + a^J In a similar manner we may transform to a single integral the triple integral a,P-l yi-i ^r-l ^^ ^^ ^^ (it + a-c + Z'j/ + c^) p+3+r » "where the integral is to be taken for all positive values of x, y, and z such that x -\-y + z is not greater than k ; the quan- tities p>, q, r, u, a, b, and c being all positive constants. Suppose that a is not less than b or c. We have u + ax + bi/ + cz — u + a {x + z) + by — {a — c) z. Proceeding as before we find that the proposed triple inte- gi-al can bo transformed into a series, each term being of the form represented by the product of DEFINITE INTEGRALS. 2G5 (j) + q + 7-){p + q + r+l)...{2y + q + r + p-l) and the triple iiite, q, ... is equal to 2, and that each of the quantities a, y3, , . . is equal to ^J{1 — x^). Thus the result is im]r^(i_^,) n-l On-ip/!^!^^ But if the variables may have negative as well as positive values, this result must be multiplied by 2""'. Thus we get m-l "-1 ^ • (1 - <) ' Hence, finally, since the limits of x^ will be — 1 and 1, the multiple integral is equal to n-l 2 n n-l TV r('!i^+i f /(fc.)(i-^.Vrf^.. This agrees with the result given by Professor Boole in the Cambridge Mathematical Journal, Vol. in. p. 280, as it may be found by integrating his equation (15) by parts. 281. It is required to transform to a single integral the multiple integral • " \r\ T-^^-. -^ dx. dx„... dx. DEFINITE INTEGRALS. 2C7 the integral being so taken as to give to the variables all values consistent with the condition that x^'' + x,^^ + ... + xj^ is not greater than unity. As in the preceding Article the integral may be trans- formed into ii ... —-Z AA T. dx^ dx^ ...dx^. First integrate with respect to the variables a;,, a*3,,.,a'„, the limits being given by the condition that x^^ + x^' ...+x^ is not greater than 1 —x^. If the variables are to have only positive values then the integral dx^ dx^ . . . dx^^ by Art. 278 would be equal to 1 (r f-^Vf""' r^"'^'' 1 "--1 r that is, to l,Jm^a_..>)^■!:£IG), (A,.t.273). that is, to 2^1 • — ^^ (1 - x{) " . r (1 But if the variables may have negative as well as positive values, this result must be multiplied by 2"'\ Thus we get 2G8 DEFINITE INTEGRALS. Hence finally, since the limits of x^ are —1 and 1, the multiple integral is equal to TT r .,-, V ,, o, „--i 282. Many methods have heen used for exhibiting in simple terms an approximate value of V (^n + 1) when n is very large : we give one of them. The product e~' x" vanishes when x = and when x = cc; and it may be shewn that it has only one maximum value, namely when x = n. We may therefore assume e X =e 11 e (1), where t is a variable which must lie between the limits — co and + CO . Thus [ e-' x" dx = e'" tf I e-^'^dt (2). Jo J -cc dt Take the logarithms of both members of (1) ; thus x — n loga:; = n — n\og n + f (.3) ; put x = n + u) thus n — n log (n + w) =f— n log n (4). But by Taylor's Theorem log {n + u) = log n + u u^ n 2 (n + Oaf ' where ^ is a proper fraction ; thus (4) becomes therefore V (n) u V(2} [n-i-eu) ^ '' DEFINITE INTEGRALS. 2G9 therefore u = ,. " ^^ ,,, (G). But from (3) ^ = ^^^ - = 2< + — = V(2^0 + 2(i-^)^ l)y(C). Hence (2) becomes /•CO (* GO Jo j - 00 and I e~^'' dt = \l{'Tr); thus But since 1 — ^ is positive and less than unity, the nume- rieal value of e~*' {1 — 0)tdt is less than I e'^'tdt, that j -00 Jo is, less than ^. Hence we conclude from (7) that as n is increased indefinitely, the ratio of F (?i + 1) to e''' if ^ {'Imr) approaches unity as its limit. We may observe that in the original equation (1) we have f and not t itself; hence the sign of i is in our power, and we accoi'dingly take it so that equation (5) may hold, sapf)osing \/n and ^2 both jDOsitive. (See Liouville's Journal de Mathematiques, Yol. x. p. 464, and Vol. xvii, p. 448.) Definite Integrals ohtained hy differentiating or integrating with respect to constants. 283. We shall now give some examples in which definite integrals are obtained by means of differentiation Avith respect to a constant. (See Art. 213.) 270 DEFINITE INTEGRALS. To find the value of I e"'*'-^' cos 2rxdx. Jo Call the definite integral u ; then du J = — 2 xe'"-''^' sin 2?-^ dx. dr Jo Integrate the right-hand term by parts ; thus we find . du 2ru dr~ a' ' therefore d log u 2?* dr d' ' therefore log zt = - -, + 0/ therefore u = Ae a"- , where yl is a quantity which is constant with respect to r. that is, it does not contain r. To determine A we may suppose ?' = 0; thus u becomes / e-"'-*^" (^^, that is, 7^, (Ai't. 272) Hence -4 = -77— , and e'^''''^" cos2rxdx = '^r~&~"'^' 2a Jo 2a 284. We have stated in Art. 214, that when one of the limits of integration is infinite the process of differentiation with respect to a constant may be unsafe ; in the present case however it is easy to justify it; we have to shew that Q-a"x- p^i^ vanishes where p is ultimately indefinitely small; it is obvious that this quantity is numerically less than /< 00 Pj I e~"'-^V,c where p^ is the greatest value of p, that is, J Q \/ IT less than — p^ ; but this vanishes since p^ does. Similar considerations apply to the succeeding cases. DEFINITE INTEGRALS. 271 sin rx dx ,— kx 285. To find the value of e Jo "^ Denote it by u, then du _ r* dr~Jo But e "^-^ cos rx dx = e *-^ e ** cos rx dx. r sin j-a; — Ic cos r.r therefore e '''^ cos rxdx — j^ ^ ,, . du k thus h dr k' + r' therefore u = tan ' , • No constant is required because u vanishes with r. This result holds for any positive value of A; ; if we suppose k to diminish without limit, we obtain r°° sin rx , tt J ^ '^ TT if r be positive ; if r be negative the result should be — ^ . "We can now determine the definite integral /, ^ sin rx cos sx , — ax; ^ for it is equivalent to 'Jo ^ ".'o ^ and the value of each of these two definite integrals can be assicmed. Thus if r + s and r-s are both positive the result is ^ ; if they are both negative it is - ^ ; if they are of con- trary signs it is zero. 272 DEFINITE INTEGRALS. 286. To find the value of f e~V^^-7dx. Jo Denote it by u, then du du _ [ -(x-+-)djc :j- = — 2a e \ «v — r ; da j *■' da assume x = - , then the Hmits of z are (^, c) dc dx J a. J a J a a J a b fP a J a (f) {x, c) dx dc ; since when the limits are constant, the order of integration is indifferent (Art. 02). We sliall now give some examples of this method. DEFINITE INTEGRAI-S. 273 r 1 288. We know that e"*' dx = j . Jo f^ . Integrate both sides with respect to k between the limits a and b ; thus dx = lo£f - j ^ ^ a f" e'""" dx ["^ e~"' dx It should be noticed that I ■ — • and I are both Jo X Jo X . . .^ . {"e-^'dx . , „ -oa P^-c 1 ['dx mnnite : lor is greater than e \ — , and I — is infinite. But this is not inconsistent with the assertion that dx is finite, and without findinsr the value Jo ^ . . . of this integral it is easy to shew that it must be finite. For ., . 1 , x-u c f''4^(^)dx J f'°(f)(x)dx , it IS equal to the sum ot — and I - ^ — where J ^ J c «-' (f){x) = e "■^ — e""^ ; the second of these integrals is finite, for it is less than - I ^ (a?) c?^, that is, less than - ( 5- . cJc c\a J [c Ay r^\ "We have then only to examine I ^-A_^ ^.j,^ Jo •^ Now by Maclaurin's Theorem " (a;) can assume for values of x less than c. Hence /. ■ax IS less than {b — a)c + —r- , '0 ^ 4 and is therefore finite. T.i.c. 18 274 DEFINITE INTEGRALS. 289. We know that h I e"^"" cos rx dx= jy Jo 1^ F + r ' Integrate both sides with respect to Ic between the limits a and b ; thus pe-^^-e-"- 6^ + r^ • cos rx ax = -h log —r. r, . Jo X " a + r r^ sin o^x f cos ?'tX' 290. Let I cZij; be denoted by A, and ^ cZ J X J L -T X by J5 ; we shall now determine the values of ^ and B ; the former has already been determined by another method in Art. 28.5. In the integral A put y for rx ; thus A = sin y dy _ 'o y this shews that A is independent of ?\ dB r°° a; sin ra; dx V\ e have 1 f^ 71 7 r"" sin r^ cZ^ and ijar = - ., , — jx ; Jo Jo a; (1+^0 r„, ^S f"'l+3^'sinr.r , , thus Bdr p-= T^— — ^dx = A; Jo dr Jo X \-\-oi? hence j^Bdr-'^-A=0 (1). Multiply by e"' and integrate ; we obtain since A is con- stant with respect to r e"' 1 1 Bdr + B-AI= constant. Now whatever be the value of r, it is obvious that the integrals represented by ^, J5, and 1 Bdr, are finite ; hence DEFINITE INTEGRALS. 275 the constant in the last equation must be zero, for the left- liand member vanishes when r is infinite. Thus I* Bdr -{■ B - A = {) (2). J From (1) and (2) ^=-^; therefore B=Ce^, where C is some constant. And from (2) therefore B = Ae~^ (3), Now when r is indefinitely diminished, B becomes r (JQ(^ ITT T-, — 2 ) that is 7i ; hence from (3) Jo l+a:"" 2 ' ^ ^ ^=^ and 5=^6-'. 2 2 We have supposed r positive ; it is obvious that if r be negative, B has the same value as if r were positive, and A had its sign changed ; that is, if r be negative B = -^ e^ TT and A= — -. {Transactions of the Royal Irish Academy, Vol. XIX. p. 277.) f COS VCC doc 77" From I ^; 5— = r: e'^ , we obtain by differentiation Jo 1+a; ^ with respect to r, I X sin rx dx tt = 77 C l + x' 2 And from the same integral by integrating with respect to r between the limits and c, we have /■" sin ex dx _'ir , _^. Jo log (1 - 2a cos a; + a') = log a^ + log f 1 - ^ cos a; + ^j , we have I log (1 — 2a cos x + a^dx = 'ir log a^ = 27r log a. JO If a = 1 it may be shewn by Art. 51 that the definite in- tegral is zero. DEFINITE INTEGRALS. 279 We may put the result in the following form ; I log (a" — 2ac cos x + &) civ = ir log k^, Jo Avhere P is the greater of the two quantities a~ and c^ and is equal to either of them if they are equal. By differentiating this result with respect to a we arrive at the result which constitutes the last Example of Art. 46. 293. By integration by parts we have log (1 — 2a cos X + a^) dx Hence, if a be less than 1, ^ J = x- log (1 + a)^ that IS, - log (1 + a) ; Jol-2acosx+a' 2a °^ ' 'a *^ if a be greater than 1, the result is - log (1 + a) — log a, that is, — log ( 1 + - ) . 294. In like manner we have, if r be an integer, I cos rx log (1 — 2a cos x-\-a^')dx = — a^ or — cC, Jo r r according as a is less or greater than unity. 295. Integrate by parts the integral in the preceding Article ; thus we find "■ sin a? sin ra: ^ic _7r „ , ir 1 — 2a cos x + a' 2 = ^a'*-i OY^a-^r+i) according as a is less or greater than unitv. 280 DEFINITE INTEGRALS, 29 G. Similarly from the known expansion 1 — 2a cos X + d^ = 1 + 2a cos ic + 2a^ cos '2x + -a? cos 3x + where a is less than 1, we may deduce some definite integrals; thus if r is an integer /. cos rx dx TTCi ji. J ' 1 — 2a cos x-\- c^ 1 — a^ for every term that we have to integrate vanishes with the assigned limits, except 2a'' / cos^ rx dx. Jo dx f 1 297. To find the value of t— ^ = — Jo J- + X 1 — 2a cos ex + a 2 The term ^^ k may be expanded as in the 1 — 2a cos ex + a "^ ^ preceding Article ; then each term may be integrated by Art. 290, and the results summed. Thus we shall obtain TT 11 + ae~' 2 ■ 1 - a- 1 - ae-" ' Similarly r°° dx log (1 — 2a cos ex + a^) ^ -, = it log (1 — ae '). Jo i + X' 298. It is also known from Trigonometry that = sin ex + a sin 2cx + a' sin Sex + . . 1 — 2a cos ex + a^ a being less than 1. Hence by Art. 290, we obtain X sin cxdx _ "^ (1 + x"") (1 - 2a cos ex + a") ~ 2(6" -a) " This also follows from the last formula of Art. 297, by differ- entiating with respect to c. DEFINITE INTEGRALS. 281 299. To find f^^dx. Jo l—X By expanding (1 — x) \ we find for the integral a series of which the type is 1 x" los: X dx. o By integi-ation by parts this is seen to be equal to 1 Hence the result is (! + «)" 'l 4-1 1 1 1 that is, by a kno'^Ti formula, — — , oOO. Let V denote e■^^^^~l\ that is, cos ic + \/(— 1) since ; then if/ denote any function, we have by Taylor's Theorem, /(a + v) +/(a + O = 2 |/(a) +/' (a) cos x +-^^ cos 2x + | . And 1 — 2c cos x + c Therefore = 1 + 2c cos £c + 2c" cos 2x + 2c^ cos '^x + j; i-2ccosUc- ^"=i:r^-|/^-H^/("Hi:^/'(^)+.--| 2-77 1-c In this result it must be remembered that c is to be less than unity, and the functions /(a + v) and /(a + v~^) must be such that Taylor's Theorem holds for their expansions, and gives convergent series 282 DEFINITE INTEGRALS. In a similar way it may be shewn that and r , ^-^^Q^^ ^ (/(a + r)+f{a + Ol dx j„ 1 — 2ccosa' + c" ^-^ ^ '^ "^ ^ '^' = 7r[/(a + c)+/(a)}. Substitution of impossible values for Constants. 301. Definite integrals are sometimes deduced from known integrals by substituting impossible values for some of the constants which occur. This process cannot be con- sidered demonstrative, but wdll serve at least to suggest the forms which can be examined, and perhaps verified by other methods (see De Morgan's Differential and Integral Calculus, page 63 0). We will give some examples of it. We have e-'"" x""-' dx=p-''V (ji). Jo For p put a + h V(- 1), and suppose r = »J(a' + h^) and tan ^ = - , so that p = r [cos 6 + VC" 1) sin 0} ; thus a .00 Jo Thus by separating the possible and impossible parts we have /. n tan"^ - e "" X" - cos Ox ax = n {a-' + by •J rOi)sin^?itan-'- 1 e~" x"'^ sin bx dx = „ • -" {a' + by For modes of verification see De Morgan, page 630. DEFINITE INTEGRALS. 283 302. In the formula Jo ■^' dx = Za , — a-I' -7— » . ^ l + V(-l) ,1 change a mto j^ — ■ c ; thus Jo 2c V2' therefore I jcos c'x" — \f{— 1) sin cV - cZ^ = ^^^ r^ ; therefore I cos c"a;^ dx = z^ — — , and | sin c^a;^ dx = - — ~ . Jo 2c V2 Jo '^cx^'2 If we write y for c^x^, these become r°° sin y dy _ /"° cos _?/ <7?/ _ lir Jo Vi/~~Jo "^"""V 2' These results may be verified in the following manner. By Alt. 272 Ave have 1 2 r" \JX f^TTJo "' f, Jo therefore I — ; — dx = — 1 cos x dx I e~ -'-^ cZ^ V^ V*"" ./ 2 '•=° , dz I e~-'^ cos a; cZj; \/7r Jo Jo 2 f " /-^, , by Art. 285, V 2 TT 9* Similarly we can shew that Jo sm a; , _ /tt 284 DEFINITE INTEGRALS. 803, In the integral 1 e~\ """^V^cZa;, suppose y = a: \/^; Jo thus the integral becomes —pr \ e \^ 'f / dy, which is known by Art. 286. Thus g-2aft Now put cos Q + \/(— 1) sin ^ for Z;; thus the right-hand member becomes 1 a a ' cos - + V(- 1) sin ^ ^'"" «-2a{cos0+V(-l)sinfl} 2 ^ that is, VTTf „„„^o.„:„^, ^\ // .^.:^f^..:^n0\] Thus [" e-("'+S)<=°^^os K^;^ + ^) sin 6'| cZ^ = '^e-2«cosecos C2a sin 6' + 1) , and I e V •<-■'/ sm ^ a; + — 2 ) sm ^ ^ aa; / 6 dn f 2a sin ^ + ^ _ V^g-2acose gi^ /o™ „:„ ^ , " Euler's Theorem. 304. We will now give a theorem which connects inte- gration with the summation of a finite number of terms, and which is sometimes employed for the approximate calculation of the value of definite integrals ; the theorem is usually called Euler's, though more strictly due to Maclaurin : see History of the Mathematical Theory of Probability, page 192. DEFINITE INTEGRALS. 285 By Taylor's Theorem we have /(a + A) -f{a) = //' ((0 + %f" («) + %,/'" («) + •••; change a successively iuto a-{-h, a + 2k, a+oh, ... a-^ ('i — 1) ^h and add ; then if we put a; for a + 7ih we obtain the following result : h' /^ 3 where S/' (x) denotes /' (a) +/' (a + A) i- ... + / (a; - A), and S/" (cc), 2/'" {x),... have similar meanings. For/' {x) put ^ («) ; thus j (x) +|2" (^) + ... , and, by transposition, S0(^) = ^j"^"%(^)^^-|2f(^)-|s"(a^)- (1). In the same way we have (2). (3), 20'" (^) = I {f {^)-" («)} -^20"" (0.) -^Sc/, (a:) - ... W, and so on. oc;5 DEFINITE INTEGRALS. Now from the series in (1) we may eliminate "Ecj)' (x), 2<^" (x),... by the aid of (2), (3) — The elimination may be effected thus: multiply (2) by AJi, multiply (3) by AJi^, multiply (4) by AJi^, and so on ; then add the results, and determine A^, A^, A^,... by the equations A, A. I ^ Hence we obtain 1 ra+nh ( ] 20(^)=^j^ cf>{x)dx+A,)^(f>{x)-cf>(a)^^ + A^ if {x) - f {a)| h^A^ If {x) - (x) = e" ; then a + nh _ a Thus e» _ e« e" - e"^ + so that + A^ (e^ - 0+ ^/* (e' - O + AJi^ (e* - e") e" Therefore ^„^ is the coefficient of 7i"' in the expansion of in ascending powers oih. The expansion is effected iu the Differential Calculus, Art. 123; it is there shewn that DEFINITE INTEGRALS. 287 1 1 1 Bh BJo' BJi' = - r. + H . V + + (-ir L2n ' •••' /)j, B^,... are called Beruouilli's J\"fi??i6e?'5 ; their values are, as far as B^, ' 6' ^~30' "■ 42' '"80' '~m' with respect to the values of the Numbers beyond B^ inform- ation and references will be found in a paper by Mr Glaisher in the Cambridge Philosophical Transactions, Vol. xii. Thus it follows that of the quantities A^, A^, A^,... those in which the suffix is an even number are zero, excejat A^ which is —I, and those in which the suffix is an odd number are determined by ^^"-^"^ ^^ I2n' We have then the following result: By the aid of this we may calculate apj)roximately the value of the definite integi'al 1 cj) (x) dx. J a The result may be put for abbreviation in the form Xcp{x)=C + jJcf>{x)dx-^cf>{x) + ^(}i'{x)-^^ 1 h^ where C represents a series of terms independent of x. 28S DEFINITE INTEGRALS. The series thus obtained for 2<^ (x) will be in general an infinite series, and as we cannot ensure that the series is con- vergent the preceding investigation is not rigorous: we shall return to the subject in Art. 332. As an example of the last formula take (x) = - , and h = l. Thus we get by adding - to both sides 111 1/7,1 .1 1 , 1+2 + 3 + +^=^ + ^"S^ + 2:^-12^^ + - Hence by making x infinite we infer that in this Example G is Elders Constant : see Ai't. 268. EXAMPLES. 1. Evaluate I \ . ,„ . . ,. . Result. — .^,3 ,» " • Jo X +0X +0 ^0 wo 2. Evaluate cos (a tan x) dx. Residt. -^ e ". Jo 8. Evaluate x""-'^ e''" dx. Result. -. Jo ^i • J {a' cos'^ x + b' sin^ xf 4 \ab' ^ a'bj IT 5. Prove f V(tan <^) # = -^ [^ + log {V(2) - 1] Jo V -^ L-' 6. ProveJV(cot(/>)dc^=;^ | + log {V(2) + 1} 7. Find the limiting value of a:e~*' I e^' c7a; when ic = co . Jo Result. ^ . EXAMPLES. 289 ^, ^, , f "" COS ax — COS hx , , 7> 8, Shew that ax = losf - . Jq X ° a 9. If F (x, -j be any symmetrical function of x and - , then 9 dx ^ fl f/.P "-^(--;) ^"-^(--;) 10. If i^(^) be an algebraical polynomial of less than n dimensions -: 11. Prove that e-^^^^cos (sin ^) cZ^ = 27r. Jo 12. Prove that !^— -~,^ =———— when c is indefinitely Jo 1 - ccos"^ V(2/t) -^ nearly equal to unity, n being a positive quantity. 13. Evaluate f (a cos ^ + ^^ sin 6) log (a cos' O + h sin' ^) (^^. Jo Result. 2b {log a - 2 + -,v -^-, , cos"^ ^^ 1 , supposing a greater than b. 14. Shew that f "" , 1 + 2?i cos a.r + if dx Jo * 1 + 2u cos bx + )i' ' X is equal to log [1 + n) log — , or log ( 1 + - J log -^ , according as n is less or greater than unity T. I. c. 19 290 EXAMPLES. 15. Find the value of /, ^ "where a and & are positive, but a and /3 positive or negative ; and shew that it is wholly real when a b IG. Prove that cot^{l — x+x'^)dx = ^— log 2. r°° dx f i\ 17. Prove that ^-^ log a; + - = tt log 2. 18. From the value of dx deduce that of Jo ^ Result. The two integrals are equal. 19. Prove that J^ ^ j dx = log j ^^^^aia+fc) • 20. Shew that [ " ^f'-'^^f ^« = tt. Jo (1 + if) 21. Shew that (e~^' - e"-^'') dx=(b- a) sJtt. Jo (Sohithns of Senate- House Problems, by O'Brien and Ellis, page 44.) /• » e^ + l TT^ 22. Shew that / log-^^ — ^ dx = -r . Jo °e"-l 4 23. Prove that —, . — = log — , and reconcile with Jo log a; X n rl x""^ dx this equation the result of transforming I y-^^ — by making ic'' = y. EXMIPLES. 291 24. Shew tliat f 'sin"^ cld = '^. ^ ^ ' . Jo 2 ^/n + -2\ 25. Shew that f'^qi^^P^ = miM (O + cx)"'' r(; + m) 6"(i-t c)' 20. Shew that f'.SS-" f ^'f :' ^f ^ T (Q r (,„) ^ Jo (a cos- + b sm'^j'^'" 2r (Z + 7?i) a' 6 2/. fcShew that ^ — , . ., ^ =- , — , Joacos-0 + hsm-d 2cosA«7r ^-^ ^~ ' n being less than unity. 28. Shew that ^^ ^"^""'^^^ -^^^ ^""^ o(a+^cos^)" r(,0 ^^._^.^. 29. Shew that f" ^"'"' "^^^ = Jo ( 1 — xr n sm — • ^ 7i SO. Shew that ' ^ ^^-^ '^ (1 + ex) (1 - xy (1 + c)" sin UTT ' OT oi ,1 , f" sinaa^sin-ra; , tt tt ol. Shew that dx = Q, or + - , or + - . ^ accordino- to the values of a and c. S2. Trace the locus of the equation CO , shi^cos 6x ,. y=\ — y — dd. 19—2 292 EXAMPLES. 83. Trace the locus of the equation I = ["log [1 - 2e"" cos + e-^] d9, Jo where u = sin - . a 34. Trace the locus of the equation 2 cc cos 6 dd J 2 in which the sign of the square root is always taken so as to make the quantity in the denominator positive. 35. Shew that It 77 r2 Ci ... TT^ TT I I sin ic sin~^ (sin x sin y) dxdy = -j^— -x^ 36. Compare the results obtained from sin ax e'^ dx dy, by performing the integrations in different orders. 37. Find the value of e "' -*" dx, and hence shew that Jo ---— 7 oa \/7r „ / fx a\ -— .-> 7 e «- •^- dx = — tV- =0 -2 - ^ e "- ■^- (7a-. 4e" J Va ic / 38. Shew that JJV(l + ^^ + 2/^) -^ 4U V' the integral being extended over all the positive values of x and y which make x^ + y" not greater than unity. r /x a Jo W ^^ EXAMPLES. 293 39. Shew that n+l ■i dx chj dz ... IT 9, the number of variables being n, and the integration being extended over all positive values which make «* + 2/' + -"' + not greater than unity. 40. If A,-^A^x^A^x^^ ^F{x), and a^ + a^c + a^x^ + = / (x), prove that A^fi^ + A^a^a? + A^a.^x^ + = ^/> (^0 + ^(^01 {/(") +/W1 ^^ - ^o«o. where u = xe^^'^~'^^ and v = xe~^'^^~'^K 41. If the sum of the series a^ + a^x + a,jK^ + can be expressed in a finite form, then the sum of the series a/ + ttjV + a^x*' + can be expressed by a definite integral. Prove this, and hence shew that the sum of the squares of the coefficients of the terms of the expan- sion of (1 + xy when n is a positive whole number, may be expressed by Q2n+2 r — '" i'^'ecos'nede-i. «r5- ^ cos-"< Jo 42. Shew that cos ex dx TT f e' Jo Fc + 'o 1 + cc^' 2 [1 + 0-' ' l + C- Shew that /, (ji (sin 2x) cos xdx=\ (p (cos'^a;) cos x dx. Jo (Liouville's Journal de 2Iathe'niatiques, Vol. XVili. page IGS.) 294< EXAMPLES. 2 ■• X X 2 n TT. 44. Shew that 1 — ^5 + -^i — cos [x sin y) dy. "JO 45. Shew that r^m-i Q-x^dx [ /-"'-' e-^" dy = — ^ 4G. Shew that r (,2cos29+2^,s>n20)COS f , ^.^^ ^^ + i^^ COS 2^ dx J -co sm ( 2x J 7U7r ?i sin — n 1 „cos ,^ . = 77-6-" . ^ + a); sm TT ^ being comprised "between the hmits + -j . rx+i 47. Shew from Art. 2G7 that I log F (.r) c/o; is equal to the J X limit when w is infinite of - log -^F {nx) (27r) 2 7?^'- 48. Hence by the aid of Ai't. 282 shew that rx+l 1 log T (x) dx = xlog X — X + ^ log 27r. 295 CHAPTER XIII. EXPANSION OF FUNCTIONS IN TEIGONOMETEICAL SERIES. 305. The subject ^xe are about to introduce is one of the most remarkable applications of the Integral Calculus, and although in an elementary work like the present, only an outline of the subject can be given, yet on account of the novelty of the methods, and the importance of the results, even such an outline may be of service to the student. For fuller information we may refer to the Differential and Integral Calculus of Professor De Morgan, and to Fourier"s Theurie... de la Chaleur. The subject is also frequently considered in the "vvi'itings of Poisson, for example, in his Traite de Meca- nique,\ol. i. pp. 64-3. ..653; in his Theorie...de la Chaleur; and in different 5lemoirs in the Journal de V Ecule Pohjtechnique. The student may also consult a Memoir by Professor Stokes, in the 8th Vol. of the Camhridge Philosophical 'Transactions, a Memoir by Sir W. E. Hamilton, in the 19th Vol of the Transactions of the Roijal Irish Academy, and a Memoir by Professor Boole, in the 21st Vol. of the same Transactions. 80G. It is required to find the values of the m constants A^, A.^, A^,...A^^, so that the expression Aj^ sin X + A^sm2x + A^sin Sx + + -^m ^^^ ^^ may coincide in value with an assigned function of x when x 77" has the values 6, 26, Sd,...m9, where d = - — — ^ . m + 1 20G EXPANSION OF FUNCTIONS Lety(j:) denote the assigned function of x, then we have by hypothesis the following m equations from which the constants are to be determined, f{e) = A^ sin e + ^2 sin 16 + A^ sin 3^ + +^„, sin mO, f (2^) = A^ sin 2^ + A^ sin 4^ + ^3 sin 6^ + + A,, sin 2m^, f{rii&) = J.,sinm^+^2sin2??z^ +^3sin 3?w^+ +^,,^sin mmO. Multiply the first of these equations by sin rQ, the second by sin 2r^, , the last by sin 7???'^; then add the results. The coefficient of A^ on the second side will then be sin rQ sin sO 4- sin %'d sin IsO + + sin mrd sin msO ; we shall now shew that this coefficient is zero if s be different from r, and equal to \ {m + 1) Avhen s is equal to r. First suppose s different from r. Now twice the above coefficient is equal to the series cos (r — 5) ^ + cos 2 (r — s) ^ + ■\- qq^ m {r — s) 6 , diminished by the series cos (r + 6^) ^ + cos 2 (r + s) ^ + + cos m {r + s) 6. The sum of the first series is known from Trigonometry to be equal to sm [zm + 1) ^ — ^ sm -- — ^— 9 ■ {r-s)d ' 2 sm-^ — j^— - {r-s)e\ . (r-s)d sm -({i — s) TT — - — ^ — Y — sm ^ I 2 that is, to ^ / N /I — — ^ — . . (r -s)0 2 sm -^^ — A-^ — This expression vanishes when r — s is an odd number, and is equal to — 1 when r — 5 is an even number. The sum of the second series can be deduced from that of the first by changing the sign of s; hence this sum vanishes IN TRIGONOMETRICAL SERIES. 207 "vvhen r + 5 is an odd munber, and is equal to — 1 wlicn r+s is an even number. Thus Avheu 5 is different from r, the coefficient of Ag is zero. When s is equal to r, the coefficient becomes sin^ r6 + sin" 2rd + + sin^ mrd, that is, 9 — 9 ] cos 2r6 + cos 4r^ + + cos 2mr6 1 . And by the method already used it will be seen that the sum of the series of cosines is — 1 ; therefore the coefficient of ^1,. is h {m+1). A = Hence we obtain 2 m + - sin ref{6) + sin 2r6'/(26^)+ + sin mref{md) , and thus by giving to r in succession the different integral values from 1 to m, the constants are determined. Now suppose m to increase indefinitely, then we have ultimately 2 f'^ A^ = - \ s'mrvf{v) dv. TT J Q TTJo And as / (cc) now coincides in value with the expression A^ sin x+ A^ sin 2x+ for an infinite number of equidistant values of x between and TT, we may write the result thus f(x) = - Zf sin nx I sin nvf (y) dv, where the symbol S" indicates a summation to be obtained by giving to n every positive integral value. 307. The theorem and demonstration of the preceding Ailicle are due to Lagrange ; we have given this demonstra- 298 EXPANSION OF FUNCTIONS tion partly because of its historical interest, and partly because it affords an instructive view of the subject. We shall how- ever not stop to examine the demonstration closely, but pro- ceed at once to the mode of investigation adopted by Poisson. 308. The following expansion may be obtained by ordi- nary Trigonometrical methods : = 1 + 2A cos ^ ^ ■(1), ^ ; IT iV — X) T ^ I 1 - 2/i cos —^ — ^ + /i* V + 2/r cos — — + 2/i^cos — —, -+ . . h being less than unity, so that the series is convergent. Multiply both sides of (1) by (f>{v), and integrate with re- spect to V between the limits — I and I; also make h approach to unity as its limit. On the right-hand side the dilferent powers of h become ultimately unity. The numerator of the fraction on the left-hand side will ultimately vanish, and thus the integral would vanish if the denominato?' of the fraction were never zero. But if x lies between — I and I, the term cos — ~ will become equal to unity during the integra- tion, and thus the denominator of the fraction will be (1 — /i)^, and will tend towards zero as h approaches unity. Hence the integral will not necessarily vanish; we proceed to ascertain its value. Let v — x = z and h=l —g, then {l-h^)4>{i^dv Cg{l + h)4>{x + z)dz ^7 TriV — x) ,,, J „ ... „1TZ 1 - 2/i COS — ^-v — '- + h' J g" ^ 4/i sm' ^ Now the only part of the integral which has any sensible value, is that which arises from very small positive or nega- tive values of z\ thus we may put . irz _iTZ and (j){x +z) = (ji {x); IN TRIGONOMETRICAL SERIES. • 299 and the intesfral becomes "O' 9{i+J0{^) — ~p=2^ W = lif W dv + '^j Sr|' (^ {v) cos '''' ^''~ '''^ dv ...(2). If however x = I or — /, then the integral on the left- hand side has its sensible part when v is indefinitely near to I and —I; we should then have to perform the above process in both cases, but the integral with rosiject to z would only extend in the former case from — /S to 0, and in the latter from to a. Hence instead of 2l(J3 {1} on the left-hand side, we should have 1(f) [l) + I(f> (— I) ; and instead of (f) [x) on the 1 1 left-hand side of (2) we should have ^ ^ (^) + ^^ (— 0- Thus we have determined the value of the right-hand member when X hes between I and —I, both inclusive; its value in other cases can be determined by the method which will be explained hereafter in Ai't. 321. 809. In the same way as the result in Art. 308 is found, we have, if we integrate between and I, ^ (^^ = \i\\^ ^'^ ^' + 7 ^^ \y ^'^ ^"^ """ V ""^ ^' ^^^' this holds if x has any value between and I; but when a; = the left-hand member must be ^ (0), and when x = I the left-hand member must be h (f>{l). Thus we have deter- mined the value of the right-hand member when x lies 800 EXPANSION OF FUNCTIONS between and I, both inclusive ; its value in other cases can be determined by the method which will be exjalained hereafter in Art. 821. Similarly Jl] ^(^)^^"+I^"j {x) = \ Ccf> {v) dv + j S;° cos^ f^cos ^<^(t;)(^v...(3). L J b b J Q b This holds for any value of x between and I, both in- clusive. From (1) and (2) by subtraction '^ 7? TT T* r Tinr }) (*') = 7 ^r s^^ ~/~ s^^ "~r" 'Pi'v) dv (4). This holds for any value of x between and I both exclu- sive ; and when x=0 or I, the left-hand member should be zero. Equation (4) coincides with Lagrange's Formula. We may observe that either of the formulae (3) and (4) may be deduced from the other. Suppose we take (8) aud write sin —j- (f) [x] instead of {x). Thus TTX X r . TTV sin -Y {x) = -j j sin -y ^ (v) dv 2.^00 nirx [^ nirv . ttw . . , , -H -J 2j cos -y- I cos — ,- sm -j (p [v] dv. ■^.-r nirv .TTV 1 . (n + l)'rrv 1 . (n—l)'rrv JN ow cos — , sm -T- = ^ sm -^ j-^ ^ sm j ; IN TRIGONOJEETRICAL SERIES. 301 and therefore it will be found that the result may be exhibited thus, sin —j- (x) need not be a continuous function; for example, from x = to x = a we might have (f> {x) —f\ (x), then from x = a to x = b we might have [x) =f^ (x), then from x = b to x = c we might have (f) (x) =f^ (x), then from a? = c to x = l we might have ^ {x) =f\ (x). The formula (3) for instance would still be true for all values of x between and I inclusive, as is evident from the mode of demonstration, except for the values where the discontinuity occuis. When for example x = a, then the value of the right-hand member would not be f^ (a) or f^ (a) but i {./^i W +/2 (^)}- If therefore for x^a we have fi (^) —f-i i^)> ^^® formula holds also when x = a. Some writers adopt a mode of expression for such a formula as (3) of Art. 309 which draws attention to the pos- sible discontinuity. Instead of (f) (x) on the left-hand side they put ^ {cf) {x + e) + (j) (x — e)], where e represents an inde- finitely small positive quantity. Thus when there is no dis- continuity the limit of (x), and so also is the limit of

(r) = c from v = to v=a, aud then from v = a to v = 1 it is zero ; therefore cos -^- 6 (v) av becomes c I cos -^- dv that is — sm — j- Jo t Jo I nir i therefore the required expression is ca 2c ( . ira irx 1 . ^ira ^ttx -— H — -^sm -y- cos -,- + - sm — r- cos — y- l IT [ L I z L L 1 . 37ra ^TTX 1 H-gsm-^cos-^- +...'; this will give \c when x = a. Or we may use formula (-i) of Art. 309. Then f"' . nvTT , d /, na7r\ c sm —7- dv = — 1 — cos — J— , Jo i n7r\ I J' and we have for the required expression 2c f 7ra . TT^ 1 27ra . 2ttx — \ vers -y- sm -^ + - vers —r- sm -r- TT ( t I 'I L L 1 Svra . Zirx ) + ^ vers — ;— sm 3 '^'^ I ^^" I ^ I' this gives when x=0, and -|c when x=a. 325. Find an expression which shall be equal to kx from a; = to ^ = ^ , and equal to k (l — x) from x = - to x = 1. u 2 Here I 9 \v) COS — j- dv = ^y cos -y- ay + ^- (i — ?;) cos -7— dv Jo t J o J I I 2 nrr k]^(l . nir 1 WTT 1 ] H' , . = ^ -^ s- sm -^ + -2- cos -?i 5- \ -\ sm mr — sm TT [271 2 wV 2 wVj ?i7r TT T T COS -r- 1 . 1 . ?i7r cos7i7r 2 - sm nir — ;r- sm — - + — r, n— kF { nir ^ 20—2 808 EXPANSION OF FUNCTIONS This is ^„ wheu n TT n other case, and is of the form 4r + 2, and in every j){v) dv = h \ V dv + h {l-v)dv=~', thus the required expression is TT -I 1 "llTX 1 1, COS ~~i 1- ^ COS I Qttx + . If we denote this by y, then from x = to x = \l both in- chisive y = kx, and from x—l^l to x = l both inclusive y =k{l — x); for values of x greater than I the values of y recur as she'wn in Art. 321. Thus the value of y is the ordinate of the figure formed by measuring from the origin equal lengths along the axis of x to the right and left, and drawing on each base thus obtained the same isosceles triangle. As another example we may propose the following : find a function ^ {x) in terms of sines which shall be equal to X from ic = to x= a, then be equal to a from a; = a to x = 7r— a, and then be equal to tt — a; from x = '7r — u. to x^tt. The result is 4 f . 1 . 1 . (b(x) = - -^sin a sin x + -xr7, sin 3a sin Sx+ ^, sin 5a sin ox + TT { 6' o this is true from a; = to ^^ = tt both inclusive. We may give the following y geometrical interpretation of this ^j q result : Let OACB be a square, such that OA = TT, and OB = ir. Take for the origin, OA for the o A axis of X, and OB for the axis of y, and let the axis of z be at right angles to the axes of x and y. Let a pyramid be formed having OA CB for its base, TT TT TT and its vertex at the point x==-^, y = -^ , z= -r: then the fol- ^ Zi Id IN TRIGONOMETRICAL SERIES. 309 lowing equation represents the four faces of the pyramid which meet at the vertex, z= - {sin a; sin ?/ + ^ sin 3^ sin 3^ + ^^sin 5a; sin 5?/+ ...}, By the mode of obtaining the result it apphes to that j)art of the surface for which y is less than - ; and then by in- spection we see it applies to that part of the surface for TT which y is between ^ and tt, "We may conveniently put I + 2 for X, and ?; + - for y. The student may verify the following examples. If X be numerically less than a the exjDression tn + 1 is equal to a — a: if ic be positive, and a + a? if ^ be negative. Prove that for values of x between — vr and ir inclusive x^ _ TT^ COS 1x cos 3x __--_cosa; + -2^ 3^"^ This may be obtained from Art. 310 by integration ; or from equation (3) of Art. 309. Integrate this result : thus x^ TT^x _ . sin 2a; sin 3a; Find an expression in terms of sines which shall be equal to sin — from a; = to x = a, and equal to from a; = a to cc = TT. The result is fsin a sin x sin 2a sin 2a; sin 3a sin 3a; TT'^-a'^ "^ 7r''-2V ^ 7r'-3V Find an expression in terms of cosines which shall be equal to -j — x^ from a; = to x= ^, and equal to from 310 EXPANSION OF FUNCTIONS X "2 to X = 77. The result is TT" 12 4 ■f -■ TT COS X- cos 3^ -1- cos5j? f .. .. fcos 2x ■ z ■< = — COS 4ic cos 6x 826. Other formulae may be given analogous to those in Art. 809 ; we will here investigate some. We have by Art. 809 (l)- In the same manner as this result was obtained we may also prove that -^ (^•) = ^ijl ^ (^) ^^^ + 7 ^^ !l ^ (^) ^°^ ^^^^^-^—^ dv... (2). This holds when x has any value between and Z; but when a; = the left-hand member must be <^ (0), and when x = l the left-hand member must be ^ (l). Subtract (1) from (2); thus (}){x)=-j2^ I (f){v)cos^ 2^ dv (.3). This holds when x has any value between and I; but when x = the left-hand member must be ^(/) (0), and when x= I the left-hand member must be |(^ (I). Now in the same manner as (8) was obtained, we may obtain the following result, starting with v + x instead of V — X, A lv<»/■^/^ {'^n-l)'7r(v + x) . ,,. ^ = -^^Tj ^^^ Pi=^> thus p^= i p dd. 312 EXPANSION OF FUNCTIONS TT f2 Similar! V, p.,= p^d6, ' Jo P3= I PM Jo P.- [%/i0 and so on. Now in formula (5) of tlie preceding Article suppose i= ^ ', then since p is some function of 6, we have p = A^cos9 + A^cos39 A- A^cos5$ + ... where A^, A^, A^,... are certain constants determined by that formula (5). Thus p^ = A^ sin ^ + - ^3 sin 3^ + - A^ sin 56 + o o P^= A^ cos 6 + — J3 cos 3^ + -2 --^5 cos 06 + o O /03 = ^4jSin^+-r^J[3sin3^ + Ti-4^sin 06 + o Proceeding thus we obtain, when n is indefinitely large, p^ = A^smd, or p„ = A^cos6; and these equations represent a cycloid; see Ai't, 105. It should be observed that the formula which we have used for p assumes that p vanishes when 6=^ '• see Art. 326. But this does not really affect the demonstration ; for by the nature of the problem p^ really does vanish when ^= -, and therefore a formula for p, like that given for it will hold, IN TRIGONOMETRICAL SERIES. 813 and the process can then be continued by which p^, p.,,... are successively obtained. In Art. 102 it is shewn that the in- vohite of an Equiangular Spiral beginning from the pole is an Equiangular Spiral ; but close to the pole this curve forms an infinite number of coils, and this singularity renders our pre- sent investigation inapplicable : thus the apparent contradic- tion between the result obtained in Ai't. 102 and the theorem here investigated is explained. We may next examine the nature of the result when the tanjrents at the extremities of the orioinal curve are not inclined at a right angle. Suppose these tangents to be in- clined at an angle a; and j^ut a for I in the formula (5) of the preceding Article. Then we have p = A^cos ~+A^cos-^- + A^cos-^ + ; and by proceeding in the same way as before we arrive at the result Pn = ^ cos — , or p^ = k sm ~ , where k = A 2a ■ - Z.X '- TT If k were a finite quantity, we should thus obtain an epicycloid if a is greater than — , and a hypocycloid in which the diameter of the revolvinfj circle is less than the radius of TT the fixed circle if a is less than - ; see Arts. 110 and 111; and this is the usual statement of the results. But it will be observed that h becomes indefinitely great in the former case and indefinitely small in the latter case ; so that in the former case the radii of the fixed and revolving circles must be supposed to increase indefinitely, and in the latter case to diminish indefinitely. 328. Suppose a, h, and J — a to be positive quantities. Consider the double integral I 1 cos ux (p(v) cos uvdudv. 814 EXPANSION OF FUNCTIONS By integration by parts we have r , , . ,

' (v) sm uv ^^ Ja U Thus tlie proposed double integral becomes , ,,. r°° cos tta; sin w6 , , , , /■°° cos Ma:; sin wa , i> (}) (^^- (a) du J U J u cos ux (f)' (v) sin uv , , —^ du dv. Ja U The first and second terms may be easily found by Ai't. 285. In the third term we can chan^je the order of integration, and apply Art. 285 to fiud the result of integra- tion with respect to u. We shall then obtain the following results, assuming cc to be positive : I. Let X be greater than b. Then each of the three in- tegrals vanishes. II. Let X be between a and b. Then the first term is TT equal to - ^ (h) ; the second term is zero. And, by Ai't. 285, r°° cosM>rsin»' {v) and integrate witli TT 77 respect to v, we obtain - cf) (b) — - (f) (x). Thus on the whole we have |c^(6)-||0(&)-^(^(^)|, TT tliat is ;^ (oo), as the value of the original double integral. IN TRIGOXOMETRICAL SERIES. 815 III. Let X be less than a. Then the first term is ^ <^ v^). the second term is | (a), and the third term is ^ {<^ (&) - 9 («) ] • Thus on the whole we have l4>{h)-l<^{a)-l{^[h)-cl>[a)], that is zero, as the value of the original double integral. TT Hence finally the double integral is equal to or to ^ ^ (^), according as x lies beyond or within the limits a and h. TT It may be conjectured that ii x=a the value is 7 ^ (cf), and \i x = h the value is ~ ^(6) ; and this conjecture is easily verified. If X is negative tlie value of the double integral is tlie same as for the corresponding positive value of x; since cos (— ux) = cos ux. 329. In like manner supposing a, h, h — a, and x to be positive we can shew that s,mux<^{v)&\nuvdudv has the same value as the former double integral. If x is nega- tive the value is numerically the same as for the correspond- ing positive value of x, but of contrary sign ; since sin (— ux) = — sin ux. 330. By combining the results in Arts. 328 and 329 we obtain the following. If a, b, b — a, and x are positive r" rb I (f){v) cos u[x— v) ill dv J J a is equal to or to tt^ (x) according as x lies beyond or with- 316 EXPANSION OF FUNCTIONS "TT . , . , TT ill the limits a and h ; and is equal to ;- (a) and ^ ^ (6) respectivelj at the limits. This result admits of extension. The limitation that x Is to be iwsitive may be removed: for, by virtue of the remarks at the ends of Arts. 328 and 329, If x is negative, so that it is beyond the limits a and h, the double integral vanishes. Again, suppose that a and h are negative quantities : put a = — h, and h = — k; also put v = —v , and x = — x. Tlieu I cos u {x — v)dv = — \ cos u {x — v) dv = I cos u {x — v) dv, J a J h Jk where h — k is positive. 331. In this way we find that, \i p — q be positive, \ i^) = ^J_^ ^ (v) dv+] ^-f ^cos "^ZA ^ (^) dv • within the limits ^j and q; and is equal to g^(j>) and TT TT , 7177 put ~j = h, -J- = nil = u; thus we have 1 r^ . , . , 1 ^ f r^ W=2|| (^(v)fZy+- S jl Go&u[v-x)j>{v)dv[h, IN TRIGONOMETRICAL SERIES. 317 u being a mnltiple of h, and the summation denoted by 2 extending for all values of u from /i to co , But if I becomes indefinitely great the difference h of successive values of u becomes indefinitely small, and the sum denoted by S be- comes an integral taken with respect to u from u = () to w = 00 . Thus if we make I— x , and put du for h, and the sign of integration instead of 2, and suppose (v) is such I ri .1 that ^, 6 [v] dv vanishes with -j , we have "1 /• 00 /«00 (b(x) = - \ I cos u (v — x) [v) du dv. 332. We shall now return to the subject introduced in Art. 804, and shall give another demonstration, due to Poisson. of the formula there obtained. In Art. 308 we have obtained the following result: \[<^{j)-\-c^{-i)]=~^^[^4>{v)dv^\^^'j{v)cos'^:^^ where the summation denoted by 2 applies to the positive integer r, and extends from 1 to oo . Change I into ^ : thus -I /•2 2 r^ cos 2 2 Change ^ [v) into {Id + v) ; then the result becomes l\ . /„ l_ 2 y I <^ [U + v) dv + jX \ (j) {Id + v) cos 2 dv. ' ' 2r7r(v-^ 818 EXPANSION OF FUNCTIONS Put z tor kl + v; then the right-hand side becomes -I (f) {z) dz + J '^ j (}) (^) cos — J- — '^^^• ki--^ ki- 2 13 5 2n — 1 Now put for k in succession the values ^ , ^ , ^ . . • — ^ — , and add the results ; thus observing that cos— — I reduces to cos — r— , we have -.-0(0) + <^(/) + (»iO nl Tm^ 9-) 1 f"' 2 p" 27'7r^ = -y (f)(z)dz+j'Z ^{z) COS ^^-j^dz, tj o J I' therefore (j){0) + (f>{l) +0(20 + ... +(f>(^nl-l) = ]j%(z)dz-l{{0)] + ^^^ f%{z)cos^-^dz. It will be seen that this result resembles that of Art. 30-i ; we shall now compare them more closely. By integi-ation by parts ■ I 0(0) cos mz dz = — 4){z) sin mz 10' (z) sin mz dz = — cf) fs) sin mz -\ :, 6' (z) cos mz — -^A 6" (z) cos mz dz. CVintinue this process, and then take the integral between n 7 n 2r7r ^, the limits and )U; put qu tor— - : thus |%(.) cos ^dz = 1 {0' {nl) - 0' (0)1 _ 1, {0'" Od) - 0'" (0)1 + . . . + ^'^^ (0^-^ {nl) - 0^-1 (0)1 (— i\« fill + ^ —„^- 02* (z) cos mz dz. m-^ Jo IN TRIGONOMETRICAL SERIES. 319 Now effect the summation with respect to r, and denote bv S the infinite series 1 +-:ri + -^ + -rz + ... Thus = ]jy(z)dz-l{(nl)-(0)] + El^'("O-f(O)}-S[f"O^O-0"'(O)} + ... + 2^.-1^2. - ,-:2. j^ ^'''(^) COS -j-dz. The fact that this result, up to the last term exclusive, agrees with that in Art. 30i depends upon the property of Bernoulli s Numbers involved in the known formula Cf _ " -"2r-l The last term in the result just obtained gives us an equiva- lent in the form of a definite integral for the remainder after a certain number of terms of the series in Art. 30-i. The property of Bernoidlis Numbers may be established thus. Use the formula for sin 6 which is given in Plane Trifjonometry, Art. 322, take the logarithms, and differentiate with respect to 6; thus we obtain an exiaression for Scot 9 2 S* in which the coefficient of 6"'' is ^r- Again we have TT COt^ = 26'\/-l Thus ^ cot ^ = ^ V- 1 + - ;— the last term can be expanded in powers of 6 by Art. 123 of i\i& Differential Calculus; and by comparing the coefficient of 6-" with that already given we obtain the required formula. 320 MISCELLANEOUS EXAIVIPLES. Let V be any integer less than n. The sum of the series (f){i'l) + ^{vl + l) + ... +(p{nl-l) may be obtained by subtracting the value of (l>iO)+(fi{l)+...+cf){vl-l), from that of <^ (0) + ^ (Z) + . . . + {nl - 1). MISCELLANEOUS EXAMPLES. Change the order of integration in the expression n (fi {a;, y) dx dy. 2a Change the order of integration in the expression '2a rV(4a.f) <^ {x, y) dx dy. J \/{2ax-x^) re rb.v 3. Transform I ^{x,y) dxdy into an integral with J J ax respect to u and v, having given u = y + x, y = tiv; and determine the limits of the new inteo-ral, 4. Transform I \ 4>{x, y) dx dy into an integral with respect to u and v, having given y -\-cx = u, y = uv ; and determine the limits of the new integral. 5. Transform j j (x— y) (y — z) {z - x) dx dy dz into an in- tegral in which u, v, lu are the independent variables, where u^ = xiiz, -=-^ 1- - , w^ = x~ -\- ir -\- z^. "^ V x y 2 ^ MISCELLANEOUS EXAMPLES. 321 G. Prove that U ) --Jo *^ n + af"Y where t = x" and t = i". (See Arts. 203 and 66 ; and transform as in Art. 242.) /T 7J-2 7. Prove that tan ^ log: cot ^ cZ^ = -r^ - Jo 48 8. Prove by transforming the expression from rectangular to polar co-ordinates that the value of the definite integral g-U*+2xVcosa+2/^) ^/.^ ^/^ -g Qonal to Jo Jo |- V^ri^ f sin - j , where i^fsin-j denotes a complete elliptic function of the first order of which sin ^ is the modulus. 9. Prove that I e-x->ncot2^ sin (,,^.2 ^ ,^) ^^ ^ g-^ (.^ ^ ^^ //7rsm2/3^^ ^ 10. Shew that TT f ' tan-^ {« V(l - tan^ x) ]dx = '^ tan"^ " V^ - ? cot"^ 'l(l±^^ ' - 2 w sin (^f tan - j 11- If ./ (B = L ,; n dO, determine the geometrical J sin u ° meaning of the equation y ='xf{sinx). 12. A curve of double curvature revolves round the axis of;r; shew that the surface generated = 27r|v'{(yc/y + ^chy + (/ + z"-) {dxy}. T. L C. 21 822 MISCELLANEOUS EXAMPLES. 13. Shew that /. dx TT a' + bx' + x* 2aV(& + 2a)' , /■ afdx _ ^"""^ Jo d' + hx' + x*~2 TT V(^ + 2a)' assirming that the denominator of the expression under the integral sign does not vanish for any real value of the variable. 14. Find an expression in terms of sines which shall be TT 9 TT TT equal to x when x lies between — ^ and — , and shall 77" be zero when x lies between — tt and — — , or between ^ and TT. 1 ( 1 1 . ' Result. ^ -^ sin 2iC — ^ sin 4a; + ^ sm 6a; — . . . |- 2 f . 1 . „ 1 . . + - ^sm a; — - sm Ja; + ^rzr sm ox — . 823 CHAPTER XIV. APrLICATIOX OF THE INTEGRAL CALCULUS TO QUESTIONS OF MEAN VALUE AND PROBABILITY. 833. We will here give a few simple examples of the application of the Integral Calculus to questions relating to mean value and to probabiliti/. Let <^ (x) denote any function of x, and suppose x succes- sively equal to a, a + h, a + 2h, ... a + {ii — 1) h. Then 4> {a) + 4> {a + h) + <^ {a + ^h) + . .. + [a + (n - 1) h} n may be said to be the mean or average of the w values which (f) (x) receives corresponding to the n values of x. Let h — a — nh, then the above mean value may be written thus, [ (f> (a) + (fi{a + h)+4>{a + 2h} + ... -f [g + ( y^ - 1) /,}] h b — a Suppose a and h to remain fixed and n to increase inde- finitely ; then the limit of the above expression is ^ (x) dx a This may accordingly be defined to bo the mean value of ^ (x) when x varies continuously between a and h. 21—2 ' o24 APPLICATION OF THE INTEGRAL CALCULUS 334. As an example we may take the following ques- tion ; find the mean distance of all points within a circle from a fixed point on the circumference. By this enunciation we intend the following process to be performed. Let the area of a circle be divided into a large number n of equal small areas; form a fraction of which the numerator is the sum of the distances of these small areas from a fixed point on the circumference, and the denominator is n ; then fintl the limit of this fraction when n is infinite. Suppose i\, r„, ... r^ to denote the respective distances of the small areas ; then the fraction required is 1 r 1 n - "' Multiply both numerator and denominator by rAOAi^ which represents the area of a small element (Art. 148), thus the fraction becomes {7\ + 7\^+ ...+7\,]rA0Ar nrAOAr The limit of the denominator will represent the area of the circle, that is, ttc", if c be the radius of the circle. The limit of the numerator will be, by the definitions of the IntegTal Calculus, llr^cWdr, the integration being so effected as to include all the elements of area within the boundary of the circle. Thus the result is IT ■ 1c COS 9 r' (Id dr ^■^ J a This will be found to srive 82c ^ '^ 9 TT 335. The equation to a curve is r = c sin 6 cos 6, find the mean lengtli of all the radii vectores drawn at equal angular intervals in the first quadrant. TO QUESTIONS OF MEAN VALUE AND PROBABILITY. 825 It easily follows, as iii tlie last Article, that the required vican lenqth is I c sin 6 cos 9 dd c , that is, — . TT vr 2 Again, suppose the portion of this curve which lies in the first quadrant to revolve round the initial line, and thus to fjenerate a surface. Let radii vectores be drawn from the ori- gin to different points of the surface equably in all directions: it is required to find the mean length of the radii vectores. The only difficulty in this question lies in apprehending clearly what is meant by the words in Italics. Conceive a spherical surfiice having the origin as centre ; then by equable angular distribution of the radii vectores, we mean that they are to be so drawn that the number of them which fiill on any portion of the spherical surface must be proportional to the area of that portion. Now the area of any portion of a sphere of radius a is found by integrating a^ iismddcpdO within proper limits: see Art. 175. Hence a'^ sin 6 Acf) A9 may be taken to denote an element of a spherical surface, and 27ra^ is the area of half the surface of a sphere. Thus we shall have as the required result d'c sin 6 cos 6 sin 6 dcj) dd 'lira:' the integration being extended over the entire surface con- sidered. Hence we obtain IT 2,1 /-2 ['[%sin''^cos^f/arallel equidistant straight lines, the distance between two consecutive lines being b ; a closed curve having no sin- gular points, whose greatest diameter is less than b, is thrown down on the area : then the chance that the curve TO QUESTIONS OF MEAN VALUE AND PROBABILITV. 327 will fiill on one of the straight lines is -. , where I is the peri- meter of the curve. Let AA be the lonsfest diameter of the closed curve, and assume that the curve is symmetrical with respect to AA'. It is easily seen that we do not alter the problem by sup- posing the point A constrained to fall on a straight line drawn between two consecutive lines of the given system, and meeting them at right angles, for the jDroportion of the favourable cases to the whole number of cases remains the same after the limitation as before. Take two such con- secutive straight lines, and consider one of them, which we will denote by MN; we shall estimate the chance that the closed curve will cross MN, and by doubling the result we get the chance that the closed curve will cross the system. Let A be at the distance x from MN; draw A Y per- pendicular to MN, so that AY = x. Suppose the curve to revolve around A, and it is obvious that in this position of A 26 the chance that the curve crosses MN is -^ , where 6 is the ZTT ang-le between AA' and A Y when the closed curve touches Ax MN; and we may denote by — the chance that A falls be- tween the distances x and x + Ax from MN: thus, as in Art. 336, we obtain finally — r l(f)dx for the required chance. Now \(pdx = X(f) — Ix defy; when a; = we have ^ = tt, and when x = AA' we have = 0; the limits of x are and AA' ; thus l(f)dx = — j xd(j)= \ x d(}>. [' 1 xd = ^lhj Art. 91 ; thus the chance of crossing MN is Jo ■" -T—T : and doubling this we obtain for the required chance — r . S28 APPLICATION OF THE INTEGRAL CALCULUS We assumed that AA' divides the curve symmetrically; but the result will be the same if this restriction be removed. Instead of the expression >,- we shall now have — — ^- where (f)^ denotes the angle between A A' and A Y when the closed curve touches MN at a point on one side of AA', and „ denotes the corresponding angle Avhen the closed curve touches MN at a point on the other side of AA' . Then finally 1 f'^ 1 P If"' instead of — r xd6 we shall have ^— , cc deb, + ---, x deb ■ TrbJo 27r67o ^' 27!*6Jo and the sum of these is — -^ as before. ztto This problem was given as an Example for the particular case of an ellipse in the first edition of the present work ; in the second edition the problem was put in the general form here discussed : a verification by simple reasoning mav be seen in Bertrand's Calcid Integral, page 484. This problem includes that of Art. 33G ; for a rod of length 2c may be regarded as a very slender oval curve of perimeter 4c ; t^hus ^-7- becomes -^ , that is -^ — , that is ^^ . TTO TTO ItTO, IT a 338. A very curious theorem in the Integral Calculus was obtained by Professor Crofton, by the aid of the Theory of Probability, and published in the Plidosophical Trans- actions for 1868; this w^e will now give. The method of the discoverer of the Theorem well deserves the attention of the student, on account of its novelty ; we will however here mainly follow that adopted by Bertrand in his Galcid Inte- gral, which involves nothing but the ordinary principles of the Theory of Probability. 339. An indefinitely large plane area is ruled with parallel equidistant straight lines ; suppose two closed curves fixed in one plane, each completely outside the other, and let them be thrown down on the area; suppose also that the distance between two consecutive parallel straight lines is such that the two curves cannot cross more than one straight line at a time : required the chance that one of the straight lines shall cross both of the curves. TO QUESTIONS OF MEAN VALUE AND TJIOBABILITY. 321) Imagine a string drawn tightly round the two curves, so as to enclose them both, and to form two common tangents ivhicli do not cross ; let l^ be the length of this string. Again, imagine a second string drawn tightly round the two curves, so as to enclose them both, and to form two common tansfents which cross ; let l.^ be the length of this string. Then the required chance is -^ — j~^ , where b is the distance between two consecutive parallel straight lines. For it is seen on investigation that — ^ ex23resses the chance of the boundary formed by the second string being crossed by a straight line; but this includes the cases in which the common tangents are crossed, and not any part of the peri- meter of the two curves : and moreover cases in which both perimeters are crossed are counted twice over. The cases not required constitute the aggregate corresponding to -~ ; and TTO thus by subtraction we obtain the result -"— r^ . TTO SiO. We now apply the general result of the precedino- Article to a particular case ; w^e supi^ose one of the two curves to become an infinitesimal straight line, that is a curve in w^hich the longest diameter is infinitesimal, and the shortest is infinitesimal compared with the longest. Let PQ denote this infinitesimal straight line, and suiDjDose its situation such that PQ produced would intersect the closed curve associated w^ith PQ : w^e proceed to estimate l^ — l^. Of the two ends, P and Q, let P be the more remote from the closed curve. Let PA and PB be the tangents from P to the curve ; let QC and QD be the tangents from Q, so that G is very near A , and D is very near B. Then 4 - /, = AC+ CQ + 2PQ + QD-\- DB-{AP+ PB) ^2PQ + AC+ CQ-AP+BD + BQ-BP = 2PQ - PQ cos a-PQ cos /3, where QPA = a, and QPB = /3. 830 APPLICATIOX OF THE INTEGRAL CALCULUS Therefore in this case the required chance = — ^ (2 — cos a — cos (S). TTO 341. Our object is now to solve this problem : tioo straight lines are drawn at random across a plane closed curve : it is required to find the chance that they ivill in- tersect within the curve. But this will require some develop- ment ; and in the first place we must explain the sense in which we use the phrase a random straight line drawn across a plane curve. Suppose a plane curve thrown doAvn on such a system of parallel straight lines as we have considered in the problems of Arts. 336... 339; and let this process be repeated until a straight line crosses the curve: the straight line which thus first crosses the plane curve is called a random straight line drawn across the plane curve, or briefly a random line. It follows from this definition that unless the curve be a circle random lines will not occur Avith equal facility in all directions with respect to the curve ; for instance, if the curve be an ellipse of great eccentricity random lines will occur parallel to the minor axis with much greater facility than parallel to the major axis. Let us determine the chance that a chord of a curve drawn at random should lie between two assigned directions including an infinitesimal angle d9 ; this may for brevity be described less accurately as the chance that a chord drawn at random should have an assigned direc- tion 6. Let p denote the breadth of the curve measured at risrht angles to the assicjned direction, that is the distance between the two tangents to the curve which are parallel to that direction ; then the required chance is obviously propor- tional to p dO, and so may be denoted by Cp dO, where G is some constant. We may determine C from the circumstance that the sum of the chances corresponding to all directions is, unity, as the chord must have some direction. Thus J Q TO QUESTIONS OF MEAN VALUE AND PROBABILITY. 331 but bv the aid of Art. 01 we see that this becomes Cl=\ 1 ' where I denotes the perimeter of the curve ; therefore G = y • 342. One chord drawn at random is parallel to a given direction : find the chance that it will be intersected by another chord drawn at random. The chance that the first chord should cross an assigned breadth j:) of the curve, which is at right angles to the given direction, within an assigned space dp of p, and fall within the angular distance dO from the given direction is f > that is -^, — . Suppose such a chord denoted by il/iVin a 6 diagram ; and let z denote the chance that it will be inter- sected by a second chord drawn at random. If we throw the curve on the system of parallel straight lines we have, as in Art. 339, the expression — j— for the chance that the chord MN is intersected. This may be con- sidered as the chance of a compound event, namely, the chance that the curve is intersected, and that it is inter- sected along il/i\\ Thus therefore z = ■ — -, — . Hence the chance that the first random chord is MN, and that this chord is intersected by a second random chord, is dpdd niN that IS — - — do dp. 343. We can now return to the problem proposed at the befrinnincT of Art. 341. If Ave sum all the values of the ex- IT pression just given we obtain the chance that two chords o32 APPLICATION OF THE INTEGRAL CALCULUS drawn at random u'ill intersect within the curve: this chance then is j,jJ3fNd0dp. But iMN'diy, between the proper limits, is equal to the area of the closed curve, which we will denote by H ; and IdO between the limits is equal to tt. Thus finally we have for the required chance — „ - . 344. We now proceed to find the chance that two random chords produced will intersect ivitliout the closed curve ; and we begin by finding the chance that the inter- section takes place Avithin a certain infinitesimal area which occupies an assigned position. We may naturally expect that this chance will be proi^ortional to the magnitude of the in- finitesimal area, and independent of its form ; but we will not assume this : the reader may draw the infinitesimal area of any form, as circular or rectangular. Consider first the direction which makes an ano-le 6 with a fixed straight line; let r denote the breadth of the infini- tesimal area, and p tlie breadth of the closed curve, both measured at right angles to the specified direction. The chance that a random chord should have this direction is ^-—j— ; and the chance that with this direction it should cross T the infinitesimal area is - ; the chance of the compound event rdd , is — — . The chance that this intersection occurs within an as- signed portion dr of r is y- , that is • — - — . Let a sti'aight line in the specified direction be denoted by 3INQP, cutting the closed curve at M and N, and the infinitesimal area at Q and P. The chance that a second random choi'd intersects the TO QUESTIONS OF MEAN VALUE AND PROBABILITY. 33o first within tho infinitesimal area is the same as the cliance that it intersects the straight line FQ. Let z denote this chance ; then, by Art. 340, and as in Art. 842, — p {'1 — cos a — cos p) = —J- X z, iro ITU PQ therefore z = — ,- (2 — cos oc — cos /3). Hence tlie chance that there will bo intersection, and that one of the chords will be PQ drcW PQ ,_ _. , ,/, = — T— z = —jf- (2 — cos a — cos p) dr do. Therefore the whole chance of intersection within the as- signed infinitesimal area is ~i JJPQ (2 - cos a - cos /3) dr dO. Now \PQ dr between the proper limits is the infinitesimal area, which we will denote by a ; thus tlie expression becomes J. I (2 — cos a — cos /3) d9. Let -v/^ be the angle which the closed curve subtends at any point of a ; then ^ + oi = -^, so that jS ^-yjr — a and we may put the expression in the form y„ {2 — cosa- COS (->/r-ct)] fZa ; f-' . and this will be found equal to ^ (i^ — sin -vlr). 345. Til us the whole chance of intersection without the closed curve is 2 f J. Ida) {'^ — sin -v^) where dco is put for a, and denotes an element of area ; the integration is to extend over the whole area outside the closed curve. The sum of this chance, and of that found in Art. 343, must obviously be unity ; thus 334- EXAMPLES. therefore I fZw (-v/^ — sin ■\/r) = - — ttO. Here /I represents the perimeter of any closed curve, D. the area, -v//^ the angle which the closed curve subtends at any- external point, d(o an element of area there ; and the integral is to extend over all the area outside the closed curve. This formula in the Integral Calculus constitutes the theorem dis- covered by Professor Crofton. 346. A large number of very interesting problems rela- ting to the subject of the present Chapter will be found in the volumes entitled Mathematical Questions, with their solu- tions. From the Educational Times.... EXAMPLES. If r=f{9) and y=f[-) he the equations to two curves, f(9) being a function which vanishes for the values 6^, 6,,, and is positive for all values between these limits, and if A be the area of the former between the limits 6 = 6^^ and d—0„, and M the arithmetical mean of all the ti-ansverse sections of the solid generated by the revolution about the axis of x of the portion of the latter curve between the limits x = a0^ and x = ad^, 27J- shew that J/ = ^ tt ^> suiwosiug &„ greater than 0^. A ball is fired at random from a gun which is equally likely to be presented in any direction in space above the horizon : shew that the chance of its reaching more than — th of its greatest range is . / 1 1 ] . m ^ V \ 111/ EXAMPLES. 335 !. From a point in the circumference of a circular field a projectile is thrown at random with a given velocity, which is such that the diameter of the field is equal to the greatest range of the projectile : find the chance of its falling within the field. ,, , 1 2 ° Iiesult. -. (V2 — 1), \ On a table a series of straight lines at equal distances from one another is drawn, and a cube is thrown at random on the table. Supposing the diagonal of the cube less than the distance between consecutive straififht lines, find the chance that the cube will rest without covering any part of the lines. Besidt. 1 , where a is the edge of the cube and c CTT the distance between consecutive straight lines. i. Prove that the mean of all the radius-vectors of an ellipse, the focus being the origin, is equal to half the minor axis, when the straight lines are drawn at equal angular intervals ; and is equal to half the major axis when the straisrht lines are drawn so that the abscissae of their extremities increase uniformly. ). An indefinite number of equidistant parallel straight lines are drawn on a plane, and a rod whose length is equal to r times the jierpendicular distance between two consecutive lines is thrown at random on the plane : find the chance of its falling upon n of the 2 straight lines. If n = ?• = 1, shew that the chance is - . 7. Two arrows are sticking;' in a circular tarfjet : shew that the chance that their distance is greater than the radius of the targ^et is *-, - . ° 47r 8. Supposing the orbits of comets to be equally distributed tlirough space, prove that their mean inclination to the plane of the ecliptic is the angle subtended by an arc equal to the radius. Result. ooQ EXAMPLES. 9. A certain territory is bounded by two meridian circles and by two parallels of latitude which differ in longi- tude and latitude respectively by one degi-ee, and is known to lie within certain limits of latitude : find the mean superficial area. 10. A straight line is taken of given length a, and two other straioht lines are taken each less than the first straight line and laid down in it at hazard, any one position of either being as likely as any other. The lengths of these straight lines are b and b' ; it is required to find the probability that they shall not have a part ex- ceedmff c in common. (^a-h-b' + cY (a-b){a-b')' {Canib. Fhil. Transactions, Vol. viil. page 386.) 11. From any point within a closed curve straight lines are drawn at equal angular intervals to the circumference: shew that the mean value of the squares on these straight lines is the product of — into the area of the curve. 12. A messenger M starts from A towards B (distance a) at a rate of v miles per hour, but before he arrives at JS a shower of rain commences at A and at all j^laces occu- pying a certain distance z towards, but not reaching beyond, B, and moves at the rate of w miles an hour towards A ; if M be caught in this shower he will be obliged to stop until it is over ; he is also to receive for his errand a number of shillings inversely propor- tional to the time occupied in it, at the rate of ?i shil- lings for one hour. Supposing the distance z to be unknown, as also the time at which the shower com- menced, but all events to be equally probable, shev/ that the value of il/'s expectation is, in shillings, nv (1 u u (u ■\- v) , u-{-v\ a 2 y V w ] EXAMPLES. 3o7 13. A large plane area is ruled with parallel equidistant straight lines, and also with a second set of parallel equidistant straight lines at right angles to the former set ; a thin rod is thrown at hazard on the area : find the chance that the rod will fall across a line. (See History of...ProhahiUty, page 347.) 14. Suppose a cube thrown on the system of lines described in the preceding Example : find the chance that the cube will fall across a line. (See History of... Probability, page 348.) 15. Let there be a number n of points ranged in a straight line, and let ordinates be drawn at these points ; the sum of these ordinates is to be equal to s ; moreover the first ordinate is not to be greater than the second, the second not greater than the third, and so on : shew that the mean value of the ?-*^ ordinate is 5 fl 1 1 1 ) n [n n — 1 n — 2 ?i — ?• + Ij (See History of ...Probability, page 545.) IG. Verifv the formula in Art. 345 by direct intefrration in the case where the closed curve is a circle. T. I. c. 22 SS8 CHAPTER XV. CALCULUS OF VARIATIOXS. Maxima and Minima of integrals involving one dependent variable with fixed limits. 347. The theory of maxima and minima values of given functions is fully considered in works on the Differential Calculus. If, for example, y denotes any given function of an independent variable x, then we can find the value or values of X which make y a maximum or minimum, or we can shew that there are no such values in some cases. We are now however about to consider a new class of maxima and minima problems. Let ;/ denote a function of x which is at present undetermined ; and let V denote a given 7 y-7- function of a*, y, -^, -~,... Suppose we wish to find the relation which must hold between x and y in order that the integral I Vdx, taken between given limits, may have a maxi- mum or minimum value. We cannot here efifect the integra- tion, because y is not known as a function of o:, and therefore V is not known as a function of x; thus the ordinary methods of solving maxima and minima problems do not apply. We require then a neAv method, which we shall now proceed to explain. 348. The department of analysis to which we are about to- introduce the student is called the Calculus of Variations ; its object is to find the maxima or minima values of inte- gral exj)ressions, the exj^ressions being supposed to vary by CAL',ULTTS OF VARIATIONS. 339 assigning d'l^evont forms to the functions denoted hj the de- pendent variables. It will be seen, as we proceed, that the method of finding these maxima or minima values is ana- logous to that of finding ordinary maxima or minima values by the Differential Calculus. » 340. It will be useful to recur to the method given in the Differential Calculus. The student will remember that the terms maximuin and minimum are technical terms, which are defined and illustrated in treatises on the Differential Calculus ; and they are used in mathematics in the sense there assigned to them. Mistakes are frequently made by confounding a maximum, value in the technical sense of the word maximum, with the greatest value in the ordinary sense of the word greatest. Suppose y a given function of an independent variable x; then if an indefinitely small change is given to x, in general an indefinitely small change is consequently given to y, which is comparable in magnitude with that given to x. The pro- cess of finding a maximum or minimum value of 7/ may be said to consist of two parts. First we determine such a value of x that an indefinitely small change in it does not produce in y a comparable indefinitely small change, but a change which is indefinitely small compared with that of x. In the second place, we examine the sign of this indefinitely small change which is produced in y by the change of x; and for a maximum this sign is to be necessarily negative, and for a minimum positive. We may therefore describe this process briefly thus; we make the terras of the first order in the change of the depend- ent variable vanish, and we examine the sign of the terms of the second order. We shall pursue a similar method with the problem wliich we have now to discuss ; we confine ourselves, however, at present entirely to the first part of the process, and shall hereafter recur to the second part. S'yO. We have first to explain the notation which will be used. Let x denote an independent variable, ?/ any func- tion of a-, and ^^, -A,-., the differential coefficients of y 340 CALCULUS OF VARIATIONS. with respect to x. We shall use hj to denote an indefinitely small quantity which may be any function of x; and if u denote any quantity whatever which depends on y we shall denote by da the increment which 2t receives when y is changed into y + hj. Thus, for example, consider the ditt'erential co- efficient -^ ; when y receives the increment S_y this differen- tial coefficient receives tlie increment -y- - , so that by S — we mean — ^ • It is often convenient to use the symbol p ax for -,- : and so also hp is a convenient symbol for -y^ . dx' -^ dx Acfain, consider the second differential coefficient -fr, : when *= ' dx y receives the increment Zy this second differential coefficient receives the increment -7-25 ^'^^ ^^ the second differential coefficient is often denoted by q we may conveniently use S^ for — r-^ . Similarly r and s may be used for the third and CLjC fourth differential coefficients of y respectively, and hr and Ss for -y-v and ~r4 respectively : and so on. ax dx The differential coefficients are also often denoted by y'y y"' y"'---5 ^^^"I ^^^^^ %' V' W'>--- "^^y ^e used as equi- valent to hp, hq, h',... resj)ectively. 351, The introduction of the symbol S is due to La- grange. The student will see that this symbol resembles in meaning the symbol d, which is used in the Differential Cal- culus. Both dy and hy express indefinitely small increments; dy however is generally used to denote the change in value of a given function consequent upon a change in the value of the dependent variable, hy is used to denote the change made by ascribing an arbitrary change to the form of a function. The quantity denoted by dy is called the variation of y. CALCULUS OF VARIATIONS. 341 7 72 352. Let V denote a given function of a;,y,~ , -^-^ , ... ; and let U= 1 Vdx, -wliere x^^ and x^ are supposed to denote fdven limits. The value of Z7 cannot' be found so lonf^ as we do not know what particular function y is of a; ; but without knowing this we are able to obtain an expression for the increment made in U by ascribing the arbitrary increment 8i/ to 1/, from which important inferences can be drawn. Suppose V=(f) {x, y, y, y", y", ...); then by definition gF= -J-;, ~T~J' j •■> ^^® complete diiiorential coefficients. Hence finally + S>l^[R~...\-Sq,[R-...], + , {^^[.^ dP d'Q d'R , \. , Here we have adopted some obvious simplifications of nota- tion ; thus we use Si/^ for {Si/)^, and Sp^ for (-p) , a-nd so on. 353. The value oi SU may be denoted thus, SU=H^-E^+ r KSy dr, J Xq where H. denotes a certain afjonreorate of terms in which x, is put for X, and H^ a similar aggregate of terms in which x^ is put for X ; these aggregates do not involve any integrations. Also dx dj? dx^ 344 CALCULUS OF VARIATIONS. Since JTj — H^ involves only the values of the variables at the limits, we shall sometimes speak of H^ — H^ as the terms at the limits. 354. We can now determine the conditions which must hold in order that U may have a maximum or minimum value. For, in order that U may have a maximum or mini- mum value, h U must vanish, whatever 5?/ may be, provided only that it is an indefinitely small quantity. This requires that 7^=0 and 11^-11^ = 0. For if K is not always zero, it will be in our power to give such a value to Sy as will make B U positive or negative at our pleasure, and not zero. Suppose, for example, that the highest differential coefficient of Zy which occurs in H^ — H^ is the w*^ Put hy = a{x — oc^Y" {x — x^Y", where a is a function of X which is indefinitely small, and is at present undeter- mined. Then this value of 8y makes H^ — H^ vanish, so that oZ7 reduces to I KZy dx. Now take a such that it is always ^ .To _ positive when K is positive, and negative when K is nega- tive ; then SC/" is necessarily positive. And if the sign of a be changed, SZ7 is necessarily negative. Thus if K is not always zero, it is in our power so to take hy as to make h U positive or negative at our pleasure. Hence for a maximum or minimum value of U we must have /v = 0; and then I KSydx vanishes, and therefore also TTj — H^ must = 0. 355. The student has now become acquainted with the essential features of the Calculus of Variations; these are (1) the reduction of BU to the form H^ — H^+ \ Khydx, J Xq (2) the principle that K must vanish in order that U may be a maximum or minimum. Although the subject admits of considerable development, by various extensions of the problem we have considered, still the two results we have already obtained are the chief results. and therefore in -rr the differential coefficient -y^ will CALCULUS OF VARIATIONS. 845 S5G. We now proceed to examine more closely the nature of the two conditions K= and 11^ - IT^ = 0. The equation K = is Avhat is called a diferential equa- tion. Suppose that ~^ is tlie highest differential coefficient which occurs in V; then this will in general occur in R also, —-^ the differential coefficient -7^ dx ax' occur, and this will be the highest differential coefficient which occurs in K, so that the differential equation /i=0 will be of the sixth order. And in general the order of the differential equation is twice the order of the highest differential coeffi- cient which occurs in V. It is shewn in treatises on Differential Equations that the solution of a differential equation involves as many arbitrary constants as the number which expresses the order of the dif- ferential equation. We must now shew how the arbitrary constants which arise from the solution of the equation K= are to be determined, so that a definite result may be ob- tained. The condition H^ — H^=0 serves for this purpose. Two cases may arise. (1) Suppose that no conditions are imposed by the pro- blem on the values of y and its differential coefficients at the limits of the integi-ation ; then Sy^, Sy^, Sp^, Sp^,... are all arbi- trary quantities, that is, we have it in our power to suppose any indefinitely small values we please for these quantities ; for example, we may suppose that as many of them as we please are zero. Since %, , 8y^, Bp^, Sp^,... are thus all arbi- trary, in order that H^ — IT^ may certainly vanish, the coeffi- cient of each of the arbitrary quantities must vanish. This furnishes for determining the constants as many equations as there are constants. (2) Suppose that conditions are imposed by the problem upon the values of 7/ and its differential coefficients at the limits of the integration; then Bi/^, By^, Bp^, Bp^,... are not all arbitrary, for some of them can be expressed in terms of the 346 CALCULUS OF VARIATIONS. rest by means of the given conditions. Let as many as pos- sible of the quantities 87/ ^, Sy,,, Bj)^, Bp^,... be eliminated from H^—IIf^, and then the coefficients of those which remain must be equated to zero. The equations thus obtained, together with those which express the given conditions, will form a system equal in number to the number of constants, and therefore will serve to determine those constants. 357. The principal difficulty in examples consists in the solution of the difterential equation K = 0, and this difficulty is frequently insuperable. We will now shew that when V does not explicitly con- tain the independent variable, one step in the solution of the differential equation can always be taken. It will be suf- ficient for practical purposes to confine ourselves to the case in which V involves no differential coefficient of y higher than the third. Since V is supposed not to involve x explicitly, we have for the complete differential coefficient of V dx dx dx dx dx And by supposition dx dx^ dx^ .(1). Thus dV _ dP dy p dp d^Q dy ^ dq d^R dy j^dr dx dx dx dx dj? dx dx dx^ dx dx Now dP dy jy dp _ d p dif dx dx dx dx dx * d^Q dy r\dq_d (dQ dy „ d^y]^ dx* dx dx dx \dx dx dx^\ * d'R dy Tfdr_d ((£R dy_dRd^y ^ d'y) dx^ dx dx dx\dx^ dx dx dx^ dx^) CALCULUS OF VARIATIONS. 347 Hence, by integ-ration, dii d Q di) „ dij d'R d>i dU d\ij d'y ^ dx dx dx dx dx dx dx dx ax where C is an arbitrary constant. The highest differential .. . , • /.x • d'li ,. , . d'R coefficient that can occur m (2) is ~ which occurs in -^ ; thus (2) is a differential equation of the fifth order, which is a first integral of the equation (1) which is of the sixth order. Particular cases may be obtained by supposing R or Q or P to bo zero. For example, the most useful case is that in which V involves only y and ^- ; so that (1) becomes dx and (2) becomes 358. The differential equation /r=0 is also susceptible of one integration when V does not contain the dependent variable. For then N= 0, and the equation becomes dP_d;Qd^_ ^^ dx dx' dx^ and therefore dQ^d'R ^ dx dx r xx r (J^^ 359. We know that I Vdx= 1 V -j- dy, supposing the limits- of the integration with respect to y taken to corre- spond to those of the integration with respect to x. And the differential coefficients of y with respect to x may be expressed in terms of the differential coefficients of*- with respect to y. f dx Thus in V J- dy we may regard y as the independent vari- able, and X as the dependent variable, and proceed to find the maximum or minimum value of the integi-al in this new 348 CALCULUS OF VARIATIONS. form. We may feel a j)viori certain, as the problem is really not changed by this change of the independent variable, that we shall obtain the same result as if we had kept the original independent variable. Hence the cases considered in Arts. 357 and 358 may be seen to coincide. SCO. Again, let us suppose that V involves only p and q. Then the differential equation K=0 reduces to dx dx^ ' therefore, by integration, ^ = f+^.- Also ^^P^ + Q'k dx ax dx dx dx dx dx therefore, by integration, Here (7, and C^ are arbitrary constants. In this case the differential equation K=0 is of ih.Q fourth order, and the result we have obtained is a differential equation of the second order ; so that we have effected two steps in the integration of the differential equation K=0. 861. We shall now proceed to consider some examples ; as we have already intimated we confine ourselves entirely to the first part of the process for finding maxima and minima values ; see Art. 349. 302. To find the shortest line between two points. This example is introduced merely for the purpose of illustrating the formulae, as it is obvious that the result must be the straight line joining the two points. CALCULUS OF VARIATIONS. 34-9 Here F=V(1+/) and U= T^/il +p'') dx. J Xq Thus F involves onlyp, auJ the equation K=0 reduces to — = ; hence P must be a constant, that is, . ,/, — jr must dx vCj-+p; be a constant. This shews that p must be a constant, and therefore the required curve must be a straight line. In this case E^ - H, =^r^^ - vIT+pT) • If now tlie two points are fixed points, we have Sij^ =- and gy^ = ; thus //j - H^ vanishes. Then the value of p must be" found from the condition that the straight line must pass through the two fixed points. Suppose however that the ordinates of the two points are not fixed ; the ahscissce are fixed because x^ and x^ are taken to be invariable. In this case %, and S?/^ are arbitrary ; and therefore i/^— ZT^will not necessarily vanish unless the coeffi- cients of Sy\ and 3?/, vanish. This requires that p, and p^ should vanish, and as p is a constant by supposition this con- stant must be zero. Thus our formulae are consistent with the obvious fact, that when two straight lines are parallel the shortest distance betw^een them is obtained by drawing a straight Line perpendicular to them both. 363. To find the curve of quickest descent from one given point to another. The following is a fuller statement of the meaning of this problem. Suppose an indefinitely thin smooth tube con- necting the two points, and a heavy particle to slide down this tube ; we require to know the form of the tube in order that the time of descent may be a minimum. The problem is known by the name of the hrachistochrone ; it was first proposed by John Bernoulli in 1696, and gave rise to the Calculus of Variations. We shall assume that the required curve lies in the ver- tical plane which contains the two given points. Let the axis of y be measured vertically do^vnwards, and take the axis of 350 CALCULUS OF VARIATIONS. X to pass through the upper given point. The particle is supposed to start from rest, and then by the principles of mechanics the velocity at the depth ?/ is i^{2gy). Thus the time of descent is / — ,,^i-\ dx. We may then take J 'To \/(2/7.5/) -^ Here V involves only y and jj ; so that, by Art. 357, for a minimum we must have that is, therefore V=Pp+C, p V(l+/) _ = G. + ^; V[2/(l+/)} Hence ?/(l +^/) = a constant = 2a suppose ; y . therefore therefore / = 2a-' y dx dij y V a-yj ^{2ay-fj' therefore x = a vers" - — \J{2ay — y^) + h, where h is another a constant. This shews that the required curve is a cycloid with its base horizontal, its vertex downwards, and a cusp at the upper point. We may suppose the origin at the ujDper point so that x^ = 0, and then 5 = 0. Ilerell^-H,= phf 1 phf V(2a) {(jP%)i-(i^^.'/)ol' As we suppose both the extreme points fixed hy^ and Zy^ vanish, and therefore H^ — H^ vanishes. \ CALCULUS OF VARIATIONS. 351 The constant a must be determined by the condition that the cycloid shall jDass through the lower given point. Suppose however that only the abscissa of the lower point is given, and not the ordinate. Then, as before, //„ vanishes, and II, = - -,,/[ -. Now S//, is arbitrary, so tbat in order that ' V(-«) //j may vanish, we must have j)^ = ; tlius the tangent to the cycloid at the lower limiting point niust be horizontal. This condition must be used in tliis case to determine the constant a. 3G4. We may modify the preceding problem by sup- posing that the particle does not start from rest, but starts with an assigned velocity. In this case we will suppose that the axis of x is not drawn through the upper point, but is so taken that the velocity at starting is that which would be gained in falling from the axis of x to the upper fixed point. The solution remains as before ; the cusp of the cycloid is however no longer at the upper fixed point, but in the axis of x. This might have been anticipated. For let ACB be an arc of a cycloid, having its cusp at A; then this is the curve of quickest descent from rest at A to B, and there- fore CB must be the curve of quickest descent from C to B, starting with the velocity at C. 8G5. To find the curve connecting two fixed points such that the area between the curve, its evolute, and the radii of ■';urvature at its extremities may be a minimum. By Art. 157 the expression which is to be made a mini- mum may be taken to be Here V involves only j) and q ; and therefore, by Art. SCO, for a minimum we must have F= Qq + Cj) + C.,, that is, (1±£l = Jl±fl + c,p + C, ; 1 9. therefore ffl^±^^)i^^> \\ S52 CALCULUS or VARIATIOXS. By integration G,i^u^p^~f^ = ^x+C,.. (1). Also _^^__,j/-^=9^; therefore by integration, C^tan"^/) ^^ ~ = ^^y + constant add Cjj to both sides of this equation, and we have 0,tan-j)4-^i^^^ = 42/+a, (2). Eliminate tj},n~'jj from (1) and (2); thus therefore ^/[l +/) = ^rT77^~ ^' Avhere B is such that 4-Y This is a differential equation for determining the required curve. T. L c. 23 Vv So-i CALCULUS OF VAEIATIONS. Integrals with limits subject to variation. 367. We have now sufficiently explained and illustrated the method of finding the maximum or minimum value of an integral expression involving one independent variable, when the limits of the integration are supposed invariable. We shall proceed to some extensions of the problem ; and we begin by considering the modification which arises from sup- posing the limits of the integration variable. Suppose, for example, that we have two given curves in one vertical plane, and that we wish to find the curve of quickest descent from one of these curves to the other, the particle starting with the velocity obtained in falling from a given horizontal straight line. Here we have to find the point at which the particle is to leave the upper curve, and the point of the lower curve towards which it is to proceed, as well as the path which it is to describe. We have therefore to effect more than in the examples hitherto considered, and we shall now explain how we may proceed. We know, from what has been already given, that the curve must be a cycloid with its base horizontal and a cusp on the given horizontal straight line. For suppose any other curve drawn from any point in the upper curve to any point in the lower ; this curve cannot be that of minimum time, for we know that, without changing the extreme points, we can find a curve of less time of descent than this curve, namel}^ a cycloid with its base horizontal, and a cusp on the given horizon- tal line. Since then we know that the required curve must be such a cycloid, the part of the problem which depends on the Calculus of Variations may be considered solved; and we may investigate, by the ordinary rules for maxima and minima, the position of the particular cycloid for which the time is a minimum. In fact, taking any arbitrary initial and final points, we may find the equation to the cycloid passing through these points ; then the time of descent will become a known function of the co-ordinates of the initial and final points, and we may determine for what values of these co- ordinates the time is a minimum. CALCULUS OF VARIATIONS. 355 S6S. We have shewn in the preceding Article that it is not absolutely necessary to make an}'- moditication in our for- muhc in order to include the case in which the limits of the integration are supposed to be susceptible of change ; for the process already given, combined with the ordinary rules of the Differential Calculus, would enable us to solve any ex- ample. It is however convenient to bring together all that is wanted for solving such examples, and accordingly we shall now supply the requisite modification of our original for- mulce. As before, let rxi U= Vdx. •_ P J Xa Suppose that in addition to the change of ;/ into y + ^y the limits x^ and x^ are changed into x^-\-dx^ and x^-\-dx^ respectively. In consequence of this change of limits f/" re- ceives the increment rx(t+dxa Vdx- Vdx, J Tn rxi+dxi fxo+dxo J Xi •' Xo that is, neglecting squares and higher powers of dxj^ and dx^, U receives the increment V^dx^ — V^dx^, If we annex this to the expression already given for BIT, we shall obtain the complete change in U consequent upon the variation of y, and the change of the limits. 369. If no condition is imposed on the limiting values of the co-ordinates, the additional terms just obtained, V^dx^ - V^dx^, can only be made to vanish necessarily by supposing V^ = and Fp = 0. We thus introduce two new equations in ad- dition to those which are obtained from H^— H^ = {)\ and at the same time we have two new quantities to determine, namely, x^ and x^. However, a more common case is that in which the limiting values have to satisfy given equations. Such a case we have already indicated in Art. 367, where a 23—2 356 CALCULUS OF VARIATIONS. curve is required, the extreme points of which are to lie on given curves. We will consider that limit of the integration for which the quantities are distinguished by the subscript 1. Let Y=y+hy, then if there had been no change of the limit, the extreme values of the variables would have been x^ and y^ before variation, and x^ and Y^ after variation. If however x^ is changed into x^-\-dx^, we have Y^ changed into that is, neglecting squares and higher powers of dx^ we have Y^ changed into Y^-\-{-j-\ dx^, that is, neglecting the product ^■p^dx^y'mio y^-^hj^-\-i^ \ dx^. Supposing then that the \CtX/ ^ given relation which is to be satisfied by the extreme values is we must have 3/i = ^ (^i)' and also to the first order. Thus 8y.={t' (-)-!}/-. This gives a relation between hj^ and dx^, so that we can eliminate one of them from the complete value of hll. Similarly, the relation can be found between hj^ and dx^. In geometrical problems (^J is the tangent of the inch- nation to the axis of x of the straight line which touches the required curve at the limiting point ; and y^' [x^ is the tan- CALCULUS OF VARIATIONS. 357 gent of the inclination to the axis of x of the straight hnc ■which touches the givox curve at that point. A particular case may be noticed which is sometimes useful. Suppose the comjilete change of y^ is to be zero ; this gives 8y^ + (—) dx^ = ; similarly if the complete change of 2/0 is to be zero, hj^ + (J^^ dx^ = 0. 870. We may illustrate the preceding Article by a figure. Let AB represent the required curve, and MEN the given curve on which the extremity B of the required curve is to lie. Let A'B' represent the curve derived from AB hy ascribing the variation hj to each ordinate y. Draw BG and B'C'D parallel to the axis of y, and BD parallel to the axis of .r. Then ultimately BC=By^, BD = dx^, B'D = f'{x;)dx^, C'D^T^ dx^. Hence B'C = l-yjr' (x^) — {-~\ y dx^. Thus the geometrical interpretation of our process is that if we reject quantities of a higher order than those we retain, we have B'C = BO ultimately. 358 CALCULUS OF VARIATIONS. 371. Let us now consider the case of the brachistochrone problem which has been enuuciated in Art. 367. Let the notation be as in Art. 363. Then + P % Lvij/(i+/)}. phj X LV{3/(i+i/)i. \y-'^>^'- dP As before from the equation N— -r- = we deduce thus V{2/(l+/)}=V(2a); Let us suppose that the equation to the fixed curve from which the particle is to start is F=%(2!), and that the equation to the fixed curve at which the particle is to arrive is Y= yjr (X). Then by the j)reccding Article we have %i = {^'G'^) -P]i dx„ Sy„ = [x{x) - p]o dx,. Thus the value of SU can be put in the form BJJ = \dx^ — \dxg ; wbere and similarly V(2a) 1 ^0 V(2a) {l+p,^'{x;)}, {l+PaX'Wl- Since dx^ and dx^ are arbitrary, Sfywill not necessarily vanish unless X^ = and \ = 0, Thus l+2^i^'('^i)=0 and l+i)o%Vo) = 0; . CALCULUS OF VARIATIONS. 359 and these shew that the cycloid must cut each of the two fixed curves at right angles. 372. We have hitherto tacitl}'' assumed that the function V does not involve the limiting values of the variables or of the differential coefficients. Suppose now however that V does involve x^, x^, t/^, y^,2'>o,l\>--- (1) Suppose that x^ and x^ are not susceptible of any change. When y is changed into y + hy, besides the varia- tion we have already investigated, V will receive an addi- tional variation arising from the change in yo,yi,--. which occur explicitly in V. These additional terms in 6 F are and consequently the following additional terms occur in Now Sj/g, By^, Bp^, S;:»,, ... ate not functions of the variable X, but only of the limiting values of x; we may therefore bring these quantities outside the integral sign and write the additional terms thus, ha T- (^^+ hi -j-dx + dpj -j-dx + Thus the occurrence of these additional terms will not affect the reasoning by which it is shewn in Art. 354 that we must have K = in order that U may be a maximum or minimum. These additional terms must be annexed to the expression H^—H^, and the whole then made to vanish. Since the relation between x and y is supposed to be found from the equation K=0, the expressions under the integral signs in these additional terms become definite functions of x, so that the integrations which are indicated can be effected, at least theoretically. 360 CALCULUS OF VAKIATIONS. (2) Suppose that x^ and x^ are also changed, and let them become x^ + dx^ and x^ + dx^ respectively. Then V receives the additional increment where dV' dx^ and ' dV dx^j 'dV' _dx^_ dx^ + dV dx 1-1 dx^, indicate complete differential coeffi- cients ; that is to say, we are to remember that x^ occurs implicitly in y^, p^, ..., and similarly for x^. Thus besides the additional terms we have already given SZ7 receives the increment dx, J Xa d_r dx + dx^ Xi ^0 dV' d X. dx, and this expression must be annexed to the aggregate formed of H^ — Hq and the additional terms already given. 373. For an example we will take another modification of the brachistochrone problem. Suppose two given curves in the same vertical plane, and let it be required to find the curve of quickest descent from one of these to the other, the motion commencing at the first curve. Let the axis of y be measured vertically downwards; let 3/0 be the ordinate of the starting point, then when the ordinate is y the velocity is ^J[2g {y—y^]. Thus we may take We have then to change y into y —y^ in the solution of Art. 371, and to add to the expression there given for SU the terms found in Article 372. Here V= ^^t ^ ; so that y„ is the only limiting value CALCULUS OF VARIATIONS. 361 which occurs in V. We are therefore to add to the former value oi BU dV' Lf^^^oj dx; and dV dx, OJ dVfdji Hence by Art. 371, after putting A' = 0, we have SU= \dx^ - X^dx^ + \Bj/o+ (-£ ) ^'^0 where \ and \ have the values assigned in Art. 371 Now in the present case dP dx' ^'^dV dx, dt/o dy~ therefore ^^dx = P,-P, =%^ ; and %o + ( / ) dx^ = -^^ {x^ dx^, as in Art. 371. Thus W=^ \dx^ - \dx, + ^^ {p, -r- p^ dx^ V(2a) -V(2^^^+M'(ac?^o Then by equating to zero the coeflficients of dx^ and dx^ we have 1 +p,f (x^) = and 1 -\-2:>jc (xj = 0, so that X (^q) = ^' (^i)- Thus the cycloid cuts the lower fixed curve at right angles, and the tangent to the upper fixed curve at the initial point is parallel to the tangent to the lower fixed curve at the final point. 362 CALCULUS OF VARIATIONS. Integrals with tiuo dependent variables. 874. We have hitherto supposed that F is a function with only one dependent variable ; let us now suppose that V is a function of two dependent variables. Let F be a function of x, y, z, and the differential co- efficients of y and z with respect to .r ; let U=\^'Vdx, and let us investigate the variation in the value of ?7when y and z receive variations. By proceeding as in Art. 352 we shall obtain the follow- ing result, hU=^H^-H, + J,-J,+ \''\Khj + Uz)dx, where the symbols have the following meanings : hj, as before, denotes an arbitrary variation given to y, that is, hy is an indefinitely small arbitrary function of x ; K, as before, denotes dJ[_ddV d^d]^_ dy dx dy dx dy" -where ^— , i-r, -i-„,..- are partial diffei'ential coefficients, dy ay ay and -^ -,-, , -T-T.-m,--' are complete differential coefficients dx dy dx dy relative to x ; hz is an arbitrary variation given to z, that is, Zz is an in- definitely small arbitrary function of x ; L is relatively to z the same as K relatively to y, that is, ~ dz dx dz (?^ dz" //, - H^ has the meaning already given, and J^ - J^ is rela- tively to z the same as H^ - H^ relatively to y. CALCULUS OF VARIATIONS. 3G3 875. "We now proceed to find a maximum or minimum value of U on the suppositions of the preceding Article. (1) If 7/ and z are independent, in order that hU may certainly vanish we must have K=Q and L = 0; and also ir,-iT, + J,-Jo^o. The values of y and z in terms of x must be found by solving the differential equations K=0, L = 0; and the arbitrary constants which occur in these solutions must be determined by equating to zero the coefficients of the arbitrary quantities %„, Bij^, (S ^^] , ... Bz^, Bz^, (B -j^j , ... which occur (2) Suppose however that y and z are not independent, but that they are connected by the relation <^ (x, y, z) — 0, which is always to hold. Since this relation is supposed to hold alwavs, we have also (ji^x, y + By, z + B2) = 0; and therefore ultimately i^By+^Bz^O. ay ^ dz rxi Thus the integral I {I^By + LBz) dx becomes r d(f) - By dx, I. dz . and in order that this i dition may vanish K L . d(f> d -^ — 7 77T~, — 'J I „'2\ > let (f) [x, y,z)=0 be the equation to the surface on which the line lies. Then by the preceding Article we have, as the con- dition for a minimum, d y' d z' d^ d^ dy dz Let s represent the length of the arc of the curve ; then y _ dy , / dz Thus the above equation may be written d\j d^z ds^ _ ds^ /^x d') dx, and the length is V(l +i^^) d^- J Xq Xo Let F= ^ y V(l +/) + a V(l H-/), then we require a maximum value of Vdx. Here by Art. 357 we must have V=Pp+C„ that is, therefore -^^ = &(7,; theretore 1 + p = ■ ' .,, .,.j , and therefore CALCULUS OF VARIATIONS. 8G9 dx 10 hence x = A\og[y + B+ sj[{y + Bf - A"]] + C,, where C^ is a new constant, and A = bC^ and B = ah. This equation shews that the required curve is a catenary. If the ends of tlie required curve are supposed fixed, the terms depending on the limits vanish, and the constants A, B, C^ must be determined by making the catenary pass through the fixed points and have a given length between them. Suppose however that instead of being fixed the ends are only con- strained to lie on fixed curves. By proceeding as in Art. 871 we obtain the following limiting terms : V,dx^ - V,dx^ + P,%i - PoS^o. Consider the terms with the suffix 1 ; we have V^dx^+P^Si/^, that is, (I + .) V(l + i^n d., + (f ■ + a) -0^^ . Now supposing y = -\^{x) the equation to the fixed curve, we have hy^ = [^'{x^) —p^ dx^, so that the term reduces to ^^ + "^ [i-\-v,^'{:x,)]dx,. To make this vanish we must have 1 + ^i'»/^'(^,) = 0, for 7/^-\-ab cannot vanish, as then x^ would be impossible. A similar result holds at the other limit ; and thus it appears that the catenary must cut the fixed curves at right angles. 881. Given the surface of a solid of revolution, to find its nature that the solid content may be a maximum. Take the axis of x as the axis of revolution. Then the surface is Stt 2/V(1 +i^^) ^-^i ^^^ ^^^ volume is tt I y dx. Let V=y" + ay>s/(l +j)^) ; then we have to find a maxi- mum value of 1 Vdx. Here by Art. 357 we must have J Xq T. J. c. 24 o70 CALCULUS OF VARIATIONS, that is, y^ + ay V(l +/) = ,,ff ... + C, therefore ^ + -j^^ = C. This is a differential equation to the curve which Avould by- revolution generate the required surface. Supposing that the ends of the generating curve are required to pass through fixed points, the terms at the limits vanish. If either of the fixed points is on the axis of revolution, the value 1/ = is to satisfy the equation to the curve ; thus (7 = 0. Then the general equation reduces to this gives a circular arc as the generating curve. 382. Given the mass of a solid of revolution of uniform density, required its form so that its attraction upon a point in its axis may be a maximum. Let the axis of x be taken as that of revolution, and the position of the attracted point as the origin. Let the solid be divided into indefinitely thin slices by planes perpendicular to the axis of x. If y represent the radius of a slice, x its distance from the attracted point, k its thickness and p its density, the attraction is (see Statics, Chapter Xiii.) 27roK -fl - "" ZTTpK { ^{x' + f Therefore the whole attraction of the solid is and the mass of the solid is X, irp j y-dx. CALCULUS OF VARIATIONS. 871 Thus let V= 1 — yj-^ 57 + ail"; then we have to invcsti- gate a maximum value of Vax. Xa dP The condition N — 7- + = reduces here to N = 0, ax that is, 2ay -\ ^ — -, = ; {x^ + y^f therefore 2a (a;' + y'f- + x = 0. If we suppose the limits a;, and x^ susceptible of change we have the limiting terms V^dx^ — Vjioc^ ; and to make these vanish we must have Fj = and Vq = 0; this leads toy^ = and 1/^ = 0. Thus the solid must be formed by the revolution round the axis of x of the whole closed curve determined by the equation 2a {x^ + y^ ^- a; = ; the value of a must be found from the condition that the mass, and therefore the volume, is given. Douhle Integrals. 382. We shall now consider the problem of finding a maximum or minimum value of a douhle integral; and we be- gin by finding the variation of a double integral. Let z he a function of the independent variables x and y at dz present unknoAvn ; let F be a given function of x, y, z, -j- /7f C^L fill and -T^; let Z7= I I Vdxdy; the integration is supposed effected with respect to y first, and the limits y^ and y^ are supposed given functions of x. It is required to determine what function z must be of x and y in order that U may have a maximum or minimum value. Let hz denote an indefinitely small arbitrary function of x and y ; let SF denote the variation made in F when z receives the variation hz, and let 8 U denote the variation in U; then we have first to obtain an expression for h U. 24—2 372 CALCULUS OF VARIATIONS. Let L denote the partial differential coefficient of V with respect to z, M the partial differential coefficient of V with dz respect to 7- , and N the partial differential coefficient of V dz with respect to -j- ; then we have dx dy ' where, as heretofore, we confine ourselves to the first power of the indefinitely small quantities. Hence The value of S F may be written thus ; and therefore The differential coefficients with respect to x and y which are here indicated are complete differential coefficients. Also [''' f "' ^ {Mz) dxdy = [''' [{Mz)^ - {mz\] dx, where (NSz)^ denotes the value of NSz when y^ is put for y, and {Ndz)^ denotes the value of N8z when y^ is put for y. And by Art. 216, where (MBz)^ denotes the value of MSz when y^ is put for y, and {M8z)^ denotes the value of MBz when y,, is put for y. CALCULUS OF VARIATIONS. 873 Therefore ^ ["' ^ {^ISz) dx dy aro -' 2/0 ^"^ Mhzd}j) -(r'lIBzdj/) 2/0 /x^Xi \' l/o J x-=-x„ Therefore S L = 1 [^ ~ 'dx ~ dTj '^ + fj' \{mz), - {Nhz)}^ dx + {\''Mhzd^ -(T'lmzdu) -[^,^mz)^^-i^Mlz)M^x. If the limits y^ and y^ are constants, the terms in the last line vanish. Of the four terms which compose SfT" it will be seen that the second is similar in character to the third, and might be expressed in a similar manner. We have supposed that the limits of the integrations are not susceptible of change ; if they are it is easy to see that we must add to the expression for 3^7 the terms dx f ''' Vdy] - (dx r Vdij) + r{V,dy,-VJy,)dx. J Xo In geometrical applications the limits of the integrations with respect to x and y will frequently be determined by the perimeter of a closed curve ; in this case y^ = y^ both when x = x. and when x = x ; and therefore MBz dy and /•ji dx " Vdy vanish when x = x^, and also when x = x^, - 2/0 S74 CALCULUS OF VARIATIONS. 884. In the value of 5C/ found in the preceding Article, there is one term which is a double integral involving Sz under the integral signs, and various single integrals de- pending upon the limiting values of Bz. By the method already used in Art. 354, it will follow that 8U wall not certainly vanish unless the coefficient of Bs under the^ double integral sign vanishes; thus for a maximum or minimum value of U we have as a necessary condition J- cUI dN _^ dx dy This is a partial differential equation for finding z in terms of x and y ; and we may say that the arbitrary func- tions which occur in its solution must be determined so that the remaining terms in Sf/" may vanish. But the dif- ficulty of integrating the partial differential equation in general prevents any practical examination of these terms at the limits. 385. As an example, let it be required to determine a surface of minimum area bounded by a given curve. Here by Art. 170, -/:7:/i-(£)*Hi)}-^-- let us put as usual dz dz d'z_ ^l__^ drz^ dx=^'' d^r^' d^''"^' d-^dy-'' df '• The condition for a minimum reduces to dM . dN dx dy = 0, ^1, ^ • + d p , d q _r. that IS, to -;: -7:^- o— ^ + -j If-, ,.-i, ^2> ~ "' that is, to r (1 +/ + 2') - {pr -\-qs)p^t{\ +/ + 2') - {ps + qt)q=^ 0, that is, to (1 + q") r - '2pqs +{l+p")t = 0. CALCULUS OF VARIATIONS, 375 It is shewn in works on Geometry of Three Dimensions tliat this equation indicates that the required surface is such that at every point the two principal radii of curvature are equal in magnitude and of contrary signs. Since we suppose the boundary of the required surface to be a fixed curve Bz vanishes all round this boundary ; thus the terms relative to the limits in BU all vanish. Discrimination of Maxima and Minima values. S8G. We shall now give some examples which illustrate the second part of the investigation of raaxirna and minima values of integrals ; see Art. 849. Consider the example of finding the shortest line between two given points. Here F=V(1+/), U=rVdcc, Suppose y changed into ?/ + S?/, and consequently p into p + Bp; put p + Bp instead of jj in V and expand ; thus V becomes ^(^+^^ + V(iT?) + 2(i+/)^ ■■•• where the terms which are not expressed are of the third and higher orders in Bp. Thus we obtain The first of these terms is what we formerly denoted by BU, and the investigation of the minimum value of U so far as it has hitherto been carried, consists in making this term vanish. Supposing then that this term vanishes, and neg- lectino- terms of the third and higher orders, we have 2.U(l+/f 376 CALCULUS OF VARIATIONS. If Xj — Xq Is positive, every element of this integral is positive ; thus 8 U is j)ositive, and therefore a minimum value of U has been obtained. S87. Again, take the case of the brachistochrone, when the extreme points are fixed. Here Change y ihto y + Sy, and j) into ^ + S^ ; and expand the new value of V. Thus F becomes V(l+p') ^ ^i}.^f)8y ^ pZp ^ ^ 8 (1 +/)^ (Sy)^ ^ j ^^g;^ ^ ^ (¥)" 8/ 2/(1+/)^ 23/^(l+FJ^ and from this we can obtain h U. Now by the process of Art. 363 the terms of the first order in 8 fare made to vanish; then, neglecting terms of the third and higher orders, we have ,^^p|3(i+rf%)-_ P^yh^ ^^ m u ixol 8/ 2/(1+/)^ 2/(1+/)*) We have now to investigate the sign of this expression when the relation between x and y is that which is deter- mined in Art. 363 ; and we shall shew by some transforma- tions that hU is> positive. Since 3/^ (1 +/)' = (2^)*, we have 3,. r--(3(2a)^ (%)^iLM^ CALCULUS OF VARIATIONS. 377 and as the extreme points ai-e supposed fixed Sij vanishes at the limits ; therefore Now -^f^-U- ?^ ^-^'' = -- --^-^-^^^^ dx\y/ y^ chj if f y' f Therefore f '* ^-^^^ da: = \ f '' {ly^ ^^ dx ; and w^-i-rm^'^ma.. Thus 8 Z7 is positive, and therefore a minimum value of U has been obtained. The discussion is much simplified by taking the axis of x vertically downwards, keeping x as the independent variable, 888. The preceding Article shews that it may be possible to change the expression of the second order to which SCT is reduced by our previous investigations, from a form in which the sio-n is uncertain to a form in which the sign is obvious. A general theory with respect to suitable transformations of Ruch terms of the second order has been given by Jacobi ; for this we refer to the works named at the end of the present Chapter. It may be observed that many of the problems discussed in the Calculus of Variations a're of a kind in which we may infer, with more or less certainty, the character of the result from the nature of the particular problem. Thus, for instance, we may perhaps see in a particular case that a least value must exist ; so that if a solution presents itself, and only one, which may be a maximum or a minimum, we infer that it must correspond to the least value. 889. In the problem discussed in Art. 385 it is easy to shew that the result really gives a minimum. Here 378 CALCULUS OF VARLiTIONS. V= V(l + F + q'), U= r r V(l +/ + ql dxdy. •1 XaJ Va Suppose z changed into z + hz, in consequence of which p becomes p + hp and q becomes q + hq. Thus V becomes W pq^p^q J (1+ 2(l+p^ + 2^)i (l+/ + 2^)t 2(1+/ + 2^)^ (1 + r) {SpY _ pqSpSq (1 +/ ) (S^)^ Then supposing the terms of the first order made to vanish, and neglecting terpis of the third and higher orders, we have ^ 1 p. fy^ {SpY+{Sqy + (qSp-pSqy Thus the term under the integral signs is necessarily positive ; so that a minimum value of U has been obtained. Condition of InUgrahility. 890. In Art. 854 we have found that /f = is a neces- sary condition for the existence of a maximum or minimum value of the integral there considered. It may however happen that in certain cases the relation K = is satisfied identically ; this case we proceed to exemplify and interpret. Suppose we are seeking a maximum or minimum value of K\!/ y yi CALCULUS OF VARIATIONS. 379 / '2 " Here F= ^ - -4" + -^ , y y y ^ dV V 2xf ^~ dy- f^ f xy" dV 1 2xj/ dy'-y / ' ^ dy" V' ^^ dx'^ dx'~ / • / y' On collectinsr the terms it will be found that dx dx' vanishes. Thus the relation /t = is an identity in this example, and we cannot obtain from it any value of y. In this example we shall find that / Fc7^ = ^, y that is, the integral Vdx can be pbtained without assigning the value of y in terms of x. Thus if we wish to find a maximum or minin^um value of Vdx, we must mvestigate a maximum or minimum value of ( -— ) — ( -^ . We are V J/ A \y K_ therefore not concerned with the maximum or minimum of an undetermined integral expression of the kind hitherto 880 CALCULUS OF VARIATIONS. considered, but with the maximum or minimum of an expres- sion free from the integral sign. This species of maximum and minimi:m problem is con- sidered in some of the exhaustive treatises on the Calculus of Variations ; as it does not present much intei'est we will refer the student to such works. 391. We shall now prove universally that the necessary and sufficient condition in order that V may be integrable without assigning the specific value of y in terms of x, is that K = should be identically true. An expression which is integrable without assigning the specific value of the depend- ent variable in terms of the independent variable is sometimes said to be integrable per se, and is sometimes said to be ira- mediately integrable. 392. We first prove that the condition is necessary. Suppose that V involves x, y and the differential coefficients of y with respect to x up to -7-^ inclusive. If the function V is immediately integrable the integral Vdx can be expressed in the form Xa whore the form of the function denoted by <^ remains un- changed whatever may be the value of y in terms of x. Now suppose that y receives such a variation as leaves the values of y and its differential coefficients at the limits unaltered ; then from the value of I Vdx it follows that J Xo B Vdx = ; f J X L CALCULUS OF VARIATIONS. 381 thus by Art. 352 r=^'. (dV d dV d' dV I , ^ But this cannot be true whatever Sy may be, unless dV__^dV _^d/F_ dy dx dy' dx^ dy" * " ' and unless this is identically true it determines yas a function of X. Thus if Fis immediately integrable the relation K= must be identically true. Next we shall shew conversely that if this condition holds V is immediately integrable. It is usually considered sufficient to say, that if this condition holds the variation of Vdx depends solely on the liviiting values of x, y, and the differential coefficients of y ; and therefore I Vdx must itself depend solely on these limiting values, that is, V must be immediately integrable. We shall however reproduce a more satisfactory demonstration which has been given of the proposition. Suppose F= (/) {x, y, y',y",...). Let u and v denote two functions of x at present unde- termined ; let a denote a quantity which we shall vary inde- pendently of x. Let -^ {ol) denote what F becomes when we put u + av instead oi y, and u +av' instead of y', and u" + av" instead of y", and so on ; thus ^Ir (y.) = (f) [x, u + av, XL + av , w" + av", . . .). Differentiate both sides with respect to a, so that we have a result which we may denote thus, ^^'^ du'^^du'^'^du"'' + 382 CALCULUS OF VARIATIONS. Integrate both sides, from a = to a = 1 ; thus that is, we have the following identically true, (j) {x, u + V, It + v, u" + v", ...) = (p {x, u, u', III' , ...) [du clu aib J Integrate both sides with respect to x ; thus I ^ {x, u + V, it' + v', il" + v" , ...) dx = I ^ (a"y M, u', u", ...) dx + I du. \~v + -j~,v' + -j^,v" + ...[dx [ait die du where in the last term the order of the independent integra- tions has been changed. By integration by parts d(b , , d(b f d d(b , -j-, V dx = V -r^,— Iv -r~ -^h dx, du du J ax du [d(f) „ , , dd> d d(b f d'^ dd) ^ j^^^^=^ d^'-'rxd^'-^rd?d^"^''^ and so on. Thus CALCULUS OF VARIATIONS. I (f) (x, u 4- V, II + v, u" + v", ...) dx 383 = l(/>(a7, u, u, ib",...)dx '^]o"\du' dxdu"^dx'du"' fi , fd(b d dcj) \ , V {Yy,--j--j-777+ •'• Ida doL + + + 1 da d(f) d d(f) d^ d(b du dx du dx^ du" d^dj) dx^ dii" dx Now by supposition the relation K= is satisfied identi- cally whatever may be the value of ?/ ; so it is satisfied if u + av be put for y. Hence d<^ d d(f> d" d(p du dx du dx^ du" 0. The functions u and v are at present in our power; put y — u for V and we have \^{^> 2/> V' y",-")dx = l(f>(x, u, u, u", ...) dx ^y '},\du' dxdu'^dx'du ; ■^^y-''^?JS-'-Tx^'^'-)^'' + 38-i CALCULUS OF VARIATIONS. Thus I Vdx is here actually exhibited as an expression consisting of terms, one involving only ordinary integration with respect to cc, and the others ordinary integration with respect to a. The function u is still in our power; it should be chosen so that none of the quantities which occur become infinite or indeterminate; it may happen that consistently with this limitation we may put u = 0. 393. It will now be easy to give the necessary and suffi- cient conditions for ensuring that a function shall be integrable per se more than once. Let Fhave the same meaning as before. We have, whatever V may be, I \ I VdxY dos = sc j Vdx — I x Vdx. In order then that V may be integrable per se twice, the condition must of course be satisfied which ensures that it is integrable per se once ; and then the only additional condition is that a; F must also be integrable ])sr se once. Thus in order that V may be integrable ^jer se twice, the necessary and sufficient conditions are that the following relations must be identically true, dV_d_dV ^dV_ dy dx d})' dx^ dy" ^ '' dVx d dVx d" dVx dy dx dy' dx^ dy" ^ '' We may modify the form of (2). For dVx_ drr dVx_ dV dVx_ dV dy '""dy' dy '"^ dy" dy"~''df' ' £dVx^ d_dV dV dx dy' dx dy' dy ' ^dVx^ ^dV_ ^ d_dV dx' dy" ' dx^ dy" " dx dy" ' CALCULUS OF VARIATIONS. 3Si dx' dy'" ~ "^ dx' dy'" ^ dx' dy" ' Substitute in (2) and omit the terms which are zero by (1) ; then we obtain dy dx dy" dx^ dy'" " ^ Thus (1) and (2) may be replaced by (1) and (3). By a formula given in Art. 55 the 'n}^ integral of any pro- posed expression is exhibited in terms of n single inte- grals. From this formula we infer that in order that V may be integrable per se n times, it is necessary and sufficient that each of the following expressions should be integrable per se once: V, xV, x'V, x^'-'V. For example, in order that V may be integrable ^jer se three times, besides the conditions (1) and (2) or (1) and (3;, the following must be identically true, dV^_±dy^^dV^_ ^^ ' dy dx dy dx^ dy" '" ^ We may modify the form of (4). For d^dVx^ _ ^d_dV dV dx ay dx ay ay ^ il^ ^ar^^^^ + ^x~^ 2 ^— ^ dx^ dy" dx^ dy" dx dy" dy" * d' dVx'^ ^ d? dV , ^ d" dV . ^ d dV — iC 7 3 7 '/' ~r VX -. 2 7 /// "r t) dx^ dy"' dx^ dy" dx^dy" dx dy'" ' Substitute in (4) and omit the terms which are zero by (1) and (3) ; then we obtain dtj" \.ld,xdy" '^l.^dx'dy ~ ^'^''• Thus (5) may be taken instead of (4), in conjunction with (1) and (2) or (1) and (3). T. I. c. 25 o 86 CALCULUS OF VARIATIONS. Addition on the Variability of the Limits. 394. In the method we have adopted of treating problems involving changes of the limits we have followed the example given in two most elaborate works on the subject, those of Strauch and Jellett ; and we decidedly recommend this method as the best. We do not ascribe any variation to the independent variable, but only to the dependent variable. Another method however has been frequently adopted, and it should be explained in order that the student may understand any reference to it which may occur in his reading. In this method a variation is ascribed both to the dependent and independent variables. Let X become x-\-hx and let y become y + Sy, ^x and hy beino- indefinitely small arbitrary functions of x) it is required to find the variations or -j- , -j~^ , ... w?/ dti "We denote the variation in -i^ by 8 -^ ; therefore ^dy ^ d(y + B y) _ dy dx d{x-^ hx) dx dy_^dB^ dx dx dy dSx dx dx _ dy d Sy dy d Sx dy dx dx dx dx dx ' neglecting small quantities of the second order. Thus adopting the usual notation for a differential co- efficient, we have ^ , d'Su , d6x d(Sy — y'Sx) , ,,-> dx "^ dx dx dx ^ , „. d(8y-y'Sx) or ^y — y ^^^ = ■■ - - ■ ■_ — . CALCULUS OF VARIATIONS. la this result change y into y ; thus hy"-y"hx = '^^^y^ dx Similarly Sy -y 6x= ^/ » and so on. Put ft) for hy — y'hjc ; thus p. // tt ft) ,/, ~, ts til (t (O rr>r<^ 387 Now let F be any function of x, y, and the differential coefficients of y with respect to x; and let t^= I Fi^a;. J rro Let it be required to express the variation of Z7 which arises from the variations Bx and By in x and y respectively. Let 8 F denote the change made in V; then Xl d (x + Bx) Xo dx BU=j {V+B]^'^'-^^^^^dx = rv^4^dx+rBVdx, J g;^ ax J X(, neglecting a term of the second order. Vdx Now [v~dx=VBx- dx 'dr dx Bx dx, therefore I V'^ dx= {VBx)^-{VBx\-^ -~ Bxdx, 25—2 388 where CALCULUS OF VARIATIONS. dx denotes the complete differential coefEcient of V with respect to x. dV' dx hx[ dx. . . »„ dV^ , dV^ dl ^ . dV ^ „ And oV=-rox + ~j~6y+-j-,by+~j-^,hy + ... , dx dy "^ dy ^ dy -dr] dV dV . dV ,. dV ,„ ^ -^dx^dyy^dy'y ^di'y +•••' dx thus and finally 'dV dx - dV dV . dV „ CX= -J- C0 + -^-7 Oi + -j^-, CO + dy dy dy SU={VSx\-{VBx),+j^(^'La,+ dV ..., ,. d.i dV "We need not proceed further as we have arrived at a result equivalent to that in Art. 368; we have here co instead of the By which occurs there, and Bx^ and Sx^ for dx^^ and dx^ respectively. In geometrical applications it will be observed that x and y become by variation x + Bx and y+ By respectively. Thus ^1 + ^^1 ^^^^ correspond to the x^^ + dx^ of Art. 369, and y^ + By^ will correspond to the iY+-7~dx\ of Art. 369. Discontinuous Solutions. 395. Some problems in the Calculus of Variations admit of discontinuous solutions, and as the subject has attracted much attention in recent times a few words may be here conveniently devoted to it. Let there be an integral I cj) dx which is required to be a maximum or a minimum, where <^ is a given function of X and y and the differential coefficients of y with respect CALCULUS OF VAEIATIONS. 889 to X. Change y into y+By; then in the usual way we obtain for the variation of the integral to the first order an expression of the form L + I MZydx, where L depends on the values of the variables and the differential coefficients at the limits of the integration. Now if hy may have either sign we must have Jl/ = as an indispensable condition for the existence of a maximum or a minimum. Suppose however that owing to some conditions in the problem w& cannot always give to hy either sign : for ex- ample suppose that throughout the whole range of the inte- gration hy is essentially positive, then it is no longer necessary that M should vanish. If M is positive througli the whole range of the integration we are sure of a minimum ; and if M is neo^ative through the whole range of the integration we are sure of a maximum. We assume here that we are able to satisfy the condition Z = ; or to ensure that L shall be positive in the former case and negative in the latter case. Next suppose that hy may have either sign through part of the range of the integration, but that it is essentially positive through the remainder of the integration. Then if M vanishes through the former part and is positive through the latter part of the range we are sure of a minimum ; and if M vanishes through the former part and is negative through the latter part of the range we are sure of a maximum. We assume as before that the condition relative to L can be satisfied. For illustration we may take the problem which first sug- gested these remarks. Required to determine the greatest solid of revolution the surface of which is given, and which cuts the axis of revolution at two fixed points. With the usual notation we have to make tt i y^ dec a, maximum while Stt | y V(1 + p"^) dx is given. Let a be a constant at present undetermined ; then we have by the well known theory to make u a maximum, where a denotes ^[if + Uy^{\+f)]dx, 390 CALCULUS OF VARIATIONS. "We obtain Bu = L + j M By dx, where M stands for 2y +2a^/{l +})') - 2 ^ J(T+P^) ' By the known principles of the subject we put 31=0, and this leads in the usual way to ,-, "j^ 2x = & - 2/^ where b is another constant, which is introduced by the integration. Since the curve is to meet the axis of x at given points we have y = at those points ; hence 6 = 0, and the equation reduces to ^ + /=0.thatis,{^ + y = 0. Take — ——ir. + w = ; this leads to a circle which has V(l+/) . its centre on the axis of x and its radius equal to - 2a. Let A and B denote the given points on the axis of x. If the given surface is exactly equal to that of a sphere on AB as diameter such a sphere fulfils all the conditions of the problem. But if the given surface be not equal to that of a sphere on AB as a diameter, suppose C and I) points on the axis such that the given surface is equal to that of a sphere on CD as diameter. Then we obtain a discontinuous solution by taking for the generating curve the part of the axis of x between A and C, the semicircle on CD as diameter, and the part of the axis of x between I) and B. This solution was first suggested by observing that the fundamental equa- tion obtained above splits into the two factors y = and 2a „ We shall see on examination that if vanishes for the semicircle on CD as diameter ; and for the parts of the axis of X which enter into the solution M reduces to 2a. Thus EXAMPLES. 391 wlicn L is made to vanish Zu reduces to 1 2a hj dx^ for limits corresponding to ^ C and DB. Now Sy is essentially posi- tive for all this range, and 2a is negative as we see from its geometrical meaning. Thus hit is a negative quantity, indi- cating the existence of a maximum. On this subject the student is referred to the Researclies in the Calculus of Vacations, principally on the theorij of Dis- continuous Solutio7is, by the present writer. 89C. For further information on the Calculus of Varia- tions the student may consult Professor Jellett's treatise, and the History of the Progress of the Calculus of Variations during the Nineteenth Century, by the present writer. The most interesting examples in this subject are those which are connected with physical science, as the problem of the brachistochrone ; accordingly we shall include some more applications of this kind in the following selection for exercise. EXAMPLES. 1. A curve of given length has its extremities on two given intersecting straight lines : determine its form when the area included between the curve and its chord is a maximum. 2. Determine a plane closed curve of given perimeter which shall include a maximum area. (See History... y page 68.) 3. Required to connect two fixed points by a curve of given length so that the area bounded by the curve, the ordinates of the fixed points, and the axis of abscissae shall be a maximum, supposing the given length greater than is consistent with the solution ob- tained in Art. 379. (See History..., page 427.) 4. A rectangular dish is to be fitted with a tin cover of given height having the ends vertical : determine the form so that the amount of material used may be the least possible. 892 EXAMPLES. 5. A mountain is in the shape of a portion of a sphere, and the velocity of a man walking upon it varies as the height above the horizontal great circle of the complete sphere : shew that if he wishes to pass from one point to another in the shortest possible time, he must walk in the vertical plane which contains the two points. G. When a curved surface can be divided by a plane into two symmetrical portions the intersection of the plane and surface, when an intersection exists, is in general a line of minimum length on the surface. (See History..., page 365.) 7. Find the minimum value of fl(^g)'sin. + < y + ".-^"'">' U. J [ \dxj sm a; J • (See Philosophical Magazine for December, 1861, and July 1862.) 8. Required the minimum value of I |-^] dx under the following conditions : 2/0 ~ -^> I ~ dx = — l. (See ZTtstory..., page 432.) 9. Required the variation of 1 Vdx, where F is a function of X, y, ~- , -j~, ... and v, where v =■ \ Vdx, and V is also a function of x, y,-j- , -y-^ , . . . (See History..., page 21.) 10. Let s denote I v'(l +P^) dx, and let j> (s) be any function JO of s ; then the relation between x and y is required which makes I (s) dx a maximum or a minimum Jo while I ^(1 +P^) d^ ^ss ^ given value, a being a con- J stant. For a particular case suppose (/> (s) = s. (See History. . ., page 453.) EXAMPLES. S93 11. Required the curve at every point of wliicli is a maximum or a minimum. (See Ilistory..., page 1.) „ , . 1 dy . 12. Required the curve at every pomt ot wIiicH 2/^ ^^ maximum or a minimum, the variations of y and being so taken that at any point yx — y a dy dx da; "J shall undergo no change by variation. (See History..., page 41-i.) 13. Apply Art. 375 to prove the point assumed in Art. 363, namely, that the required curve in thebrachistochrone problem lies in the vertical plane which contains the two given points. 14. The form of a homogeneous solid of revolution of given superficial area, and described upon an axis of given length, is such that its moment of inertia about the axis is a maximum : prove that the normal at any point of the generating curve is three times as long as the radius of curvature, 15. A given volume of a given substance is to be formed into a solid of revolution, such that the time of a, small oscillation about a horizontal axis perpendi- cular to the axis of figure may be a minimum : de- termine the form of the solid. (See History..., page 891.) IG. A vessel of given capacity in the form of a surface of revolution with two circular ends, is just filled with inelastic fluid which revolves about the axis of the vessel, and is supposed to be free from the action of gravity. Investigate the form of the vessel that the whole pressure which the fluid exerts upon it may be the least possible, the magnitudes of the circular ends being given. Result. The generating curve is a catenary. ^O O' S94 EXAMPLES. 17. Find the equation given by the Calcuhis of Variations for the transverse section of a straight and uniform canal, when one of the three quantities, the surface, the capacity, and the normal hydrostatic pressure, is either a maximum or a minimum, and the other two are given, the terminal surfaces and pressures not being taken into account. Shew also that when the surface is a minimum and the capacity only is given, the section is circular; and when the normal pressure is a minimum the section is a catenary or two sti-aight lines, according as the surface or tlie capacity is given, 18. If there are two curves with their concavities down- wards and terminated in the same extremities, a par- ticle moving under the action of gravity will take a longer time to describe the upper curve than the lower curve, the initial velocity being supposed the same in the two cases. (See History..., page 848.) 19. Assuming that a ship's rate of sailing is a function of the angle which the direction of its course makes with the direction of the wind, shew that the bra- chistochronous course between two given positions is rectihnear, and that unless it be in the straight line joining the positions it is in two directions always making the same angle with the direction of the wind. (See Philosophical Magazine for September, 1862.) 20. A solid of revolution is to be formed on a given base with a given volume so as to experience a minimum resistance when it moves through a fluid in the di- rection of its axis : determine the figure of the solid. (See i^esearc/tes... Chapter X.) 395 CHAPTER XVI. MISCELLANEOUS PEOPOSITIONS. 397. In the present Chapter we shall investigate a few miscellaneous propositions of interest. 398. It is required to transform the series 1^ _ 1 X 1 x^ _ _1 x' VI m+ll-x'^ m + 2{l-xy m+ 3 (1 -;r)'"^ "* into a series arranged according to powers of a;; it being supposed that ^ is less than unity. JL ^~ X X ?/ • • Put -, = ?/, so that X = -, -— . The given series 1-a; '^' 1+// ° y Jo 1 {y-ir^dy Jo 1 + z/ io-^ \y y+y '' ^f:_rrdy 111 Jo .V+ 1 * 396 MISCELLANEOUS PROPOSITIONS. Then by repeated integration by parts we have y+1 (m + l)(2/+l) «2 + lJo(_y+l/ 1 p ■ v"^^ ^ _ y {in+V){y + V) (m + l)(m + 2)(y+l)' 2 fy ^v^'^^ dxj and so on. Thus we see that 1 [yy-'dy f'U 1+2/ 1 f X a? 2x' ' / ... , t \ r ... . ON < m [m + l (m+l)(m4-2) (w+l)(m + 2)(7/i + 3) 2.3^* ] '^ (m + l) {m + 2) {m + 3) {m + 4) ■^' " * " j Hence the required transformation is effected 1 2 For example put m = -x, and divide both sides of the equation by 2 : thus , 1 a; 1 ic'^ 1 x^ l-r.-. + 3 1-a; ' 5 {\-xf 7 (1-^)' \x 2^2 2 Ax .3 ~^ |3"''3.5"^3.5.7'^* If we put sin'^^ for x this gives a known transformation a for ;r : see Differential Calculus, Art. 374. tan d "" 399. In Art. 62 it is shewn that if we integrate a function of two independent variables, with respect to both variables, between fixed limits we obtain the same result when we adopt either order of integration, provided the function remain finite between the assigned limits. Con- versely if by changing the order of integration we change MISCELLANEOUS PROPOSITIONS. 897 the result it follows that the function must have been in- finite within the range of the integrations. This principle has been applied by Gauss to shew that every rational inte- gral equation has a root real or imaginary. Consider the expression Put r (cos 6 + J— 1 sin 6) for x ; then the proposed expres- sion takes the form P +QJ— 1, where P =zr" cos nO + py~^ cos {n — \)6 +. . .+p„_ir cos 9 + /?„, Q = r" sin nO -{-p^i-'^'^ sin (n — 1) ^+...+^„_irsin d. _ p Let F= tan * ^^ ; then dV ^ dd pdQ dd dd dV P'^ pdQ '^dr dr P' + Q' Hence Y~Ja iii"^*ol^Gs (P^+ Q")^ in the denominator; and d'^V if we can shew that -5 — tt, becomes infinite "within a certain drdo range of values for 6 and r, it follows that P and Q must simultaneously vanish. We shall take and Itt for limits of 6, and and a for limits of r, where a is large but finite ; and we shall d^V . . cZ^F integrate -, — t-, between these limits. Integrate , — 7-, first "^ drdd ° drdO dV with respect to 6 ; thus we obtain -y— : now take this be- tween the limits and lit, then the result is zero, for P and Q and their ditferential coefficients have the same value when 6 = '2.77 as when (9=0. Hence by adopting this order 398 MISCELLANEOUS PROPOSITIONS. of integration we obtain zero as the result of the first inte- gration, and therefore zero also as the final result. Now adopt the other order. Integrate —j — j-3 first with drew d V dP dO respect to r ; thus we obtain -^-r . Now Q ^n and P -7^ ad dO dU dV both vanish when r = 0, so that -j-r vanishes when r = 0. do When r = a we have for the value of Q -j-r — P ,^ a series do dO proceeding according to descending powers of a ; the first term of which is — 7ia'^" (cos** nd + sin^ n6), that is — ^m^"* : and a may be taken so large as to render all the other terms insignificant in value compared with this. In like manner P^ + (^ may also be made to differ as little as we please from its first term, that is from a^". Hence I ^ — rpi dr = — n, Jo drdO that is we have a result differing as little as we please from this by taking a large enough. Then, integrating with resjject to 6 between and 27r, we obtain — 2?i7r. Thus by performing the integrations in different orders we obtain two different results ; and therefore the function must become infinite within the range of the integrations : and therefore P and Q must simultaneously vanish within that range. Bertrand's Calcid Integral, page 188. 400. It is shewn in Art. 177 that if a curve havins; the equation y = A+Bx+Gx^ + Dx^ be made to pass through three given points the ordinates of which are equidistant, the area bounded by the curve, the extreme ordinates, and the axis of x is equal to ^ [y^ + 4y,^ + y^ ; where y^, y^, and y^ are the ordinates and h the distance between two consecutive ordinates. It will be observed that an infinite number of such curves can be drawn, since there are four coefficients A, B, C, D at our disposal, and only three conditions to de- MISCELLANEOUS PROPOSITIONS. S0.9 termine them : thus wc might make the curve pass througli any fourth point we please. Nevertheless the area men- tioned remains always the same. This result admits of gene- ralisation into the following theorem : Let a curve having the equation y = A^ + A^x + Ajf + ... + A^^_^x'"~^ be made to pass through 2/i — 1 given points, of which the ordinates are equidistant, then the area bounded by the curve, the extreme ordinates, and the axis of x is always the same. The demonstration is of pi-ecisely the same kind what- ever may be the positive integral value of n ; we will suppose for simplicity that n = 3. Let 2/j, 7/2, y^, y^, y^ denote the ordinates of the given points ; and let h be the distance between two consecutive ordinates. Suppose the first ordinate to correspond to the abscissa x=0. Then from the elements of the Theory of Finite Differences we have X . x{x — Ji) . „ X (x — h) (x — 2h) . , ^ = ^' + A ^y- + Ani~ ^^- + h' \3 — ^ ^- x{x-h)(!c-2k)(!c-3h) ,, + - ^Mi ""'^ X (x - h) (x - 2/0 (x - 370 (x - 4/0 where ^y,=y,-y„ ^"y, = y,-y,-{v,-y^ = y,-^u, + y„ and so on. Thus the value of A*_y^ involves y, , y„, ... up to y^; and the value of A^?/j involves y^, y^,--- np to y^, where y^ is the ordinate of any arbitrary sixth point, corresponding to an abscissa oh. Now the area which we require = / ydx, so that the Jo term which involves A^y^ is ^\' cc {x - h) (x - 2A) {x - 3/0 (x - 4/0 dx, a |£.'o 400 MISCELLANEOUS PROPOSITIONS. In the integral put f + 2A for x, then it becomes rth -2/1 2fe [■•ih that is, (I' _ 47,2^ (^_ 7,^)1^^. J -2fe and this vanishes by first principles : see Art. 42. Hence I ydx does not involve A'^/f ^^^ o^lj ^2/1' ^^I/i> '•• "P to A^y^; and so when expressed in terms of y^, y„,... involves these ordinates up to y^ inclusive. This establishes the re- quired result. It is scarcely possible that a result so general and so simple has not been already given ; but the writer has not met with it. 401. From Wallis's Formula we may deduce in an ele- mentary way the formula for the approximate value of 1.2.3 ... X, when x is very large. Professor De Morgan seems to have first noticed this in his Differential and Inte- gral Calculus, page 293 ; and the process has been put in a very simple form by Serret : see his Cours de Calcul Dif- ferentiel et Integral, Vol. Ii. page 206. According to "Wallis's Formula, as given in Art. 36, we have 7r^ 2.2.4.4...f2a;-2)(2.r-2) 2x 2 ~ 1. 3. 3. 5.. .(2^-3) (2a;- 1) 2a;- 1 ^ '' when X is infinite. 1 . 2 3 ... .a; Now let dt (x) stand for -1^^ ; then it will be e"'' x'' sj2Trx found that (1) gives, when x is infinite, 1 2 ~ ^ I {i>(2x)y MISCELLANEOUS PROPOSITIONS. 401 and therefore by extracting the square root we have, when x is infinite, From the form of ^ {£) we obtain ■ -^- = 1(1 + 1)-^ p); therefore log ^^ = " 1 + (^ + g) '"S (l + J.) In this series the terms are alternately positive and negative. The numerical value of the ratio of the terra rt" 1 which involves a;""*"^ to the preceding- term is -^. r . - , which is certainly less than unity Avhen n is greater than 2, provided x is not less than unity. Hence the value of the series is less than t-— ^ , and therefore 12a;- ° ' where 6^, 6„, d^,--. are all positive fractions less than -- . T. I. c. 2Q 402 MISCELLANEOUS PROPOSITIONS. Hence the sum of the terms on the right-hand side of (G) is less than z _-„ x ^> that is less than ^-^t— . (b Lv) . 1 Therefore log , ,\\ is less than :r^— , and therefore when cc is infinite = 1 (T). From (2) and (7) we have when x is infinite and therefore r= = 1 + /5» e "x"' J'Ittx where ^ vanishes when x is infinite. Thus the required formula is established. 402. We proceed to some further developments which are due mainly to Serret. A limit closer than that assigned by (4) may be found for log -J-. — —^. » {x) (1 + x) log <^(a;+l) ^ ^^\l2x' 12a,''^40x-^ '"^ 2?i(?z + l)^" ^•••j J ^^ , (n-2)(-ir ■ 12.^- 120a,'' "• 2«(n + l)i?i + 2)a;" In this series the terms are alternately positive and negative. The numerical value of the ratio of the terra which involves ^"'^'^ to the preceding term is n{n — V) 1 n{n-l) + ''i{n-:i)'~x' MISCELLANEOUS PROPOSITIONS. 408 wliich is certainly less than unity if n is greater than 2, pro- vided X be greater than unity. Hence (1 + x) log -,--— — 3-r is less than q-^r- , ^ ' °0(.c+l) 12j7 and therefore log -~ — r^ IS less than 4>{x + l) l^x{x + l)' 403. "We have identically loo- 6 U) = loo- _\r (x) ; hence A and B must be zero, and therefore equation (10) must hold for all positive values of X. 405. By Art. 403 we see that log {x) is equal to the sum of a series of quantities, which are all positive by equation (5). Hence log [x) is positive. Hence by equa- tion (9) it follows that log r {x + 1) is greater than ^ log "^tt — x ■\- ix + ~\ log x ; and therefore V {x -\-V) is greater than e~''x'' J'lirx. We shall now find an opposite limit for F (^ + 1). By Arts. 402 and 403 we see that log (h (x) is less than i^^ 2 , r-/ -, \ » MISCELLANEOUS PROPOSITIONS. 405 that is lofr (b (ic) is less than ^^ ^ •] -I ; °^^ ^ 12 \x + n x + n+1}' therefore loir cb (cv) is less than r— — . Hence by equation (9) it follows that log r (x + 1) is less than - log 27r — x + (x + -j log x + —^ , -J \ «-/ i^x and therefore r (.r + 1) is less than e"'^'^ x"" J 'lirx. 40C. We proceed to an investigation of Stirling's Theorem, which amounts to an expansion of logr(a;+l) in a series proceeding according to inverse powers of x. From equation (8) we obtain by differentiating twice d-'\og<^{x) _ 1 1^ ^ ^ 1 dx^ X 2x' "* (^ + ny ' But for any positive value of z we have -= e~'^~da, —.= I e~°-ad'X. z Jo z Jo Therefore, if x is positive. But ^e-"°= _ _^ , so that Integrate twice with respect to x, observing that log ^ (x) and -J- log (j) (x) both vanish when x is infinite. Thus CLOD iog^w=/3(r^.-|-i)''-"'«- 406 IVUSCELLANEOUS PROPOSITIONS. Therefore by (9) we have logr (a; + 1) = ^\og2'rr- X + (x + ~j log x +r?(?^-'^'»'^'"'=' '"^- Now suppose — — r expanded in powers of a. By Arts. 05 and 123 of the Differential Calculus the result is ^"2^+|2°^"g'°^ + |6 ^— + |27T2~^ • here B^, B^, ... are the Numbers of Bernoulli, as in Art. 304, and their values are ■^^^6' ^^^30' ^^^42' ^'"SO' •^«"6U'""' and /-"^ (^a) denotes that -^;^ — =- is to be differentiated 2r + 2 times with respect to a, and then 6x put for a, where ^ is a positive proper fraction. Now, observing that by Arts. 259 and 260, Jo x"'^' we have finally logr (a; + 1) = 2 log 1'Tr-x + (x+^ \ogx '^2x '6Ax'^ --^ {±r-\)2rx^'-' + 2^r^""^'^^^'^'"^^^"^^'- This formula includes Stirling s Theorem; for that amounts in fact to removing the definite integral at the end of the expression just given, and allowing the series to continue indefinitely. MISCELLANEOUS PROPOSITIONS. 407 With respect to the early history of Stirling's Theorem see the Ilistorij of...Prubabiliti/, page 188. 407. As an example of the formula obtained in the pre- ceding Article suppose r = () ; thus log r (^ + 1) = ^ log 277 - .c + Ix + -] log X ^J a ., e« (1 - 7) - 1 _ e"l(a-2)e- + a + 2} ,,„ , , e" [(3-a) e-'^-4xe°-a-3} / W = i^^r^Ty • It is easy to shoAv, by expanding the numerator of /""'(a) in powers of a, that /'"{''■>■) is always negative so long as a. is positive. Hence /" (a) continually diminishes as a increases from to 00 ; and we can sheAV that _/"' (a) is positive so long as a is : hence the greatest value of /"(a) for positive values of a is when a = 0. By evaluation we find that f"{'J-) is - when a = 0. Therefore log r (^ + 1) = ^ log 27r - a; + (^^ + - j log « + — , where X is some positive proper fraction. This result includes the two limits obtained in Art. 405. 408. Differentiate equation (11) ; thus - 1 + |j e-'^da 00 '°'^"-'„ a 1.-^:^1- 1'^""''^- 408 MISCELLANEOUS PROPOSITIONS. But, by Art. 288, log x= - (e~" — e-°^) dz ; Jo CO therefore ^-logr(.+ l)= j^ Cf-S)''^ (I-)- Therefore by putting .r = we obtain r (1) Jo Va e"-l/ Jo \oL l-e"", Hence, by Art. 268, we have another form for Eiders constant, namely L •-- e-»