ON FLUCTUATING FUNCTIONS. William Rowan Hamilton. (Transactions of the Royal Irish Academy, 19 (1843), pp ) Edited by David R.

Transcription

1 ON FLUCTUATING FUNCTIONS By Willim Rown Hmilton (Trnsctions of the Royl Irish Acdemy, 19 (1843), pp ) Edited by Dvid R. Wilkins 1999

2 NOTE ON THE TEXT The pper On Fluctuting Functions, by Sir Willim Rown Hmilton, ppered in volume 19 of the Trnsctions of the Royl Irish Acdemy, published in The following obvious typogrphicl errors hve been corrected: in rticle 8, n upper limit of integrtion of hs been dded to the integrl which in sin βα the originl publiction ws printed s dα α(1 + α 2 ) ; in rticle 13, the right hnd side of eqution (d ) ws printed in the originl publiction s (α x) 1 ψ k 1(α x) ; full stop (period) hs been inserted fter eqution (d IX ). Dvid R. Wilkins Dublin, June 1999 i

3 On Fluctuting Functions. By Sir Willim Rown Hmilton, LL. D., P. R. I. A., F. R. A. S., Fellow of the Americn Society of Arts nd Sciences, nd of the Royl Northern Society of Antiquries t Copenhgen; Honorry or Corresponding Member of the Royl Societies of Edinburgh nd Dublin, of the Acdemies of St. Petersburgh, Berlin, nd Turin, nd of other Scientific Societies t home nd brod; Andrews Professor of Astronomy in the University of Dublin, nd Royl Astronomer of Irelnd. Red June 22nd, 184. [Trnsctions of the Royl Irish Acdemy, vol. xix (1843), pp ] The pper now submitted to the Royl Irish Acdemy is designed chiefly to invite ttention to some consequences of very fertile principle, of which indictions my be found in Fourier s Theory of Het, but which ppers to hve hitherto ttrcted little notice, nd in prticulr seems to hve been overlooked by Poisson. This principle, which my be clled the Principle of Fluctution, sserts (when put under its simplest form) the evnescence of the integrl, tken between ny finite limits, of the product formed by multiplying together ny two finite functions, of which one, like the sine or cosine of n infinite multiple of n rc, chnges sign infinitely often within finite extent of the vrible on which it depends, nd hs for its men vlue zero; from which it follows, tht if the other function, insted of being lwys finite, becomes infinite for some prticulr vlues of its vrible, the integrl of the product is to be found by ttending only to the immedite neighbourhood of those prticulr vlues. The writer is of opinion tht it is only requisite to develope the foregoing principle, in order to give new clerness, nd even new extension, to the existing theory of the trnsformtions of rbitrry functions through functions of determined forms. Such is, t lest, the object imed t in the following pges; to which will be found ppended few generl observtions on this interesting prt of our knowledge. [1.] The theorem, discovered by Fourier, tht between ny finite limits, nd b, of ny rel vrible x, ny rbitrry but finite nd determinte function of tht vrible, of which the vlue vries grdully, my be represented thus, fx = 1 π dα dβ cos(βα βx) fα, with mny other nlogous theorems, is included in the following form: () fx = dα dβ φ(x, α, β) fα; (b) the function φ being, in ech cse, suitbly chosen. We propose to consider some of the conditions under which trnsformtion of the kind (b) is vlid. 1

4 [2.] If we mke, for bridgment, ψ(x, α, β) = β dβ φ(x, α, β), (c) the eqution (b) my be thus written: fx = dα ψ(x, α, ) fα. (d) This eqution, if true, will hold good, fter the chnge of fα, in the second member, to fα + fα; provided tht, for the prticulr vlue α = x, the dditionl function fα vnishes; being lso, for other vlues of α, between the limits nd b, determined nd finite, nd grdully vrying in vlue. Let then this function f vnish, from α = to α = λ, nd from α = µ to α = b; λ nd µ being included, either between nd x, or between x nd b; so tht x is not included between λ nd µ, though it is included between nd b. We shll hve, under these conditions, = µ λ dα ψ(x, α, ) fα; the function f, nd the limits λ nd µ, being rbitrry, except so fr s hs been bove defined. Consequently, unless the function of α, denoted here by ψ(x, α, ), be itself =, it must chnge sign t lest once between the limits α = λ, α = µ, however close these limits my be; nd therefore must chnge sign indefinitely often, between the limits nd x, or x nd b. A function which thus chnges sign indefinitely often, within finite rnge of vrible on which it depends, my be clled fluctuting function. We shll consider now clss of cses, in which such function my present itself. [3.] Let n α be rel function of α, continuous or discontinuous in vlue, but lwys comprised between some finite limits, so s never to be numericlly greter thn ±c, in which c is finite constnt; let nd let the eqution m α = α m α =, dα n α ; in which is some finite constnt, hve infinitely mny rel roots, extending from to +, nd such tht the intervl α n+1 α n, between ny one root α n nd the next succeeding α n+1, is never greter thn some finite constnt, b. Then, = m αn+1 m αn = αn+1 (e) (f) (g) α n dα n α ; (h) nd consequently the function n α must chnge sign t lest once between the limits α = α n nd α = α n+1 ; nd therefore t lest m times between the limits α = α n nd α = α n+m, this ltter limit being supposed, ccording to the nlogy of this nottion, to be the m th root of 2

5 the eqution (g), fter the root α n. Hence the function n βα, formed from n α by multiplying α by β, chnges sign t lest m times between the limits α = λ, α = µ, if* λ > β 1 α n, µ < β 1 α n+m ; the intervl µ λ between these limits being less thn β 1 (m + 2)b, if λ > β 1 α n 1, µ < β 1 α n+m+1 ; so tht, under these conditions, (β being >,) we hve m > 2 + βb 1 (µ λ). However smll, therefore, the intervl µ λ my be, provided tht it be greter thn, the number of chnges of sign of the function n βα, within this rnge of the vrible α, will increse indefinitely with β. Pssing then to the extreme or limiting supposition, β =, we my sy tht the function n α chnges sign infinitely often within finite rnge of the vrible α on which it depends; nd consequently tht it is, in the sense of the lst rticle, fluctuting function. We shll next consider the integrl of the product formed by multiplying together two functions of α, of which one is n α, nd the other is rbitrry, but finite, nd shll see tht this integrl vnishes. [4.] It hs been seen tht the function n α chnges sign t lest once between the limits α = α n, α = α n+1. Let it then chnge sign k times between those limits, nd let the k corresponding vlues of α be denoted by α n,1, α n,2,... α n,k. Since the function n α my be discontinuous in vlue, it will not necessrily vnish for these k vlues of α; but t lest it will hve one constnt sign, being throughout not <, or else throughout not >, in the intervl from α = α n to α = α n,1 ; it will be, on the contrry, throughout not >, or throughout not <, from α n,1 to α n,2 ; gin, not <, or not >, from α n,2 to α n,3 ; nd so on. Let then n α be never < throughout the whole of the intervl from α n,i to α n,i+1 ; nd let it be > for t lest some finite prt of tht intervl; i being some integer number between the limits nd k, or even one of those limits themselves, provided tht the symbols α n,, α n,k+1 re understood to denote the sme quntities s α n, α n+1. Let f α be finite function of α, which receives no sudden chnge of vlue, t lest for tht extent of the vrible α, for which this function is to be employed; nd let us consider the integrl αn,i+1 α n,i dα n α f α. (i) Let F be the lgebriclly lest, nd F the lgebriclly gretest vlue of the function f α, between the limits of integrtion; so tht, for every vlue of α between these limits, we shll hve f α f <, f f α < ; * These nottions > nd < re designed to signify the contrdictories of > nd <; so tht > b is equivlent to not > b, nd < b is equivlent to not < b. 3

6 these vlues f nd f, of the function f α, corresponding to some vlues αn,i nd α n,i of the vrible α, which re not outside the limits α n,i nd α n,i+1. Then, since, between these ltter limits, we hve lso n α <, we shll hve αn,i+1 dα n α (f α f ) < ; α n,i αn,i+1 dα n α (f f α ) < ; α n,i the integrl (i) will therefore be not < s n,i f, nd not > s n,i f, if we put, for bridgment, s n,i = αn,i+1 (k) α n,i dα n α ; (l) nd consequently this integrl (i) my be represented by s n,i f, in which f < f, f > f, becuse, with the suppositions lredy mde, s n,i >. We my even write f > f, f < f, unless it hppen tht the function f α hs constnt vlue through the whole extent of the integrtion; or else tht it is equl to one of its extreme vlues, f or f, throughout finite prt of tht extent, while, for the remining prt of the sme extent, tht is, for ll other vlues of α between the sme limits, the fctor n α vnishes. In ll these cses, f my be considered s vlue of the function f α, corresponding to vlue α n,i of the vrible α which is included between the limits of integrtion; so tht we my express the integrl (i) s follows: αn,i+1 α n,i dα n α f α = s n,i f α n,i ; (m) in which α n,i > α n,i, < α n,i+1. (n) In like mnner, the expression (m), with the inequlities (n), my be proved to hold good, if n α be never >, nd sometimes <, within the extent of the integrtion, the integrl s n,i being in this cse < ; we hve, therefore, rigorously, αn+1 α n dα n α f α = s n, f α n, + s n,1 f α n, s n,k f α n,k. (o) But lso, we hve, by (h) = s n, + s n,1 + + s n,k ; (p) 4

7 the integrl in (o) my therefore be thus expressed, without ny loss of rigour: αn+1 α n dα n α f α = s n, n, + + s n,k n,k, (q) in which n,i = f α n,i f αn ; (r) so tht n,i is finite difference of the function f α, corresponding to the finite difference α n,i α n of the vrible α, which ltter difference is less thn α n+1 α n, nd therefore less thn the finite constnt b of the lst rticle. The theorem (q) conducts immeditely to the following, β 1 α n+1 β 1 α n dα n βα f α = β 1 (s n, δ n, + + s n,k δ n,k ), (s) in which δ n,i = f β 1 α n,i f β 1 α n ; so tht, if β be lrge, δ n,i is smll, being the difference of the function f α corresponding to difference of the vrible α, which ltter difference is less thn β 1 b. Let ±δ n be the gretest of the k + 1 differences δ n,,... δ n,k, or let it be equl to one of those differences nd not exceeded by ny other, bstrction being mde of sign; then, since the k + 1 fctors s n,,... s n,k re lterntely positive nd negtive, or negtive nd positive, the numericl vlue of the integrl (s) cnnot exceed tht of the expression ±β 1 (s n, s n,1 + s n,2 + ( 1) k s n,k )δ n. (u) But, by the definition (l) of s n,i, nd by the limits ±c of vlue of the finite function n α, we hve ±s n,i > (α n,i+1 α n,i )c; (v) therefore ±(s n, s n,1 + + ( 1) k s n,k ) > (α n+1 α n )c; nd the following rigorous expression for the integrl (s) results: (t) (w) β 1 α n+1 β 1 α n dα n βα f α = θ n β 1 (α n+1 α n )cδ n ; (x) θ n being fctor which cnnot exceed the limits ±1. Hence, if we chnge successively n to n + 1, n + 2,... n + m 1, nd dd together ll the results, we obtin this other rigorous expression, for the integrl of the product n βα f α, extended from α = β 1 α n to α = β 1 α n+m : β 1 α n+m β 1 α n dα n βα f α = θβ 1 (α n+m α n )cδ; (y) in which δ is the gretest of the m quntities δ n, δ n+1,..., or is equl to one of those quntities, nd is not exceeded by ny other; nd θ cnnot exceed ±1. By tking β sufficiently lrge, 5

8 nd suitbly choosing the indices n nd n + m, we my mke the limits of integrtion in the formul (y) pproch s nerly s we plese to ny given finite vlues, nd b; while, in the second member of tht formul, the fctor β 1 (α n+m α n ) will tend to become the finite quntity b, nd θc cnnot exceed the finite limits ±c; but the remining fctor δ will tend indefinitely to, s β increses without limit, becuse it is the difference between two vlues of the function f α, corresponding to two vlues of the vrible α of which the difference diminishes indefinitely. Pssing then to the limit β =, we hve, with the sme rigour s before: dα n α f α = ; which is the theorem tht ws nnounced t the end of the preceding rticle. And lthough it hs been here supposed tht the function f α receives no sudden chnge of vlue, between the limits of integrtion; yet we see tht if this function receive ny finite number of such sudden chnges between those limits, but vry grdully in vlue between ny two such chnges, the foregoing demonstrtion my be pplied to ech intervl of grdul vrition of vlue seprtely; nd the theorem (z) will still hold good. [5.] This theorem (z) my be thus written: nd we my esily deduce from it the following: lim β= dα n βα f α = ; ( ) lim β= dα n β(α x) f α = ; (b ) the function f α being here lso finite, within the extent of the integrtion, nd x being independent of α nd β. For the resonings of the lst rticle my esily be dpted to this cse; or we my see, from the definitions in rticle [3.], tht if the function n α hve the properties there supposed, then n α x will lso hve those properties. In fct, if n α be lwys comprised between given finite limits, then n α x will be so too; nd we shll hve, by (f), α dα n α x = α x x (z) dα n α = m α x m x ; (c ) in which m x is finite, becuse the suppositions of the third rticle oblige m α to be lwys comprised between the limits ± bc; so tht the eqution α dα n α x = m x, (d ) which is of the form (g), hs infinitely mny rel roots, of the form α = x + α n, (e ) 6

9 nd therefore of the kind ssumed in the two lst rticles. Let us now exmine wht hppens, when, in the first member of the formul (b ), we substitute, insted of the finite fctor f α, n expression such s (α x) 1 f α, which becomes infinite between the limits of integrtion, the vlue of x being supposed to be comprised between those limits, nd the function f α being finite between them. Tht is, let us inquire whether the integrl dα n β(α x) (α x) 1 f α, (f ) (in which x >, < b), tends to ny nd to wht finite nd determined limit, s β tends to become infinite. In this inquiry, the theorem (b ) shows tht we need only ttend to those vlues of α which re extremely ner to x, nd re for exmple comprised between the limits x ɛ, the quntity ɛ being smll. To simplify the question, we shll suppose tht for such vlues of α, the function f α vries grdully in vlue; we shll lso suppose tht n =, nd tht n α α 1 tends to finite limit s α tends to, whether this be by decresing or by incresing; lthough the limit thus obtined, for the cse of infinitely smll nd positive vlues of α, my possibly differ from tht which corresponds to the cse of infinitely smll nd negtive vlues of tht vrible, on ccount of the discontinuity which the function n α my hve. We re then to investigte, with the help of these suppositions, the vlue of the double limit: x+ɛ lim. lim. ɛ= β= x ɛ dα n β(α x) (α x) 1 f α ; (g ) this nottion being designed to suggest, tht we re first to ssume smll but not evnescent vlue of ɛ, nd lrge but not infinite vlue of β, nd to effect the integrtion, or conceive it effected, with these ssumptions; then, retining the sme vlue of ɛ, mke β lrger nd lrger without limit; nd then t lst suppose ɛ to tend to, unless the result corresponding to n infinite vlue of β shll be found to be independent of ɛ. Or, introducing two new quntites y nd η, determined by the definitions y = β(α x), η = βɛ, (h ) nd eliminting α nd β by mens of these, we re led to seek the vlue of the double limit following: η lim. lim. dy n y y 1 f x+ɛη 1 ɛ= η= y; (i ) η in which η tends to, before ɛ tends to. It is nturl to conclude tht since the sought limit (g ) cn be expressed under the form (i ), it must be equivlent to the product f x dy n y y 1 ; (k ) nd in fct it will be found tht this equivlence holds good; but before finlly dopting this conclusion, it is proper to consider in detil some difficulties which my present themselves. 7

10 [6.] Decomposing the function f x+ɛη 1 y into two prts, of which one is independent of y, nd is = f x, while the other prt vries with y, lthough slowly, nd vnishes with tht vrible; it is cler tht the formul (i ) will be decomposed into two corresponding prts, of which the first conducts immeditely to the expression (k ); nd we re now to inquire whether the integrl in this expression hs finite nd determinte vlue. Admitting the suppositions mde in the lst rticle, the integrl ζ ζ dy n y y 1 will hve finite nd determinte vlue, if ζ be finite nd determinte; we re therefore conducted to inquire whether the integrls ζ dy n y y 1, ζ dy n y y 1, re lso finite nd determinte. The resonings which we shll employ for the second of these integrls, will lso pply to the first; nd to generlize little the question to which we re thus conducted, we shll consider the integrl dα n α f α ; (l ) f α being here supposed to denote ny function of α which remins lwys positive nd finite, but decreses continully nd grdully in vlue, nd tends indefinitely towrds, while α increses indefinitely from some given finite vlue which is not greter thn. Applying to this integrl (l ) the principles of the fourth rticle, nd observing tht we hve now f α n,i < f αn, α n,i being > α n, nd α n being ssumed < ; nd lso tht we find nd consequently ±(s n, + s n,2 + ) = (s n,1 + s n,3 + ) > 1 2 bc; (m ) αn+1 ± dα n α f α < 1 2 bc(f α n f αn+1 ); (n ) α n αn+m ± dα n α f α < 1 2 bc(f α n f αn+m ). (o ) α n This ltter integrl is therefore finite nd numericlly less thn 1 2 bc f α n, however gret the upper limit α n+m my be; it tends lso to determined vlue s m increses indefinitely, becuse the prt which corresponds to vlues of α between ny given vlue of the form α n+m nd ny other of the form α n+m+p is included between the limits ± 1 2 bc f α n+m, which limits pproch indefinitely to ech other nd to, s m increses indefinitely. And in the integrl (l ), if we suppose the lower limit of to lie between α n 1 nd α n, while the upper limit, insted of being infinite, is t first ssumed to be lrge but finite quntity b, lying between 8

11 α n+m nd α n+m+1, we shll only thereby dd to the integrl (o ) two prts, n initil nd finl, of which the first is evidently finite nd determinte, while the second is esily proved to tend indefinitely to s m increses without limit. The integrl (l ) is therefore itself finite nd determined, under the conditions bove supposed, which re stisfied, for exmple, by the function f α = α 1, if be >. And since the suppositions of the lst rticle render lso the integrl dα n α α 1 determined nd finite, if the vlue of be such, we see tht with these suppositions we my write ϖ = dα n α α 1, (p ) ϖ being itself finite nd determined quntity. By resonings lmost the sme we re led to the nlogous formul nd finlly to the result ϖ = ϖ = ϖ + ϖ = dα n α α 1 ; (q ) dα n α α 1 ; (r ) in which ϖ nd ϖ re lso finite nd determined. The product (k ) is therefore itself determinte nd finite, nd my be represented by ϖf x. [7.] We re next to introduce, in (i ), the vrible prt of the function f, nmely f x+ɛη 1 y f x, which vries from f x ɛ to f x+ɛ, while y vries from η to +η, nd in which ɛ my be ny quntity >. And since it is cler, tht under the conditions ssumed in the fifth rticle, ζ lim. lim. dy n y y 1 (f x+ɛη 1 y f x ) =, (s ) ɛ= η= ζ if ζ be ny finite nd determined quntity, however lrge, we re conducted to exmine whether this double limit vnishes when the integrtion is mde to extend from y = ζ to y = η. It is permitted to suppose tht f α continully increses, or continully decreses, from α = x to α = x + ɛ; let us therefore consider the integrl η ζ dα n α f α g α, (t ) in which the function f α decreses, while g α increses, but both re positive nd finite, within the extent of the integrtion. 9

12 By resonings similr to those of the fourth rticle, we find under these conditions, αn+1 ± dα n α f α g α < bc(f αn g αn+1 f αn+1 g αn ); (u ) α n nd therefore ± 1 bc αn+m α n dα n α f α g α < f αn+m 1 g αn+m f αn+1 g αn +(f αn f αn+2 )g αn+1 + (f αn+2 f αn+4 )g αn+3 + &c. +(f αn+1 f αn+3 )g αn+2 + (f αn+3 f αn+5 )g αn+4 + &c. (v ) This inequlity will still subsist, if we increse the second member by chnging, in the positive products on the second nd third lines, the fctors g to their gretest vlue g αn+m ; nd, fter dding the results, suppress the three negtive terms which remin in the three lines of these expression, nd chnge the functions f, in the first nd third lines, to their gretest vlue f αn. Hence, αn+m ± dα n α f α g α < 3bc f αn g αn+m ; (w ) α n this integrl will therefore ultimtely vnish, if the product of the gretest vlues of the functions f nd g tend to the limit. Thus, if we mke f α = α 1, g α = ±(f x+ɛη 1 α f x ), the upper sign being tken when f α increses from α = x to α = x + ɛ; nd if we suppose tht ζ nd η re of the forms α n nd α n+m ; we see tht the integrl (t ) is numericlly less thn 3bc αn 1 (f x+ɛ f x ), nd therefore tht it vnishes t the limit ɛ =. It is esy to see tht the sme conclusion holds good, when we suppose tht η does not coincide with ny quntity of the form α n+m, nd where the limits of integrtion re chnged to η nd ζ. We hve therefore, rigorously, η lim. lim. dy n y y 1 (f x+ɛη 1 y f x ) =, (x ) ɛ= η= η nowithstnding the gret nd ultimtely infinite extent over which the integrtion is conducted. The vrible prt of the function f my therefore be suppressed in the double limit (i ), without ny loss of ccurcy; nd tht limit is found to be exctly equl to the expression (k ); tht is, by the lst rticle, to the determined product ϖf x. Such, therefore, is the vlue of the limit (g ), from which (i ) ws derived by the trnsformtion (h ); nd such finlly is the limit of the integrl (f ), proposed for investigtion in the fifth rticle. We hve, then, proved tht under the conditions of tht rticle, lim β= dα n β(α x) (α x) 1 f α = ϖf x ; (y ) 1

13 nd consequently tht the rbitrry but finite nd grdully vrying function f x, between the limits x =, x = b, my be trnsformed s follows: f x = ϖ 1 dα n (α x) (α x) 1 f α ; (z ) which is result of the kind denoted by (d) in the second rticle, nd includes the theorem () of Fourier. For ll the suppositions mde in the foregoing rticles, respecting the form of the function n, re stisfied by ssuming this function to be the sine of the vrible on which it depends; nd then the constnt ϖ, determined by the formul (r ), becomes coincident with π, tht is, with the rtio of the circumference to the dimeter of circle, or with the lest positive root of the eqution sin x x =. [8.] The known theorem just lluded to, nmely, tht the definite integrl (r ) becomes = π, when n α = sin α, my be demonstrted in the following mnner. Let = b = sin βα dα α ; dα cos βα 1 + α 2 ; then these two definite integrls re connected with ech other by the reltion ( ) β = dβ d b, dβ becuse β dβ b = d dβ b = dα dα sin βα α(1 + α 2 ), α sin βα 1 + α 2 ; nd ll these integrls, by the principles of the foregoing rticles, receive determined nd finite (tht is, not infinite) vlues, whtever finite or infinite vlue my be ssigned to β. But for ll vlues of β >, the vlue of is constnt; therefore, for ll such vlues of β, the reltion between nd b gives, by integrtion, {( ) } β e β dβ + 1 b = const.; nd this constnt must be =, becuse the fctor of e β does not tend to become infinite with β. Tht fctor is therefore itself =, so tht we hve ( ) β = dβ + 1 b, if β >. 11

14 Compring the two expressions for, we find nd therefore, for ll such vlues of β, b + d b =, if β > ; dβ be β = const. The constnt in this lst result is esily proved to be equl to the quntity, by either of the two expressions lredy estblished for tht quntity; we hve therefore b = e β, however little the vlue of β my exceed ; nd becuse b tends to the limit π 2, we find finlly, for ll vlues of β greter thn, s β tends to These vlues, nd the result = π 2, b = π 2 e β. dα sin α α = π, to which they immeditely conduct, hve long been known; nd the first reltion, bove mentioned, between the integrls nd b, hs been employed by Legendre to deduce the former integrl from the ltter; but it seemed worth while to indicte process by which tht reltion my be mde to conduct to the vlues of both those integrls, without the necessity of expressly considering the second differentil coefficient of b reltive to β, which coefficient presents itself t first under n indeterminte form. [9.] The connexion of the formul (z ) with Fourier s theorem (), will be more distinctly seen, if we introduce new function p α defined by the condition n α = α dα p α, ( ) which is consistent with the suppositions lredy mde respecting the function n α. According to those suppositions the new function p α is not necessrily continuous, nor even lwys finite, since its integrl n α my be discontinuous; but p α is supposed to be finite for smll vlues of α, in order tht n α my vry grdully for such vlues, nd my ber finite rtio to α. The vlue of the first integrl of p α is supposed to be lwys comprised between given finite limits, so s never to be numericlly greter thn ±c; nd the second integrl, ( α 2 m α = dα) p α, (b ) 12

15 becomes infinitely often equl to given constnt,, for vlues of α which extend from negtive to positive infinity, nd re such tht the intervl between ny one nd the next following is never greter thn given finite constnt, b. With these suppositions respecting the otherwise rbitrry function p α, the theorems (z) nd (z ) my be expressed s follows: ( b ) βα lim. dα dγ p γ f α = ; () β= nd f x = ϖ 1 dα ϖ being determined by the eqution dβ p β(α x) f α ; (x >, < b) (b) ϖ = dα 1 dβ p βα. (c ) Now, by mking p α = cos α, ( supposition which stisfies ll the conditions bove ssumed), we find, s before ϖ = π, nd the theorem (b) reduces itself to the less generl formul (), so tht it includes the theorem of Fourier. [1.] If we suppose tht x coincides with one of the limits, or b, insted of being included between them, we find esily, by the foregoing nlysis, f = ϖ 1 dα dβ p β(α ) f α ; (d ) in which so tht, s before, f b = ϖ 1 dα ϖ = ϖ = dα dα 1 1 ϖ = ϖ + ϖ. dβ p β(α b) f α ; (e ) dβ p βα ; (f ) dβ p βα ; (g ) 13

16 Finlly, when x is outside the limits nd b, the double integrl in (b) vnishes; so tht = dα dβ p β(α x) f α, if x < or > b. (h ) And the foregoing theorems will still hold good, if the function f α receive ny number of sudden chnges of vlue, between the limits of integrtion, provided tht it remin finite between them; except tht for those very vlues α of the vrible α, for which the finite function f α receives ny such sudden vrition, so s to become = f for vlues of α infinitely little greter thn α, fter hving been = f for vlues infinitely little less thn α, we shll hve, insted of (b), the formul ω f + ω f = dα dβ p β(α α )f α. (i ) [11.] If p α be not only finite for smll vlues of α, but lso vry grdully for such vlues, then, whether α be positive or negtive, we shll hve nd if the eqution lim.n αα 1 = p ; (k ) α= n α x = (l ) hve no rel root α, except the root α = x, between the limits nd b, nor ny which coincides with either of those limits, then we my chnge f α to (α x)p f α, in the formul (z ), nd we shll hve the expression: f x = ϖ 1 p n α x dα n (α x) n 1 α xf α. (m ) Insted of the infinite fctor in the index, we my substitute ny lrge number, for exmple, n uneven integer, nd tke the limit with respect to it; we my, therefore, write (2n+1)(α x) Let then f x = ϖ 1 p lim dα n= (2n+1)α (2n 1)α α x dα p α dα p α f α. (n ) α dα p α = q α,n dα p α ; (o ) (2n+1)α 1 + q α,1 + q α,2 + + q α,n = α dα p α dα p α, (p ) 14

17 nd the formul (n ) becomes ( f x = ϖ 1 p dα f α + (n)1 b dα q α x,n f α ) ; (c) in which development, the terms corresponding to lrge vlues of n re smll. For exmple, when p α = cos α, then ϖ = π, p = 1, q α,n = 2 cos 2nα, nd the theorem (c) reduces itself to the following known result: ( f x = π 1 dα f α + 2 (n)1 b dα cos(2nα 2nx)f α ) ; (q ) in which it is supposed tht x >, x < b, nd tht b > π, in order tht α x my be comprised between the limits ±π, for the whole extent of the integrtion; nd the function f α is supposed to remin finite within the sme extent, nd to vry grdully in vlue, t lest for vlues of the vrible α which re extremely ner to x. The result (q ) my lso be thus written: f x = π 1 b dα cos(2nα 2nx)f α ; (r ) nd if we write it becomes φ y = 1 2π (n) α = β 2, x = y 2, f y 2 = φ y, (n) 2b 2 dβ cos(nβ ny)φ β, (s ) the intervl between the limits of integrtion reltively to β being now not greter thn 2π, nd the vlue of y being included between those limits. For exmple, we my ssume 2 = π, 2b = π, nd then we shll hve, by writing α, x, nd f, insted of β, y, nd φ, f x = 1 2π (n) π π dα cos(nα nx)f α, (t ) in which x > π, x < π. It is permitted to ssume the function f α such s to vnish when α <, > π; nd then the formul (t ) resolves itself into the two following, which (with slightly different nottion) occur often in the writings of Poisson, s does lso the formul (t ): π π dα f α + dα f α + π (n)1 π (n)1 dα cos(nα nx)f α = πf x ; (u ) 15 dα cos(nα + nx)f α = ; (v )

18 x being here supposed >, but < π; nd the function f α being rbitrry, but finite, nd vrying grdully, from α = to α = π, or t lest not receiving ny sudden chnge of vlue for ny vlue x of the vrible α, to which the formul (u ) is to be pplied. It is evident tht the limits of integrtion in (t ) my be mde to become l, l being ny finite quntity, by merely multiplying nα nx under the sign cos., by π 1, nd chnging the externl fctor l 2π to 1 2l ; nd it is under this ltter form tht the theorem (t ) is usully presented by Poisson: who hs lso remrked, tht the difference of the two series (u ) nd (v ) conducts to the expression first ssigned by Lgrnge, for developing n rbitrry function between finite limits, in series of sines of multiples of the vrible on which it depends. [12.] In generl, in the formul (m ), from which the theorem (c) ws derived, in order tht x my be susceptible of receiving ll vlues > nd < b (or t lest ll for which the function f x receives no sudden chnge of vlue), it is necessry, by the remrk mde t the beginning of the lst rticle, tht the eqution α dα p α =, (w ) should hve no rel root α different from, between the limits (b ). But it is permitted to suppose, consistently with this restriction, tht is <, nd tht b is >, while both re finite nd determined; nd then the formul (m ), or (c) which is consequence of it, my be trnsformed so s to receive new limits of integrtion, which shll pproch s nerly s my be desired to negtive nd positive infinity. In fct, by chnging α to λα, x to λx, nd f λx to f x, the formul (c) becomes ( λ 1 b f x = λϖ 1 p λ 1 dα f α + (n)1 λ 1 b λ 1 dα q λα λx,n f α ) ; (x ) in which λ 1 will be lrge nd negtive, while λ 1 b will be lrge nd positive, if λ be smll nd positive, becuse we hve supposed tht is negtive, nd b positive; nd the new vrible x is only obliged to be > λ 1 nd < λ 1 b, if the new function f x be finite nd vry grdully between these new nd enlrged limits. At the sme time, the definition (o ) shows tht p q λα λx,n will tend indefinitely to become equl to 2p 2nλ(α x) ; in such mnner tht lim.p q λα λx,n = 1, (y ) λ= 2p 2nλ(α x) t lest if the function p be finite nd vry grdully. Admitting then tht we my dopt the following ultimte trnsformtion of sum into n integrl, t lest under the sign ( lim.2λ 1 λ= 2 p + ) p 2nλ(α x) = dβ p β(α x), (z ) (n)1 16 dα,

19 we shll hve, s the limit of (x ), this formul: f x = ϖ 1 dα dβ p β(α x) f α ; (d) which holds good for ll rel vlues of the vrible x, t lest under the conditions ltely supposed, nd my be regrded s n extension of the theorem (b), from finite to infinite limits. For exmple, by mking p cosine, the theorem (d) becomes f x = π 1 dα dβ cos(βα βx)f α, ( ) which is more usul form thn () for the theorem of Fourier. In generl, the deduction in the present rticle, of the theorem (d) from (c), my be regrded s verifiction of the nlysis employed in this pper, becuse (d) my lso be obtined from (b), by mking the limits of integrtion infinite; but the demonstrtion of the theorem (b) itself, in former rticles, ws perhps more completely stisfctory, besides tht it involved fewer suppositions; nd it seems proper to regrd the formul (d) s only limiting form of (b). [13.] This formul (d) my lso be considered s limit in nother wy, by introducing, under the sign of integrtion reltively to β, fctor f kβ such tht f = 1, f =, (b ) in which k is supposed positive but smll, nd the limit tken with respect to it, s follows: f x = lim k=.ϖ 1 dα ( ) dβ p β(α x) f kβ f α. (e) It is permitted to suppose tht the function f decreses continully nd grdully, t finite nd decresing rte, from 1 to, while the vrible on which it depends increses from to ; the first differentil coefficient f being thus constntly finite nd negtive, but constntly tending to, while the vrible is positive nd tends to. Then, by the suppositions lredy mde respecting the function p, if α x nd k be ech different from, we shll hve β β dβ p β(α x) f kβ = f kβ n β(α x) (α x) 1 k(α x) 1 dβ n β(α x) f kβ; (c ) nd therefore, becuse f =, while n is lwys finite, the integrl reltive to β in the formul (e) my be thus expressed: β the function ψ being ssigned by the eqution dβ p β(α x) f kβ = (α x) 1 ψ k 1 (α x), (d ) ψ λ = 17 dγ n λγ f γ. (e )

20 For ny given vlue of λ, the vlue of this function ψ is finite nd determinte, by the principles of the sixth rticle; nd s λ tends to, the function ψ tends to, on ccount of the fluctution of n, nd becuse f tends to, while γ tends to ; the integrl (d ) therefore tends to vnish with k, if α be different from x; so tht lim k= dβ p β(α x) f kβ =, if α > < x. (f ) On the other hnd, if α = x, tht integrl tends to become infinite, becuse we hve, by (b ), lim.p dβ f kβ =. (g ) k= Thus, while the formul (d ) shows tht the integrl reltive to β in (e) is homogeneous function of α x nd k, of which the dimension is negtive unity, we see lso, by (f ) nd (g ), tht this function is such s to vnish or become infinite t the limit k =, ccording s α x is different from or equl to zero. When the difference between α nd x, whether positive or negtive, is very smll nd of the sme order s k, the vlue of the lst mentioned integrl (reltive to β) vries very rpidly with α; nd in this wy of considering the subject, the proof of the formul (e) is mde to depend on the verifiction of the eqution ϖ 1 dλ ψ λ λ 1 = 1. (h ) But this lst verifiction is esily effected; for when we substitute the expression (e ) for ψ λ, nd integrte first reltively to λ, we find, by (r ), it remins then to show tht dλ n λγ λ 1 = ϖ; (i ) dγ f γ = 1; (k ) nd this follows immeditely from the conditions (b ). For exmple, when p is cosine, nd f negtive neperin exponentil, so tht then, mking λ = k 1 (α x), we hve p α = cos α, f α = e α, dβ e kβ cos(βα βx) = (α x) 1 ψ λ ; ψ λ = dγ e γ sin λγ = 18 λ 1 + λ 2 ;

21 nd ϖ 1 dλ ψ λ λ 1 = π 1 dλ 1 + λ 2 = 1. It is nerly thus tht Poisson hs, in some of his writings, demonstrted the theorem of Fourier, fter putting it under form which differs only slightly from the following: f x = π 1 lim dα k= nmely, by substituting for the integrl reltive to β its vlue k k 2 + (α x) 2 ; dβ e kβ cos(βα βx)f α ; (l ) nd then observing tht, if k be very smll, this vlue is itself very smll, unless α be extremely ner to x, so tht f α my be chnged to f x ; while, mking α = x + kλ, nd integrting reltively to λ between limits indefinitely gret, the fctor by which this function f x is multiplied in the second member of (l ), is found to reduce itself to unity. [14.] Agin, the function f α retining the sme properties s in the lst rticle for positive vlues of α, nd being further supposed to stisfy the condition f α = f α, (m ) while k is still supposed to be positive nd smll, the formul (d) my be presented in this other wy, s the limit of the result of two integrtions, of which the first is to be effected with respect to the vrible α: f x = lim.ϖ 1 dβ k= dα f kα p β(α x) f α. (f) Now it often hppens tht if the function f α be obliged to stisfy conditions which determine ll its vlues by mens of the rbitrry vlues which it my hve for given finite rnge, from α = to α = b, the integrl reltive to α in the formul (f) cn be shown to vnish t the limit k =, for ll rel nd positive vlues of β, except those which re roots of certin eqution Ω ρ = ; (g) while the sme integrl is, on the contrry, infinite, for these prticulr vlues of β; nd then the integrtion reltively to β will in generl chnge itself into summtion reltively to the rel nd positive roots ρ of the eqution (g), which is to be combined with n integrtion reltively to α between the given limits nd b; the resulting expression being of the form f x = ρ dα φ x,α,ρ f α. (h) 19

22 For exmple, in the cse where p is cosine, nd f negtive exponentil, if the conditions reltive to the function f be supposed such s to conduct to expressions of the forms dα e hα f α = ψ(h) φ(h), (n ) dα e hα f α = ψ( h) φ( h), (o ) in which h is ny rel or imginry quntity, independent of α, nd hving its rel prt positive; it will follow tht dα e k α 2 (cos βα 1 sin βα)f α = ψ(β 1 + k) φ(β 1 + k) ψ(β 1 k) φ(β 1 k), (p ) in which α 2 is = α or α, ccording s α is > or <, nd the quntities β nd k re rel, nd k is positive. The integrl in (p ), nd consequently lso tht reltive to α in (f), in which, now p α = cos α, f α = e k α 2, will therefore, under these conditions, tend to vnish with k, unless β be root ρ of the eqution φ(ρ 1) =, (q ) which here corresponds to (g); but the sme integrl will on the contrry tend to become infinite, s k tends to, if β be root of the eqution (q ). Mking therefore β = ρ + kλ, nd supposing kλ to be smll, while ρ is rel nd positive root of (q ), the integrl (p ) becomes k λ 2 ( ρ 1b ρ ), (r ) in which ρ nd b ρ re rel, nmely, ρ = ψ(ρ 1) φ (ρ 1) + ψ( ρ 1) φ ( ρ 1), b ρ = ( ψ(ρ 1) 1 φ (ρ 1) ψ( ρ ) 1) φ ( ρ ; 1) φ being the differentil coefficient of the function φ. Multiplying the expression (r ) by π 1 dβ (cos βx + 1 sin βx), which my be chnged to π 1 k dλ (cos ρx + 1 sin ρx); integrting reltively to λ between indefinitely gret limits, negtive nd positive; tking the rel prt of the result, nd summing it reltively to ρ; there results, (s ) f x = ρ ( ρ cos ρx + b ρ sin ρx); (t ) 2

23 development which hs been deduced nerly s bove, by Poisson nd Liouville, from the suppositions (n ), (o ), nd from the theorem of Fourier presented under form equivlent to the following: f x = lim.π 1 dβ dα e k α 2 cos(βα βx)f α ; (u ) k= nd in which it is to be remembered tht if be root of the eqution (q ), the corresponding terms in the development of f x must in generl be modified by the circumstnce, tht in clulting these terms, the integrtion reltively to λ extends only from to. For exmple, when the function f is obliged to stisfy the conditions f α = f α, f l α = f l+α, (v ) the suppositions (n ) (o ) re stisfied; the functions φ nd ψ being here such tht φ(h) = e hl + e hl, ψ(h) = l therefore the eqution (q ) becomes in this cse dα (e h(l α) e h(α l) )f α ; cos ρl =, (w ) nd the expressions (s ) for the coefficients of the development (t ) reduce themselves to the following: ρ = 2 l l dα cos ρα f α ; b ρ = ; (x ) so tht the method conducts to the following expression for the function f, which stisfies the conditions (v ), f x = 2 l (n)1 (2n 1)πx l cos dα cos 2l (2n 1)πα f α ; (y ) 2l in which f α is rbitrry from α = to α = l, except tht f l must vnish. The sme method hs been pplied, by the uthors lredy cited, to other nd more difficult questions; but it will hrmonize better with the principles of the present pper to tret the subject in nother wy, to which we shll now proceed. [15.] Insted of introducing, s in (e) nd (f), fctor which hs unity for its limit, we my often remove the pprent indeterminteness of the formul (d) in nother wy, by the principles of fluctuting functions. For if we integrte first reltively to α between indefinitely gret limits, negtive nd positive, then, under the conditions which conduct to developments of the form (h), we shll find tht the resulting function of β is usully fluctuting one, of which the integrl vnishes, except in the immedite neighbourhood of 21

24 certin prticulr vlues determined by n eqution such s (g); nd then, by integrting only in such immedite neighbourhood, nd fterwrds summing the results, the development (h) is obtined. For exmple, when p is cosine, nd when the conditions (v ) re stisfied by the function f, it is not difficult to prove tht 2ml+l 2ml l dα cos(βα βx)f α = 2 cos(2mβl + βl + mπ) cos βl l cos βx dα cos βαf α ; (z ) m being here n integer number, which is to be supposed lrge, nd ultimtely infinite. The eqution (g) becomes therefore, in the present question nd by the present method, s well s by tht of the lst rticle, cos ρl = ; nd if we mke β = ρ + γ, ρ being root of this eqution, we my neglect γ in the second member of (z ), except in the denomintor nd in the fluctuting fctor of the numertor cos βl = sin ρl sin γl, cos(2mβl + βl + mπ) = sin ρl sin(2mγl + γl); consequently, multiplying by π 1 dγ, integrting reltively to γ between ny two smll limits of the forms ɛ, nd observing tht the development lim. 2 ɛ sin(2mlγ + lγ) dγ m= π ɛ sin lγ f x = 2 l = 2 l, l ρ cos ρx dα cos ρα f α, which coincides with (y ), nd is of the form (h), is obtined. [16.] A more importnt ppliction of the method of the lst rticle is suggested by the expression which Fourier hs given for the rbitrry initil temperture of solid sphere, on the supposition tht this temperture is the sme for ll points t the sme distnce from the centre. Denoting the rdius of the sphere by l, nd tht of ny lyer or shell of it by α, while the initil temperture of the sme lyer is denoted by α 1 f α, we hve the equtions which permit us to suppose f =, f l + νf l =, ( IV ) f α + f α =, f l+α + f l α + ν(f l+α + f l α ) = ; (b IV ) 22

25 ν being here constnt quntity not less thn l 1, nd f being the first differentil coefficient of the function f, which function remins rbitrry for ll vlues of α greter thn, but not greter thn l. The equtions (b IV ) give so tht l+α (β cos βl + ν sin βl) l α = (β sin βl ν cos βl) if ρ be root of the eqution dα sin βα f α α+l α l dα cos βα f α cos βα (f α+l + f α l ); (c IV ) α+l (ρ sin ρl ν cos ρl) dα cos ρα f α = cos ρα (f α+l + f α l ), (d IV ) α l ρ cos ρl + ν sin ρl =. (e IV ) This ltter eqution is tht which here corresponds to (g); nd when we chnge β to ρ + γ, γ being very smll, we my write, in the first member of (c IV ), β cos βl + ν sin βl = γ{(1 + νl) cos ρl + ρl sin ρl}, (f IV ) nd chnge β to ρ in ll the terms of the second member, except in the fluctuting fctor cos βα, in which α is to be mde extremely lrge. Also, fter mking cos βα = cos ρα cos γα sin ρα sin γα, we my suppress cos γα in the second member of (c IV ), before integrting with respect to γ, becuse by (d IV ) the terms involving cos γα tend to vnish with γ, nd becuse γ 1 dγ sin γα cos γα chnges sign with γ. On the other hnd, the integrl of is to be replced γ by π, though it be tken only for very smll vlues, negtive nd positive, of γ, becuse α is here indefinitely lrge nd positive. Thus in the present question, the formul f x = 1 π. lim. dβ α= l+α l α dα sin βα f α, (g IV ) (which is obtined from ( ) by suppressing the terms which involve cos βx, on ccount of the first condition (b IV ),) my be replced by sum reltive to the rel nd positive roots of the eqution (e IV ); the term corresponding to ny one such root being if we suppose ρ >, nd mke for bridgment r ρ sin ρx (1 + νl) cos ρl ρl sin ρl, (hiv ) α+l r ρ = (ν cos ρl ρ sin ρl) dα sin ρα f α + sin ρα (f α+l + f α l ). (i IV ) α l 23

26 The equtions (b IV ) show tht the quntity r ρ does not vry with α, nd therefore tht it my be rigorously thus expressed: we hve lso, by (e IV ), ρ being >, l r ρ = 2(ν cos ρl ρ sin ρl) dα sin ρα f α ; (k IV ) 2(ν cos ρl ρ sin ρl) cos ρl + l(ν cos ρl ρ sin ρl) = And if we set side the prticulr cse where 2ρ ρl sin ρl cos ρl. (liv ) νl + 1 =, (m IV ) the term corresponding to the root ρ =, (n IV ) of the eqution (e IV ), vnishes in the development of f x ; becuse this term is, by (g IV ), ( x β ) l+α dβ β dα sin βα f α, (o IV ) π l α α being very lrge, nd β smll, but both being positive; nd unless the condition (m IV ) be stisfied, the eqution (c IV ) shows tht the quntity to be integrted in (o IV ), with respect to β, is finite nd fluctuting function of tht vrible, so tht its integrl vnishes, t the limit α =. Setting side then the cse (m IV ) which corresponds physiclly to the bsence of exterior rdition, we see tht the function f x, which represents the initil temperture of ny lyer of the sphere multiplied by the distnce x of tht lyer from the centre, nd which is rbitrry between the limits x =, x = l, tht is, between the centre nd the surfce, (though it is obliged to stisfy t those limits the conditions ( IV )), my be developed in the following series, which ws discovered by Fourier, nd is of the form (h): f x = ρ l 2ρ sin ρx dα sin ρα f α ρl sin ρl cos ρl ; (p IV ) the sum extending only to those roots of the eqution (e IV ) which re greter thn. In the prticulr cse (m IV ), in which the root (n IV ) of the eqution (e IV ) must be employed, the term (o IV ) becomes, by (c IV ) nd (d IV ), { } 3x α+l πl 3 dα f α αc l(f α+l + f α l )αc, (q IV ) α l 24

27 in which, t the limit here considered, c = but lso, by the equtions (b IV ), (m IV ), α+l α l dθ vers θ θ 2 = π 2 ; (riv ) l dα f α α l(f α+l + f α l )α = 2 dα f α α; (s IV ) the sought term of f x becomes, therefore, in the present cse, 3x l 3 l dα f α α, (t IV ) nd the corresponding term in the expression of the temperture x 1 f x is equl to the men initil temperture of the sphere; result which hs been otherwise obtined by Poisson, for the cse of no exterior rdition, nd which might hve been nticipted from physicl considertions. The supposition νl + 1 <, (u IV ) which is inconsistent with the physicl conditions of the question, nd in which Fourier s development (p IV ) my fil, is excluded in the foregoing nlysis. [17.] When converging series of the form (h) is rrived t, in which the coefficients φ of the rbitrry function f, under the sign of integrtion, do not tend to vnish s they correspond to lrger nd lrger roots ρ of the eqution (g); then those coefficients φ x,α,ρ must in generl tend to become fluctuting functions of α, s ρ becomes lrger nd lrger. And the sum of those coefficients, which my be thus denoted, φ x,α,ρ = ψ x,α,ρ, (i) ρ nd which is here supposed to be extended to ll rel nd positive roots of the eqution (g), s fr s some given root ρ, must tend to become fluctuting function of α, nd to hve its men vlue equl to zero, s ρ tends to become infinite, for ll vlues of α nd x which re different from ech other, nd re both comprised between the limits of the integrtion reltive to α; in such mnner s to stisfy the eqution µ λ dα ψ x,α, f α =, (k) which is of the form (e), referred to in the second rticle; provided tht the rbitrry function f is finite, nd tht the quntities λ, µ, x, α re ll comprised between the limits nd b, which enter into the formul (h); while α is, but x is not, comprised lso between the new limits λ nd µ. But when α = x, the sum (i) tends to become infinite with ρ, so tht we hve ψ x,x, =, (l) 25

28 nd x+ɛ x ɛ ɛ being here quntity indefinitely smll. conducts to the development (y ), we hve dα ψ x,α, f α = f x, (m) For exmple, in the prticulr question which φ x,α,ρ = 2 l cos ρx cos ρα, (viv ) nd (2n 1)π ρ = ; (w IV ) 2l therefore, summing reltively to ρ, or to n, from n = 1 to ny given positive vlue of the integer number n, we hve, by (i), nπ(α x) nπ(α + x) sin sin ψ x,α,ρ = l + l ; (x IV ) π(α x) π(α + x) 2l sin 2l sin 2l 2l nd it is evident tht this sum tends to become fluctuting function of α, nd to stisfy the eqution (k), s ρ, or n, tends to become infinite, while α nd x re different from ech other, nd re both comprised between the limits nd l. On the other hnd, when α becomes equl to x, the first prt of the expression (x IV ) becomes = n, nd therefore tends to become l infinite with n, so tht the eqution (l) is true. And the eqution (m) is verified by observing, tht if x >, < l, we my omit the second prt of the sum (x IV ), s disppering in the integrl through fluctution, while the first prt gives, t the limit, nπ(α x) x+ɛ sin lim dα l f n= x ɛ π(α x) α = f x. (y IV ) 2l sin 2l If x be equl to, the integrl is to be tken only from to ɛ, nd the result is only hlf s gret, nmely, ɛ lim. dα n= sin nπα l 2l sin πα 2l f α = 1 2 f ; (z IV ) but, in this cse, the other prt of the sum (x IV ) contributes n equl term, nd the whole result is f. If x = l, the integrl is to be tken from l ɛ to l, nd the two prts of the expression (x IV ) contribute the two terms 1 2 f l nd 1 2 f l, which neutrlize ech other. We my therefore in this wy prove, à posteriori, by considertion of fluctuting functions, the truth of the development (y ) for ny rbitrry but finite function f x, nd for ll vlues of the rel vrible x from x = to x = l, the function being supposed to vnish t the ltter limit; observing only tht if this function f x undergo ny sudden chnge of vlue, for ny vlue x of the vrible between the limits nd l, nd if x be mde equl to x in the development (y ), the process shows tht this development then represents the semisum of the two vlues which the function f receives, immeditely before nd fter it undergoes this sudden chnge. 26

29 [18.] The sme mode of à posteriori proof, through the considertion of fluctuting functions, my be pplied to gret vriety of other nlogous developments, s hs indeed been indicted by Fourier, in pssge of his Theory of Het. The spirit of Poisson s method, when pplied to the estblishment, à posteriori, of developments of the form (h) would led us to multiply, before the summtion, ech coefficient φ x,α,ρ by fctor f k,ρ which tends to unity s k tends to, but tends to vnish s ρ tends to ; nd then insted of generlly fluctuting sum (i), there results generlly evnescent sum (k being evnescent), nmely, f k,ρ φ x,α,ρ = χ x,α,k,ρ, (n) which conducts to equtions nlogous to (k) (l) (m), nmely, ρ µ lim k= λ x+ɛ lim k= x ɛ dα χ x,α,k, f α = ; (o) lim χ x,x,k, = ; (p) k= dα χ x,α,k, f α = f x. It would be interesting to inquire wht form the generlly evnescent function χ would tke immeditely before its vnishing when nd ρ being root of the eqution φ x,α,ρ = f k,ρ = e kρ, 2ρ sin ρx sin ρα ρl sin ρl cos ρl, ρl cotn ρl = const., nd the constnt in the second member being supposed not greter thn unity. [19.] The development (c), which, like (h), expresses n rbitrry function, t lest between given limits, by combintion of summtion nd integrtion, ws deduced from the expression (m ) of the eleventh rticle, which conducts lso to mny other nlogous developments, ccording to the vrious wys in which the fctor with the infinite index, n (α x), my be replced by n infinite sum, or other equivlent form. Thus, if, insted of (o ), we estblish the following eqution, 2nα (2n 2)α we shll hve, insted of (c), the development: (q) α dα p α = r α,n dα p α, ( V ) f x = ϖ 1 p b (n)1 dα r α x,n f α ; (r) 27