PROF. A. ('. lIXi>\ ON STI'IIM l.lorvil.l.K HARMONIC i:\lv\\slo\s. 425 



where tin- last eight integrals all tend to /.fro when K is increased without limit, and, 

 therefore, 



(II) 



K -> " 



\\ hidi was to IK- proved. 



From this result one of the expansions of SrritM and Liorviu.E can be deduced by 

 considering the singularities of the subject of integration in F (t, x, R), that is, the 

 values of X for which >fr(l,0) = 0. 



. When *(l,0) = we have \!s(x,0) and \Js(x,l) the same but for a constant 

 factor, since 



*(l,0)*(aj,0)-*(l,0)*(*,0) -*(*,!) ....... (12) 



Also 



, 



it A .'i> p 



1 1. Mice, the residue of V(t,X, It) is 



- 1 y, (r, 0) y, (t, 0) - f 1 {+ (x, 0)}'dx., 



pi ' p 



and we have 



I OH'rfar . . . (13) 



//"' xiumnntion referring to the infinite series of ralnex of \ for which ^(l,0) = 0, 

 t tiken in ascending order of magnitude; thus f(jr) ix expanded in a series of 

 f n,ift ions <p satisfying (3) and such that 4> ranittltes at each of the extreme rallies 0, 1. 

 In order to investigate the validity of the expansion when x = 1 we need to discuss 



which 



fF(,l,B)ft 



Ja 



Here the only term of importance is the second part of the rirst integral, which has 

 the value 2nr. From this it follows, in the same way, that the expansion holds 

 good at the upper limit, and a like result can be proved when .r = ; in each case it 

 is supposed that f(x) is continuous and of limited total fluctuation in the 

 neighbourhood. 



The course of the proof, moreover, shows that the series is uniformly convergent so 

 long as x lies within an interval which is contained within another interval in which 

 /'(.'> is continuous and of limited total fluctuation. 



It ha l>een supposed that p is limited lx>th ways. When this is not so, but the 



integrals of p and - exist, the argument still applies if we detach the factor from 



f Pt 



vol.. rex i. A. 3 i 



