FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 177 



When the pair of boundary conditions for (0, IT) is ;B', (27) must be replaced by 



sin mf - ....... (32) 



IT I 



The GREEN'S function of (22) for the Ixtundary conditions 



u = at * = 0, u - at .< = IT, 



will lx? found to be 



, t) = 2 -= - - sin .< sin nt, 



= 1 n \ 7T 



from which it is easily proved that, as n tends to <x>, (32) converges to zero for 

 unequal values of a and t. We deduce that, if fa, (a) is the sum of the first n terms 

 of the series 



- sin .< [ f(t) sin tdt + - sin 2* \ fit) sin 2< dt + . . . + - sin n* (' fit) sin n* <ft + . . . (33)* 



IT JO IT Jo IT JO 



then, as n tends to oo, 



<r. () - ; (*) 



converges uniformly to zero in the interval (0, IT). 



13. We proceed to investigate the behaviour of the differences between the 

 various pairs of sums which we have denoted by 5. (.<?), <,<; H (s), "? (*), and fa, (), as ?i 

 tends to oo. The reader will easily prove that these sums are 



s , = 



1 , 

 J 



(l? (.) = _L f' fit) r 8inrt (*-0 _ sin < , 



W ' 



"o /,\ - _L f * f( t \ fain j(2n+ !)(*-<) _ 8inj(2n+l)(<+t) 

 SJ /W L Bin *(.-) 8ini 



Let us define a function /, (s) for all values of . by the rules 



= OS.o*7r, =0 -w< 



and a function f a (s) by the rules 



* This is, of course, FOURIER'S sine series corresponding to /(s). 

 VOL. COXI. A. 2 A 



