FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 173 



Again the solution of (22) which satisfies the conditions u = 1, -j- = 0, at s = is 



f/W 



COB \/\s ; and the. solution which satisfies the conditions u = 1, -3- = 0, at s = IT is 



cos </\(ir-s). It follows (cf. III., 4) that the GREEN'S function of (22) which 

 corresponds to the boundary conditions (11) is 



cos v/Xs . cos </X(ff t) , cos y/Xit . cos x/X(n a) / .\ 



~~ - ~p ~* - / - \^ ^ ^ ), ~~" - "/- ' - / \ ^~ / 



v/X sin X/XTT v X sin V/XTT 



Comparing this with the formula obtained in III., 7, we see that, as X = />', the 

 GREEN'S function is T x (*, t). 



Thus we have 



1 12 



F> (s, J) = + 2 -5 r - cos ns cos n<. 

 XTT * = in*-\ir 



From the lemma of III., 17, we see that 

 - XK A (s, t) EEEi 



ll/t 1 "!! A 



or 



N B 1 71 ~~ 



Hence, using the result obtained above, 



-x[K A (s, )-r A (, )] *^E 



+ S -^-(^ +1 (*)^. +l (0--coswco8n). . . (23) 



= 1 W A \ 7T 



Now, when s ^ t, the series 



/i (*) Ai (0- - + S (^+t() *.+i(0- - cos n * cos w< ) 



7T = 1 \ W / 



has been shown to be convergent ( 8). It follows by an argument similar to that 

 employed in II., 3, 4, that, as X tends to - oo, the right-hand member of (23) 

 converges to the sum of this series, that is to say to 



lim H (s, t, n). 



-VO 



Further, the left-hand member is 



pl\ (s, t) a. (p, s, t), 



which, for unequal values of * and t, has been shown to converge to the limit zero, as 

 p tends to oo (III., 8). It follows that 



lim H (s, t, n) = (s* t). 



