FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 171 



Evidently 



2 h 



m , = N m 



i 





1 ,s ^)1 



j 



From the remarks made above, it will be seen that g (s, t, * n) is limited for 



s . ^ TT, ^ ^ TT, S .v, ^ 7T, and ?i S N ; moreover, if 



9 (s, t, *,) = lim g (s, t, *,, n), 



-- 



</ (., *, * n) converges uniformly to" g (s, t, *,) for those values of s, t, s,, which satisfy 

 the conditions 



|ii*J<|Si?, |*ii*i-ir|i7, 

 and 



Applying a result obtained in I., 4, we see that, as n tends to oo, (20) converges 

 uniformly for S s ^ TT, S S TT ; and hence that (19) converges uniformly for these 

 values of s and t. It may be proved in a similar way that 



i cos ms cos mt A, (m, t) 



m = N 



converges uniformly for ^ s S TT, ^ / S w. 

 8. We have 



cosmt A. a (m.s) Iff';/ j i \ f; / 



!_/ = j s \h (s, t, n) ?1 ^-(TT-S) /< (a, , n) ?1 d, I (21), 



Wl> /7T I .' JO J 



m = N 



where 



8ip 



m 



i sin m(2s l s-\-t) sin m(2*, s <)1 



i=N W w = N W J 



It is easily seen that h (s, t, s lt n), like g (s, t, s } , n), satisfies the requirements of the 

 theorem of I., 4 ; and hence, from the remarks made at the end of this paragraph, 

 that the left-hand member of (21) converges uniformly for S* S TT, S t S IT. It 

 may be shown in a similar way that 



cos ms sin mt A a (m, t) 

 m = N m 



has the same property. 



z 2 



