FUNCTIONS IN THE THKORY OF INTEGRAL EQUATIONS. 179 



By employing the method of the preceding paragraph, and using the substitution 

 t s = w instead of s-f t = w, it may l>e shown that 



r. W = 



With the same convention as to the values of and a, it is easily shown that 



/.(-) =/(-)=/(-) 



(y^s^S, OSiSa) 



/. (*+o=yi (+)=/(* +0- 



Hence 



The first integral on the right-hand side of this equation is 



which may be written 



" cos n< rf<- *-+*+0 tau i f sin 



hence, applying the first corollary of II., 9, we see that it converges uniformly to 

 zero, for values of s in (-y, 8). Since the two other integrals have this property in 

 virtue of the same corollary, we conclude that, as n tends to oo, 



converges uniformly to zero, for values of s in (y, 8). As a corollary we deduce that 



has the same property. 



15. Referring to the formula} of 13, it is evident that 



s t s \ = _L |T 

 Hence 



2 A 2 



