FUNCTIONS IN THE THEORY OF INTEGRAL EQUATIONS. 137 



for all values of Si which helong to the interval (0, s), it appears that the third term 

 on the right-hand side of (12) is of the form 



a. (p, s) cosh ps ; 



and it is evident that the fourth is of the same form. Thus (12) gives 



&\ (*') = p sinh ps + a(p, s) cosh ps. 



Proceeding in a similar manner, we may obtain analogous formulae for fa (s) and 

 <f>\(s). It is, however, more expeditious to deduce these from the formula already 

 obtained for & (s) and ff^ (s) by a device which we proceed to explain. Putting in 

 evidence the argument of q, let u (s) be the solution of 



+&(-)+*] -o 



in the interval (0, TT), which satisfies the conditions u = 1, -=- = H', at s = 0. 



as 



Clearly u (TT s) is the solution of 



in this interval which satisfies the conditions u = 1, '-^ = H', at s = TT. Recalling 



as 



that the asymptotic formula (11) is valid for all values of h', and for all continuous 

 functions q, we deduce from it the formula 



and similarly, in virtue of the relation 



^[tt(ir-*)] = -' (*). 



we obtain 



tf>\ (s) = p sinh p (TT s) a. (p, .s) cosh p(irs) 



from the formula given for 0V(<). 



7. Supplying the formulae of the preceding paragraph in 



4 (X) -4* ()?() -& (Iftft), 



we obtain 



S (\) = cosh p (rrs) [p sinh ps + a. (p, s) cosh ps]( 1 + ^Ifiiiz J 



+ cosh ps [p sinh p (IT s) + a. (p, s) cosh p (TT .)] (l + a '^' S 'J , 



where it must be borne in mind that the various symbols a (p, s) do not necessarily 

 refer to the same function On multiplying this out it appears that the terms which 

 VOL. ccxi. A. T 



