I ('NOTIONS IN Till. THEORY OF INTEGRAL EQUATIONS. 151 



formula for K A (x, t) \H of the same character as that obtained in 7. Thus, when the 

 pair of conditions for (0, IT) is B', we have 



where r A (s, t) is the symmetric function of * and t which is such that 



r A (, o = si " h p*- c <* h M"-0 ( , ^ A 



p cosh pir 



When the pair of conditions is *B', the function F A (s, t) in (41) must be replaced 

 by T A (ft, t), where T A (s, t) is the symmetric function such that 



-0 , ^ t) * 



p cosh pn 



Finally, when the pair of Ixiundary conditions is *B', it will be seen that O l\ ( s 

 must be replaced by ;i\ (s, t), where the latter function is symmetric and such that 



T ( f\ - sinh P s s '"h P (ir- / ^ f \ 



u 1 x l*i l ) : i s (S ^ 1 1. 



p sinh pn 



From these formulae it may be deduced that the results obtained in 13, 16 are 

 still applicable. At the end point * = 0, it will be found that the first of the 

 inequalities of 15 is applicable when the boundary condition at this point is 



du , , 



li'u = 0. 

 as 



When the boundary condition is u = 0, we have 



K A (0, = (0 < t S TT), 



and the inequality is no longer true, save under special circumstances. Corresponding 

 remarks apply to the end point .s- = n. 



The result obtained in 17 also requires modification. Using the same notation, 

 when the pair of boundary conditions is either B', or "B', the inequality (38) must be 

 replaced by 



From this it may be shown that, in both cases, the difference between 



A. i / \ i /j\ /o/\ 



V* v"v r w> (""/ 



ami 



i(-i)*-X 

 * The reader will perceive that this result may at once be deduced from the former one (cf. 6). 



