22 Proceedings oj the Royal Irish Academy. 



the differentiations required in the left-hand members of these equations can 

 be performed under the integral sign. And this can be shown to be true for 

 the set of terms depending on values at the boundary 6, and separately for 

 the set depending on values at a. 



It is already known that, for all values of x, t, each of the two parts of 

 (22) can be differentiated once with respect to x or t under the integral sign ; 

 and it is to be proved that for x= a each can be so differentiated twice with 

 respect to t and for all values of t. And this will follow if it is shown that 

 the integrals so oljtained converge uniformly for all values of t. Now, when 

 we do so differentiate, in the set of terms which involve Pj - jIIi, on 



replacing 



f'''^-"' Faific, - ix) - e-''(^-°) Fainc, ft) 

 by 2fiTAu, the polynomial in the numerator is two dimensions lower than 

 those in the denominator, irrespective of exponentials in the denominator, 

 so that this part is uniformly convergent. And in the set of terms which 

 involve J^\ -Pa the integrand, after the double differentiation, is asympto- 

 tically a multiple of e'"''^"VZju, and therefore, as in the preceding, the integral 

 is uniformly convergent. For (22) the desired result is therefore established. 



It has next to be proved that, for x = a, (23) can be differentiated twice 

 with respect to t under the integral sign ; this follows if the integral so 

 obtained converges uniformly for all values of t. This is true for the part 

 which involves Pi - illi as fxAnlPi - jll,) is four dimensions lower than 

 the polynomials in the denominator. 



In the remaining part replace 



„n, - P„ . Ax,/ An by („n3 - aDi . A,,IAni + («n, - Pa) A,,/A,„ 

 and consider first the portion involving oil, - nfli . ^i.,/^u- The expression 

 Au aTlz - Aua^i is the product of A„ and another determinant; and the 

 operation Aa when performed on e'^'^^'''^ Ftdic, - fj.) - e'l^^"-''') Fi,(nc, n) makes 

 it become identical, save for sign, with the denominator. For this portion, 

 then, the contour may be replaced by a finite one which contains the zeros of 

 Au, and the integral is therefore uniformly convergent. As (alli - Pa) AnlAn 

 is three dimensions lower than either polynomial in the denominator, the 

 portion of the integrand which involves this expression is asymptotically a 

 multiple of 0^''n'^d/i, and therefore, as before, this portion also of the integral 

 is uniformly convergent. Thus the desued result has been established 

 for (23). 



In the preceding it has been supposed that \p", x have no internal 

 discontinuities, and if there are such discontinuities it has not been made clear 

 that the second derivates of <pa, Fj, etc., can be obtained by differentiation 

 under the sign of integration at the instants when these discontinuities 

 reach a directly or after reflexions. 



