20 Proceedings of the Royal Irish Academij. 



under the integral sign. For let ua so differentiate, substitute in (7\ and 

 write Ji = a. Bearing in mind that 2fjiTAu{n) is the value of 



at a, we see that on doing so the coefficient of the integral in 21 in the 

 numerator of the integrand may be written in the form 



2^iTe>^^hl,i^a(iJL) {e'^'~'-">Fa{iJiC, - ^) - e-'^(--'-)Fa{^c, ^)},.„, (19; 



which is identically zero. This, of course, applies to the coefficient of jlli also. 

 As for the coefficient of 



e>^'*dp.[e-'''-^-iFt{pc, - ,x) - e-''!'-')i^6(^c, yi) \ (20) 



it is 2Jia X (13) ; but, when this operation is performed on (20), and x 

 subsequently replaced by a, the result is 



- l0'<'*d^ X (1.3) X the denominator of (16), (21) 



so that the contour integral is zero. 



Similarly for any of the other simultaneous equations. 



It has now to be shown that the left-hand members of (7) et seq. can be 

 legitimately obtained by differentiation under the sign of integration. 



We could add to (p terms of the type obtained from (6), allowing for the 

 differences in notation, by multiplying each element of the integrand by c'', 

 and such as would allow the sum to be differentiated term by term twice 

 in succession at all points, including a, h* To discover such terms we do 

 differentiate separately in this fashion, twice with respect to x, the two parts 

 of <p which involve e)^ and e'l^, and then in each case integrate by parts twice 

 in succession that part of the integral in u which involves 1//, but once only 

 that which involves ^ ; we thus obtain a double integral which, when t = Q, 

 would constitute a g'?<«si-Fourier expansion of \-i^/"iXj + |-x'(^')' ^^'^ ^^s° single 

 integrals in^ohlng values of i//, i//', ;)^, at the boundaries. The additional terms 

 required are then so chosen that, when they are differentiated twice, their 

 integrands annul the integrands in the above single integrals except the terms 

 which are requisite to make the Fourier expansions of I'li-'X^) ±ix'(^) ^'alid 

 at the boundaries. For the double integrals, being uniformly convergent at 

 t = (save near discontinuities in \p"{3;) or x'{^^))' remain uniformly conver- 

 gent, save for isolated points, when each element is multiplied by e^"', and the 

 same must therefore be true when some or all of the differentiations are with 

 respect to t. Thus the second derivates of the sum of ^ and these added terms 

 may be obtained by differentiation under the integral sign, except possibly at 

 isolated instants when discontinuities in ip" or x reach a directly or after 

 any number of reflexions. It will be assumed for the present that t/-", x ^^e 

 continuous ; and it will be shown later that in the event of discontinuity the 

 exceptions are only apparent. 



* Without any such addi lion the operation in question would be admissible escept at certain 

 instauts, one of which is < = 0. 



