246 



Proceedings of the Royal Irish Academy. 



omitting for the sake of brevity the factor which is a function of \,a,h,r, 

 involves 



a 



In virtue of the equation 



{didp} {XpJ'„(Xp)} = - {y~p-nY')dn{Xp), 

 if we subtract from (62) 



[ n'Xp-'J„{Xpmp)dp, 

 we can integrate by parts, and thus obtain 



(62) = (64) - 

 If we now add 



X^pd'.{Xp)^(p) 



x'pr„{Xp)ip'(p)dp. 



xj„{Xp)4.'(p)dp, 



and write XpJ'„{^p) + Jn{Xp) = {djdp){pj„{^p)), 



we can again integrate by parts ; and, doing so, we obtain 



(62) 



(63) 

 (64) 



(65) 



(66) 

 (67) 



X'pJ„(\p)x(,(p)dp 



AVA(Ap)^(p)-Ap/„(A;a)f(p) 



XJjXp) [-p4^"{p) - xPXp) + n'p'4^{p)] dp. (68) 



Consider now the order of magnitude of the contributions which the 

 several parts of the right-hand member of (68) make, when multiplied by 

 the omitted function of X,a,h,r and taken along with the corresponding 

 parts involving J-n{Xp), to the second differential coefficient of the term 

 of (61). 



The portion arising from the integral and the corresponding integral 

 involving J_„{Xp) is eventually the term of a Fourier expansion of 



^P'\r)+ r-'xp\r) -n^-rmr). ■ (69) 



Such an expansion is, under the conditions stated, uniformly convergent, 

 save near the boundaries and the discontinuities in \p", and its term is of 

 order not exceeding X~\ 



The portion obtained from the boundary terms on the right of (68), 

 and their analogues involving J-„{Xp) is obtainable from the left-hand 

 member of (61), on replacing the integral by 



- A>j',.(Ap)^(p) - Xpj,.[Xpy4.'{p) 



{XJ'_„{Xa)-hJ_„{Xa)} 



X'pd'_„{Xp)xp(p) - XpJJXp)xP\p) \Xd'n{Xa) - JhJ„{Xa)\. (70) 



