218 



Proceedings of the Royal Irish Academy. 



where a < r <h; 



Lt. 



\.pKm{i\r)d\ 



,Kn (- i\9) P<P ip) dp = i-y^^ ~<p(r + e), (48) 



where a "^r < h; 

 Lt. "' 



C 



Lt. 



X . pKm(- i'^r) dX I qKn (- i^p)p(ji {p) dp = 0, 



A . pK„, (- iXr) qKn (- i^p] pc})(p)dp = 0, 



(49) 



(50) 



where a ^ r ^ &. When these equations are translated into -/ functions, 

 they give different expressions for the element of the integral, according as 

 p > or < r, so that they are not of much interest. 



Aet. 10. The Cases in vjhich a is zero and h infinite. 



It has hitherto been supposed that ct is not zero. When a is zero, how- 

 ever, but r not zero, it appears that, if 



\pH(p)dp\ . _ . 



converges, equation (14) holds for + n, and when n '^ j for - n ; while, if 



p'-"<P(p)\dp 



converges, it holds for - oi when n > ^ (provided, in all cases, Dirichlet's 

 conditions are satisfied for points not in the immediate neighbourhood of 

 zero). 



This may be shown, for instance, as follows : — The equation has been 

 established for both + n and - n when a is replaced by e, where e is any 

 positive quantity, however small. It suffices, then, to show that in each case, 

 under the conditions stated, the integral is zero if the limits of jo be and e. 



As the result is to be made good for + n when circumstances may be such 

 that it is not true for - n, I first establish it in the former case by the method 

 which is indicated in Art. 7. 



The question thus reduces to proving that 



-th [ ' kdXKni- iXr) I ' Jn{Xp)p<l> (p) dp = 0. (51) 



cJ-h Jo 



And it evidently suffices to establish a similar equation in which JTis replaced 

 by the dominant term in its asymptotic expansion. As for the J function, we 

 have, throughout the range, an inequality of the form 



I Jn{\p) - 23(7rV)-i COS {(2n + l)7r/4 - V} | < .4 | (V)"'^^"'^'' I > (52) 



where A is some finite constant independent of Xp. Lor, if we consider the 



