222 Proceedings of the lloijol Irish Academy. 



differentiation under the sign is legitimate. For, failure of uniformity in 

 convergence near isolated points does not invalidate the process. 



Similarly for the succeeding differentiations, when the Fourier conditions 

 are satisfied. 



Akt, 13. Simple Example of Bcssel Expansion analogous to tliat in Art. 1 : 

 Expression of <p{r) as Integral v:liose Element, as a function of r, is 

 a Multiple of 



\JJM)I-n(y''-)-I-n(^r)J„(\ay]dX 



I proceed now to consider expansions in Bessel functions analogous to 

 that in sines and cosines which is expressed by equations (9), (10), (12). 

 In the analogous physical problems it is required to expand, for values of r 

 between a and h, where a < h, an arbitrary function, ^(r), as an integral, each 

 element of which, as a function of (r), is of the form 



{CiX)Jn{Xr)^C"iXiJ.,{Xr)\clX (58) 



where the ratio of to 6" is expressed by a fraction whose numerator and 

 denominator involve Bessel functions in a manner which probably will be 

 best illustrated by taking a simple example. The discussion is most con- 

 veniently conducted in terms of the K functions. Suppose the element is 

 required to be, in so far as it depends on r, a multiple of 



Kn(iXr)X„ {- iXa] - K,, (- iXr) K^ [iXcc], (.59) 



or, in other terms, a multiple of 



Jn (Xr) J_y, [Xa) - J_n 'Xr] /„ (Aft). 



It is supposed, as usual, in translating from K to J functions, that in the first 

 instance n is not an integer : so long as we keep to K functions this question 

 does not arise, howe\er. 



The required expansion in this case is given hx the equation 



^ ,^ fi : AVi\r K„{ri.Xa)-K„{-iXr)KJiXa) \ \ JT,/ iXp)K„{-iXa)-X„{~iXp)KJiXa) \ 



A«A ^ ' ^ :— -r — '—p(b(p)dp 



= - ly {'K''^ ' t) + (p(r -1- e) ; , r > a 

 = 0, /' = a 



(60) 



In this, the numerator of the fracti^ai in the integrand may be expressed 

 otherwise, as 



-^/,/Ar)J:4AfO-J_,,(A/^/,(Aa,||J",,(Ap)J:4.VO-•-^-»(Ap)^.(Aa)|, (61) 



4 sin 71 7T 



