238 Proceedings of the Royal IrisJi Academy. 



If X = rt, the value of the left-hand member is 



provided either that the contour cuts the imaginary axis at equal distances 

 from the origin, or that the fraction Fi (go , a)/i^2 (- 00 , a) is numerically equal 

 to unity, the multiplier in the latter case being either zero or 2. 

 If X = h, its value under similar suppositions is 



f F^(cc,h) Fi{-'jz,h)i - 



-*(S-0|l-2^-^^)-2^-^}- (26) 



And the left-hand member may be written in the form 



'B 



&^^^'i)Fl~^i,h)-eM--^)F,{^,l)] {e'^i^^~-^Fo{-^i,a)-e-'^''"--^F,(^,a)} 



e>^'-^'"^F^{^,h)Fl-f_i,a)-&^^--i)Fl-f^i,lj)F,{fx,a) -. <Mdu,. 



(27 



where ^B is used to denote the sum of the residues of a function at cdl its 





 poles. 



Thus a sum theorem of the type desired is rigorously established. 



It seems unnecessary to translate into trigonometrical notation. 



It is noteworthy, however, that the problem proposed does not appear to 

 admit of a unique solution, unless the equations (5), (6) are of the simple 

 forms (2), (3). In other cases, any one term in (27) can be expressed in 

 terms of the others by means of the theorem itself. Apparently a necessary 

 and sufficient condition for uniqueness of expansion is that the terms should 

 be "orthogonal" functions, i.e. that for every two different values of fx 



Uiii^doi 



a 



sliould vanish, where Ui, u^ denote the corresponding expressions of the 

 type (4). 



A similar remark applies to the Bessel expansions below. 



Art. 2. Fxample of Bessel Exijansion : ^ (r), a < r < h, exirressed as a siwn, each 

 term of which is a Multiple of J„ (Xr) J_„ (\a) ~ J_„{Xr)J,f\a), 

 X being so chosen that each term vanishes at r = h. 



Instead of proceeding at once to consider any very general sum theorem 

 in Bessel functions analogous to that just discussed, it may be desirable to 

 illustrate the argument by a comparatively simple example. 



