Bessel- Clifford Function, and its applications. 527 



The solution of the equation (\/ 2 -hm 2 )cf) = is given 

 in solid spherical harmonics S = S„ by terms of the form RS, 

 where 



„d 2 R -. _ dR 



r-V? + 2(n + 1> -r 4- mVR = 0, . , . . (4) 

 R n = CWlW = ^ +i (imV). ... (5) 



This equation arises in the solution of 



3 = « 2 V^, ...... (6) 



with a periodic time factor, so that -y \ 2 = — mV<£, for the 



propagation of spherical waves : or else in the conduction of 

 heat, with 



da .,—,0 ,_ x 



w = /tVM ' (7) 



and an exponential time factor of decay at compound 

 discount, u = e~ mm <f> ; and the surface conditions are to 

 be adjusted. Then we take 



P _ sin mr ~ _ sin mr cos mr 

 u mr m^ m 2 r 2 



o ~ t \-iii~'r~' *'\ 

 R 2 = t-5 - sin (mr + e) V-rCOS (mr + e) ; (8 ) 



and for n=— 1, S_i = — > R_i=cos (mr + e). 

 mr 



. A cos (fm'-f-e) /n \ 



</> = A ^ -—■' cos met (9) 



mr 



d 2 (f> 



is a solution of ^ = <rV 2 <£> the same as for n = Q. 



So also the form of Macdonald's electromagnetic equation 

 for 7p (Phil. Trans. 1910) shows that yp is a Stokes-stivam- 

 f unction (S.F.) composed of terms 



R n = ■*K»+ 1 )0n+ i (*)l w (Ai), or B+l (imV)V> +1 I n <>), . (10) 



