12 Proceedings of the Royal Irish Academy. 



The harmonicoid fuuctions used al1o^•e have no simple physical meaning, 

 but, by the aid of a theorem which can be easily verified, we can construct 

 harmonicoid functions which represent potential of spherical shells having 

 a given assigned surface charge represented in spherical harmonics. This 

 theorem is as follows: — If dP be the potential due to an element of unit 

 density according to any law of force which depends only on the time and 

 the relative position of element and attracted point, and if /„(p) be a solid 

 spherical scalar harmonic in wliicli, for simplicity, we may regard the 

 coefficients as constants in time, then 



^;i"/"(V) [ " «..,^'^..-. f"""V'„.,*T,., . . . [\IP 



"■ Jo Jo Jo 



(where dP means the potential of a sphere of radius «,) is the 



potential of a surface-distribution of amount fn{Up) over a sphere of 

 radius «, and this holds both for external and internal points. 



We have also the fact that the potentials P and H, and a uniform 

 spherical shell of charge E, are given by 



" = 2;jf2H!-UJ " — V — -' ^"^™""y- 



From these formulae we find for the electric force t inside such a shell 

 2B 



'3^"' 





2E ._. , / 3 3tt' 3 4tt' \ 



3«c' 



and we find for the magnetic force ij the equation 



2E ,, ... /3 1m= 3 2m' 



,^ 2E ., ,.. /3 1m= 3 2m' \ 



/ nn = — — — ■ rr I a-' a \ -^ • - — + -• — . . .l- 



■■'">'" \2 3 c' 2 5 c* J 



