6 Proceedings of the Royal Irish Academy. 



and for the vector potential 



n-l 



more generally if /m(V) in any scalar harmonic function of V homogeneous 

 and of m dimensions, and if V/,„(/)) =f'Jp), so that Spf'mip) = - mf,„{p), 

 then scalar and vector potentials are given by the equations (u being any 

 arbitrary function of t) : 



•P. - 2-^--(|) («v).-'«). (^) 



"■ = -i-'!-.7f(l)""<«^"-">- <"> 



The corresponding electrical force 



---^2^-4f^(|)"(^FVr„,V.-«). (C) 



and the magnetic force 



ir 



1 ^" (-)"c-"/a 



- -2 



m 





In these expressions the coefficients of the operational function /,„(v) niay 

 be functions of the time. If these coefficients, or m, are such that the 

 diffei-entiatious witli respect to t bring in powers of C, we can arrange the 

 series differently ; for example, if u = c*", we find, on collecting, 



We shall call the solutions (A), (B), (C), (D), (E), and (F), karmonicoid 

 solutions of the first ki7id. 



We have also a second class of solutions of a conjugate type, in which 

 the electric and magnetic forces ('„ and ij'm are connected by the equations 



t m = ~ 1m ) 1 m °° C " Cm- 



We shall call these harmonicoid aolviions of the second kind. 



If the centre of the sphere is at rest, these solutions assume well-known 

 forms. We may first notice that if F(r) is any function of r 



MV)F(r)=f„(p)(^^jF{ry 



* Hobson : Proceedings of London liUthematical Society, vol. zziv. 



