Hence 



Proceedings of the Royal Irish Academy. 



0. 



V^iA-F-'-|^+47rpF= 



With practically the same analysis, we find 

 or, if /-, I2, 1-i denote the real current, 



V^ - y-" ^3) (^> ^> ^) + 47r (/:, /„ /a) = 0. 



We shall now pass on to the general case in wdiich the sphere has, in 

 addition to the motion of its centre, a variable motion of rotation. Let us 

 take axes fixed in the sphere, the centre being origin, and let. the direction- 

 cosines of these axes with reference to axes fixed in space be, at the time ty 



U{t), nh(t), 7h{t), 

 kit), on^it), n.{t), 

 k{t), iiH{t), n,{t). 



Also let the coordinates of any point in the sphere referred to the moving 

 axes be Xo, iJq, Zq, then 



^ = 



pr,\lr, Viiriydio d(o{V'{t-uY 



- {x- x,{tt) - x,h{u) - yok[u) - Soh{u)Y 



- (y - 2/1 00 - Xomi{u) - y(P7H{u) - z^iiu{tL))- 

 -{z - Zi{u) - XQni{u) - yon2{ti) - Zoni{ti)f}'' 



' pr.^'dro [ Viniy die [VH - uf - r(«y- - r,,^ 



- 2x^{li{u) (x - Xi(t()) + iiu{u) {y - 2/1 00) + n,{u) (z - Zi(u))) - &c.}- 



pro'dvo 



du 



V\niy , V'(t- uY - (r (u) - r,y- 



2r,r(u) ^ r\t-uy-{r{u) + r,f 



so that a motion of rotation does not alter the scalar potential. 



The corresponding vector potential may be divided into two parts {Fi,Gi,Si) 

 and {F2, G2, Hi), the former depending on the velocity of translation of the 

 origin Xi{t), x^' {t), o:^[t), and satisfying from the preceding analysis the 



