8 Proceedings of the Royal Irish Academy. 



an electric force c'-\ and magnetic force c^fio, such that 



c--{a! + 71 + rj/3o + A„) 

 is normal; and we thus get 



e" = Oo + C-'(ai + 7„) + C-'ia, + 71 + X„) + C-'foa + 72 + Ai) 



„"= c-'(io + c-'O, + 80) + c-'{fi, + S, + 8„). 



The electromotive intensity is normal as for the first, second, and third terms ; 

 and we can thus carry the solution to any degree of approximation. The 

 case in which the external field can be expanded in powers of c can be solved 

 by treating separately each term. 



As a firet example consider the case of a sphere moving without rotation 

 under the action of no external electromagnetic field, the charge on the 

 sphere being £ in electrostatic units. The cquQibrium state comes from a 

 potential Er'' ; and its continuation gives 



= E)-' + iJE'c-'jr-'(- i' + Spa) - r-'iSpaY) 

 - ^Ec-^2Spa - 2Sou\, 

 and the vector potential 



do = Ec^ar'^ - Ec-'a + . . . 

 To the first approximation the electric force e which is - VP„ - ro„ 

 = Er^'p + Ec-^\p{hr-\- a* + Spa) - ^r^Sp'oY] - haT']. 

 To get electromotive intensity we add 



VaVVzs, or Ec-*r^(pa' - iSpa). 

 Wo have now the following teiin : — 



Ec-^[- U»- + r'aSpa], 

 which is not normal to the sphere. 



Terms must be added such that to the first approximation the non-normal 

 terms are annulled. Assume a scalar potential 



P, = AS'aV.- + B(siv\' -, 



and we find by taking 



A = - - Ec^ and B = — -— 

 Z 



that the conditions are satisfied. We thus get for the complete scalar 



potential as far as c' 



