•i Proeeedings of the Royal Irish Academy. 



If the electricity appears as a surface-density, 



where civ is a directed surface-element at the place p. In this case, we have 



<r = -a.P = - ffo J Teh' [e'] T{p - p')-\ 

 If we take the point p just outside the surface (which is supposed fixed), and 

 if we let ui denote the value of tr at a neighbouring point just outside the 

 surface, then, in calculating a - a,- from the formulae just written down, the 

 only part which need be considered is that arising from the part of Vj 

 estimated normal to the surface, so that 



0- - <7i = - v-'^vV / Tdv [e] T{p - p')'^ 

 + ir>,SW,-jTrf,/[e]r(p,-p'r, 

 where, in the second integral, p is replaced by p,- ; and, by the usual theory of 

 such integrals, suitable restrictions beuig placed on the function e, and the 

 form of the surface near p, we find 



(7 - (T,- = iirl/ve. 



If the surface is a conductor, aud if there is no electric or magnetic force 

 inside, the boundary conditions are all summed up in a = iwUve, or 



£ = iTrUvC, J) = 4:1tC~-iUv. 



In the more complicated case of a surface moving with a velocity r, we 

 can take the origin moving with the surface, aud the vector p is a function 

 of the time, so that dp/dt = Srp, and the operator d/dt = StV, when applied 

 to functions of p. As before, it is only the part of V normal to the surface 

 which gives any result, so that we can replace Sr^ by Stv'^SvV. Hence the 

 effective part of ff^ is (ir' + hcSrv~^)Si'V, and we get for a conductor, under 

 the same conditions as before, 



a = {Uv + hcSTdv) e. 



(In the application of this formula, we must take care to include the convection 

 current in the current relative to the surface.) 

 If we consider the quaternion 



- (87r)-'<T(7, = CL (say), 

 we have the theorems that Sq^ is the electromagnetic energy per unit- volume, 

 and Fq has the direction of the Poyntiug flue aud of the electromagnetic 

 momentmn. Since in free space 



(Zo'^o = <'Do = 0, 

 we have in such space 



- (Stt)-' J <TD,,T,dv = 0, 

 or - (Sjt)-' J aAcj^dv - hc^ J d(l!dtodv = 0, 



