Conway — On the Motion of an Electrified i^plicrc. 3 



are connected by the relation a'"P = (SVot, and arc solutions of the equation 



V= + c-*3V9(!» = 0. 



The vectors £ and »; satisfy 



c-'i = V„, (3) 



- 7) = Vf. (4) 

 If, however, there exists a current i, these latter equations become 



c-=£ + 47rt = Vi), (5) 



- ^ = Vf. (6) 

 If the origin is moving with velocity a, we may write them 



c'-'i + c'-ShV, £ + 4i7Ti = FV)j 

 -7] - SaV. >) = VVt. 



Suppose that the current becomes confined to an infinitely thin sheet, 

 the unit normal to which is l/v, and that the sheet moves with velocity a as 

 if rigidly attached to tlie origin, and let e, r/ be the values of the vectors on the 

 positive side of the surface (i.e. containing v) and infinitely close to it, and let 

 t', 1} be the corresponding values on the negative side ; then by integration we 

 obtain the following boundary conditions* : — 



c-2(£ - i')SoUv + 47rt = V.Uv{,}- 7,'), (7) 



- („ - ,,'),'SaV'v = V.Uv{t - £'). (8) 



where i is now the surface current density. In fact, since it is only the 

 normal component of V which causes the discontinuity, we can replace V 

 by - UvSUvV and integrate. The above then represent the boundary 

 conditions at any moving current sheet where 



- SaUv 



is the velocity at the point of the sheet normal to itself. If (o denote the 

 relative current density, and c the electrical surface-density, we have 



I = to + eir where Stav = 0. 



* Royal Dublin Society Transactions, vol. viii, sei'. ii, 7. Macdonalil, "Electric Waves,'' 

 pp. U, 15. 



