518 PROCEEDINGS OF THE AMERICAN ACADEMY. 



SO that — V-Pot(E + pP) = VP = 0; 



47r 



+ 

 and since, whatever P is, 



VxVxP - — v-p + V(V -P), 



(19) VxVxP = (E + pP) = VxH, 

 which, combined with (5), gives 



(20) • VxP = H. 



(21) Similarly VxP = H. 

 Equation (12) now takes the form 



(22) / / / / }(VxP-Vx5P — 6VxP-Vx6P) 



^' °^ — ^5(E2 — GW + 2 U)}dTdt = 0. 



(23) But 



I VxP- Vx6Pf// = -VxP- Vx5P| — / 





1 • I r •■ 



= -VxP-VxSP — / [v{Px(Vx5P)} + P.VxVx5P](fi. 

 (24) .-. 

 / VxP-Vx6Pf/< ^ \ I I I VxP-Vx5PrfT I 



- y y j JiP'VxVxdTdrdt -J J'f^ f j ipx{Vx5F)'(}&dt, 



where *S is any closed surface that may recede indefinitely in all 



directions from any interior point, and of which dS is an element 



considered as a vector in the direction of the exterior normal. If 



we now let 51. = 0, then 6P = at ti and to, and the first term on 



the right side of equation (24) drops out, and so does the surface 



integral when there is only a finite amount of charge in the universe. 



+ 

 Treating P in the same way, we obtain, if 



81= 81 = 0, 



