WEBSTER. — PKOPERTIES OF THE ETHEK. 517 



and then write 



JJJ-^' 



as 



(IT 



• / 



00 



J f j [k/ + 2 H,..Hz, + kL')dT. 



But by Green's Theorem, whatever these parts are, it' both vanish at 

 infinity, 



(13) j f j ks'kjjir ^ 0. 



(X) 

 + ^ 



In this case Hs is completely determined by equation (3), so that, if 



+ ^ "*' . 



5E, 5E, and 8U are zero, SHg is zero, therefore 



(14) j j Jbkhlr = 2 J Jj (ks'SkL + Hl-5HJ (It 



= 2J j J kL-dKLflT. 



00 CC 



00 



But this must be zero whatever 5H/, is, therefore Hz, is zero, as is H^, 

 also. Therefore 



+ - + 



(5) V-H = 0, (6) VH = 0. 



To derive equations (7) and (8) we may introduce vectors I, I, P, 

 and P, defined by the equations, 



(15) 



(16) ■ P = -^ Pot (E + i), P = -1 Pot (E + i). 



47r 47r 



From these equations we may infer 



(17) V^P = - (E + i), 



V-P = -(E+ I) = -(E + pP). 

 From equation (3) we know that 



(18) V(E + pP) = 0, 



