Conway — A Theorem on Moving Distributions of Electricity. 5 



In the same way for a component F of the vector potential, we would 

 obtain the value 



rr[t) 



2Tr pVrQclrQ 



r du r{ii) 



r{t) Jt du r{u) 



On differentiating and omitting parts which cancel, we have 



dx 



dt 



rr(t) 



277 p V%dr, 



1 di 



1 dr 



\r{T'') dx r{T) dx 



r[t) \r{T)dx r{T)dxJ 



■>■{() 



2irp V'^r^dr^ 



— -— du + 

 r dx r{u) 



2irp V\dro — du. 



r{t) Jt 9^ r(u) 



' d 1 



fW. ^.3 , / 1 8r" 1 a: 



2Trp VXdr, ( — ^ :^ - ZT^^, 

 jr{t) '^ V{-r)^^ r{T)dt 



?-f=2.,F=*-« 



a«' 



dx 



dx 



dr^ 

 dx 



■r{t) 



1 d 



dr" d 1 



1 8'-7 



8r a 1 



^''^^'''»*'«\r(r") dx' ^ ^ dx dxrij") r{r)dx' ^ dx dx ^(0 



r(t) 



\_ d\' ^dr d 1 



r" 92 1 



1 8^ _2?!: 9 1 



T dx' r{iC) 



a^ 

 a?!' 



2itpV 



2Trp V\dro ^ ^^ ^^ cZw + 27rp FVo(f?'o 

 drit)ffdT'\ fdr' 



dt \\dt 



dt 



rr{t) 



+ 2iTp V^r^dr^ 



+ 2 



1 av ar a i 



r{T") df dtdtr{T'') r(T)dt- dt dt r (r) 



r{t) 



9 TZ3 /7 ( ^ ^^^' o ^'"' ^ -^ 1 av ^ ar a i 



where outside the signs of integration t = t" = t. 



On forming, by addition, the expression Vip - V-'d~4^ / dt', we find that 

 all the expressions inside the signs of integration vanish, and there remains, 



on putting in the values of ( — )-{—), &c., 



\ dx J \dx J 



-,.,v'i(^)\(^f^^'.f^'-m 



v--{?rjf>)'\('"-'m\].-i,,r' 



\ dt 



dt 



