FTTXDAMENTAL FORMULATTOXS OF ELECTRODYXAMTCS. 
237 
that is to 1)6 made mill, afterwards determining the functions (p.^, Aj, Ag to satisfy 
the restrictions which necessitated their introduction. The variation can now he 
affected in the usual way, and the condition that it Aainishes leads to the following 
equations 
E + grad ^i+ - = 0 , 
^ c dt 
B—grad 0 ,— - = 0 , 
® c dt 
with 
Curl Ai —grad ^ 
' ^ c dt 
with three equations of each of the types 
d / 
BL _ 
-c('^ + - 
1 (ZAiA 
dt\dxj 
dx^ 
\0a3. 
c dt / 
d /0L \ 
0L 
nj/'i-fe + i 
d Asj. \ 
d.AdxJ 
dt 1 
+ - [i/. Curl A,], 
c 
The first and third of these equations show that <pi and Ai are the usual scalar and 
vector potentials ; in fact from the third we have 
Curl Aj = grad f/)^ + 
1 
c dt 
so that Aj is the vector potential and then is the scalar potential. 
The fourth equation thus determines the usual expression for the reaction forces on 
the moving electron ; the fifth equation determines similarly the force on the moving 
magnetic pole in the form 
I (LA,\ m 
m (grad 0^+ y ^ Curl AJ = 7 u( B— - Curl AJ^ 
We have yet to determine Curl As; we have 
B = grad 02 + - 
1 tZAs 
so that 
Now 
c dt 
Curl B = - 4 Curl As. 
c dt 
Curl B = -C, + 47 rCurl 1 + - —C„ 
C G dt C 
1 
c dt ’ 
