ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD. 297 
L' — — fx j| \^dxdydz 
M'= —jj^^dxdydz 
W=-M- r dxdydz 
H = 
Y- 
( 4 ) 
477 
then the following will be solutions 
dL , dM , dN\l 7 7 7 
^+^+ltP lxd,jdz 
r _ t / rfH 
” _ ~dx 
M = M'— 
dy 
N=N' — 
dH 
( 5 ) 
It is evident that we may add to the right-hand side of equations (5) f y, ( y- 
ax cty ccz 
respectively, where <£ is any function of x, y, z, since these will disappear from (3) and 
also from (l). 
The electric intensity, in so far as it depends upon magnetic motions, will consist of 
two terms, one depending upon the motion of the material at the point (its 
components being found as in Maxwell, vol. ii., p. 227, note), the other upon the 
motion of magnetic induction about the point. We may add a third term, arising 
from any electrical distribution with a potential xjj. 
If there is no material motion we shall have 
p dJL d\Jr 
dt dx 
„ dM d-yjr 
dlt~ ~dy 
p dN d-ty 
dt dz 
y 
( 6 ) 
Substituting from (4) and (5) we get 
P=- 
=-dff 
du' 1 
dt r 
du 1 
■-dxdydz-jz - - - + - ) : dxdydz■ 
JL §L 
477 dx , 
J_ £ 
477 dx , 
it -dxdydz--j x jJJ(^ + ^+ <feA 
d dL , d dM , d dN\l 
— - I ' ‘ ■ j™ J ( 
dx dt dy dt dz dt Jr 
dP . dQ , dRU 
d\p 
dx 
dxdydz 
_ 1 <L[[[(*l± . . ^t^dxdvdz-^ 
477 dx] J J \dic 2 dy 2 dz 2 Jr ^ dx ’ 
dL 
substituting for —, &c., from (6). 
