ELECTRIC AND MAGNETIC INDUCTIONS IN THE SURROUNDING FIELD. 305 
It may be noticed that in a steady field 
dA' dW cdy 
clt ’ dt ’ dt 
are all zero, so that 
where 
V=- 
d dp d dM. dp d dM. dp 
dx dt dx' dy dt dy'dz dt dz 
We may obtain equations of the same form as those given in (14) without any 
hypothesis as to the movement of electric induction tubes, merely assuming that 
the total current through a curve is equal to 477 X line integral of magnetic intensity 
round the curve. 
We start with the following definitions. Let A, B, C be the time integrals of the 
components of magnetic intensity since the origin of the system. 
Then 
A=ja dt, B=j/3d£, C = jycZ£ 
and 
a 
B= 
dt ’ p 
dt ’ 
dC 
We have the equations 
dy d/3 
7711 dy ~dz 
and two others. 
Integrating with respect to t we have 
also 
dC 
dB 
dy 
~ dz 
dA 
dC 
dz 
dx 
d,T> 
dA 
dx 
dy 
which are of the same form as (12). 
Hence exactly as before we obtain equations (14) and their solutions (15). 
The equations for the magnetic intensity are now 
a — 
dA 
dt ’ 
P= 
d B 
dt’ 
dC 
y=Tt 
If we differentiate (14) with respect to t, and substitute from these equations for 
magnetic intensity, we obtain 
