XVII
285
FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS
where we now have for U, V, W
A² d²
A d¹
U = ū
-
π +
4π dt
1672 dt
...
A² d²
A¹ d¹
V = v
4π dƒ³v+
dt2
1672 dt+
A2 d2
A¹ d¹
W = w
-
4π dt200
ow +
w
162 dt
au av aw
= 0.
+ +
дх oy Əz
Corresponding equations hold for the magnetic currents. If
the series are convergent, there is no reason to doubt that they
give us the true values. But in general they will converge.
For let us consider that element of the integral U which is
due to the current u in a certain element of space. We resolve
this current into a series of simple harmonic functions of the
time and suppose u。 sin nt to be the term involving sin nt.
Then the element of U due to this term will be given by the
equation
dUdr
do sin nt
T
1
-
1 A2
1.24π
+
1
A4
1.2.3.4 1672
+...).
If n and rare
few will be in-
Hence also the
This series converges to a limit easily found.
not very large, then every term after the first
finitesimal compared with the preceding one.
integral of the elements of U will have a determinate value.
Since the same is true of V, W, P, Q, and R, we may expect
to find, in the equations (9) (10) and the corresponding ones
for magnetic currents, a system of forces in complete agreement
with all our requirements.
3. It is obvious that this system may be represented, or in
technical terms described, more simply than by the equations.
(9) and (10). By these equations we have
²U
= -
2
d2
ū
dt2