280
FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS XVII
consisting only of electric currents which still neutralises the
forces of system A everywhere. Such a system is possible;
we need only choose as current-components u, v, w where
4πи = ▼²Ù₁, 4πv = V²V₁, 4πw = V²W₁· If we now move
electric currents about under the action of both systems A and
B, there is no work done in this motion. Hence the electro-
motive force necessary to maintain the currents must be
independent of the motion, so that the induced electromotive
force is zero. But the system B by itself exerts inductive
actions; hence the system A must exert inductive actions
equal and opposite to those of B, and therefore equal to
those of a purely electrical system which exerts the same
magnetic forces as A. What is true of inductive actions due
to motion must also be true of those due to variations of
intensity; both are most simply determined in terms of each
other by the principle of the conservation of energy. Hence
from the existence of magnetic forces of the form (1) we may
directly infer that of electric forces of the form (2), whatever
may be the origin of those magnetic forces.
Let us now consider magnetic currents. Let λ, μ, v be
the components of magnetisation throughout space, and let
In
მე
+
dμ
dv
+ =
მიყ Əz
0, and A=X, Mμ, N = v.
=
These quantities are to be measured in absolute magnetic
units. It follows from the forces (1) by the principle of the
conservation of energy, and is indeed generally accepted in
electromagnetics, that the electric force produced by variations
of λ, μ, v has for components
dt dz
X₁ = A
(
d/ƏN
ƏM
djan ON
Y₁ = A
dt dy
Iz
IN).
Z₁
A
1
d /ƏM A
(
-
dt ax ду
We now put, in accordance with our notation,
αλ
p =
dt
>
q
=
αμ
dt
>
r=
dv
>
dt
1
¹ Cf. v. Helmholtz, Wissenschaftliche Abhandlungen, vol. i. p. 619.