286
FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS XVII
and
hence
dt2
1
d2
A2
વેઢે .
V²U - Ad U
dt2
=
-
4πu.
The other components of the vector-potentials, both
of electric and magnetic currents, satisfy analogous differential
equations. Since u, v, w, p, q, r vanish in empty space, the
distribution of these potentials is there given by the equations
dt2
V²V – A ?d²V
=
▼²U – A²ď²U
0,
V2P-A2
2d2p
dt2
=
0
= 0,
▼²Q - A ⁹d²Q
0
dt2
(11).
0,
-
V²R – A 2ď²R
=
0
-
dt2
V2W-A2d2W
dt2
au av aw
+ +
Əx ду Əz
=
=
=
0,
ӘР
dt2
ƏPƏQƏR
+ +
Эх ду Əz
=
0
The vector-potentials now show themselves to be quantities
which are propagated with finite velocity—the velocity of light
--and indeed according to the same laws as the vibrations of light
and of radiant heat. Riemann in 1858 and Lorenz in 1867, with
a view to associating optical and electrical phenomena with one
another, postulated the same or quite similar laws for the pro-
pagation of the potentials. These investigators recognised
that these laws involve the addition of new terms to the forces
which actually occur in electromagnetics; and they justify this
by pointing out that these new terms are too small to be ex-
perimentally observable. But we see that the addition of
these terms is far from needing any apology. Indeed their
absence would necessarily involve a contradiction of principles
which are quite generally accepted.
The vector-potentials of electric and magnetic currents have
hitherto occurred as quite separate, and from them the electric
and magnetic forces were deduced in an unsymmetric
manner. This contrast between the two kinds of forces dis-