XVII
287
FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS
appears as soon as we attempt to determine the propagation of
these forces themselves, i.e. as soon as we eliminate the vector-
potentials from the equations. This may be performed by
differentiating equations (9) with respect to t and removing
the differential coefficients of U, V, W with respect to t by
equations (10). It may also be performed by differentiating
equations (10) with respect to t, remembering that, e.g.
alau av aaw au
2d² U
dt2
V2U
=
-
by by дх
-
Əz
მე
Əz
and removing the functions of U, V, W in the brackets by
means of equations (9). In this way we get six equations
connecting together the values of L, M, N, X, Y, Z in empty
space, viz. the following:-
dL
A
dt
dX ƏM ƏN
ǝz ay
=
A
dy Əz
dt
dM
ǝx əz
dY
=
A
Əz
дх
dt
ΟΥ
-
әх
dy
dz
A
at
A
dt
dN
A
=
=
=
Əz
ΟΝ
ƏN
-
dy
ƏL
ax Əz
ƏL ƏM
(12).
dt
=
дх
dy
მე:
These same equations connect together the forces produced
by magnetic currents, for they are got by eliminating P, Q, R
as well as U, V, W. Hence they connect together the electric
and magnetic forces in empty space quite generally, whatever
the origin of these forces. The electric and magnetic forces are
now interchangeable. If we eliminate first one set and then the
other we obtain the following system, which, however, does not
completely represent the system (12):-
V²L-A2dL
= 0,
dt2
0,
V²M - Ad²M
dt2
V²N — A½d²N
-
dt2
ƏL ƏM ƏN
=
=
0,
әм
+
+
0,
дх dy Iz
V²X - Ad²X
dt2
▼²Y – A ²d²Y
dt2
42027
V2Z-A2
dt2
=
0
=0
(13).
= 0
=
=
X ƏYƏZ
+ + = 0
даду Əz