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XVII
FUNDAMENTAL EQUATIONS OF ELECTROMAGNETICS
and (3), as distinct from these latter in nature, we need only
form a system of electric or magnetic currents in which the
forces (1) or (3), as the case may be, are zero. Any endless
electric or magnetic solenoid will serve as an example.
We see at once that we cannot regard the result obtained
as final.
Indeed we deduced forces (5) from forces (2); but
now the forces (2) have been found inexact and have been
replaced by the forces (6). Hence we must repeat our
reasoning with these latter forces. The result is easily seen;
we obtain it if we everywhere replace the index 2 by 3
and put
A2
=
U-U-AU-U-AU + Ad
=
2
U
dt 1,
and similar expressions for the other components of the vector-
potential. The terms in A5 which here appear in the ex-
pressions for the magnetic forces of electric currents, and the
electric forces of magnetic currents, may be perceived apart
from the terms of lower order. We need only take an
ordinary electric or magnetic solenoid, which may be called a
solenoid of the first order, and roll it up into a solenoid which
may be called a solenoid of the second order, in order to get
a system in which the forces here calculated are the largest of
those occurring. From the consideration of such solenoids we
may demonstrate the existence of the separate terms, inde-
pendently of the fact of our admitting or not admitting that
they are simply added together to give the final result.
Our reasoning prevents us from stopping at any stage and
constantly adds, as before, more and more terms, thus leading
to an infinite series. To represent the final result we denote
by L, M, N, X, Y, Z the completely corrected forces and obtain
Vav
aw
L-A
Əz
dy
aw
M = A
Əx
2T) (9),
au
av
N = A
dy
діс
X =
-
dt
dt
(10).
Y = - Adv
Z= - AdW
dt