346
PONDEROMOTIVE FORCES
Si
z
A
%
W
df
.
Now we
have
Si
z
'dS
+
d%
dy
dz
dSjft
H-Ä
s.
i
y
dm
~chi
+
2
~d z
dSi
z
fS
v
~W.
However,
the
two
terms
6hyhz/6y+1/2gz
vanish
on
integration.
Using
Maxwell's equations,
the
term
hyghz/gy
can
be
transformed
to
H
dSj
0 +
TV
X
0
z
I
so
that finally
we can
write
equation
(10)
as
R
=
-
1
Si
y
ds)
X
+
1
Hz
df
+
\
5xVf
1
c
dsi
ß
~ji~
df
=
y
o z
J
1
Tc
-
dm
if
df
The
last integral
equals
zero,
because the forces vanish
at
infinity.
-
Thus having
ascertained the force that
acts
on
matter
traversed
by a
conduction
current,
we
obtain the force that
acts
on
a
body
permeated
by
a
polarization current
by
noting
that
from
the
standpoint of
the
theory of
electrons the
polarization
current
and
the conduction
current
are
completely
equivalent
with
regard
to
electrodynamic
action.
By
taking
into
account
the
duality
of
magnetic
and
electric
phenomena,
one
also obtains the force exerted
on a body
permeated
by
a
magnetic
polarization current in
the electric field.
In
this
way
we
obtain the
following equations
as an
overall
expression
for the forces that
depend
on
the
velocity of
the
elementary
particles:
(11)
it10
+
1
£
sa
Ji