204
MOTION OF CENTER OF GRAVITY
since
we assume
according
to
the above
that the individual material
points
mv
change
their
energy,
and thereby
also their
mass,
only
by
taking
up
electro
magnetic
energy.
If
we
assign to
the
electromagnetic
field
too
a mass
density
(pe),
which
differs
by a
factor
1/V2
from
the
energy
density,
then the
second term
of
the
equation
takes
the
form
yi
d
It
'
xpdrre
If the
integral in the
third
term
of
equation
(2)
is denoted
by
J,
then
this
equation
becomes
(2a)
i
x
dwv
v
dt
d
+
It
'
xpedr
1
Ü7
J
=
0
We
now
have to
find the
meaning
of the
integral
J. If
one
successively
multiplies
the second,
third, fifth,
and
sixth
of
equations (1)
by NV, MV,
ZV,
YV,
adds them and integrates
over
the
space,
one
obtains, after
a
few
integrations
by
parts,
(3)
dJ
It

4iV If I
+
yJV

yM
dr
=

4
VR
X
where
Rx
is the
algebraic
sum
of the
Xcomponents
of
all
forces exerted
by
the
electromagnetic
field
upon
the
masses
m1...mn. Since the
corresponding
sum
of
all forces
due
to
the conservative
interactions vanishes,
Rx
is
at
the
same
time the
sum
of
the
Xcomponents
of
all forces
acting
upon
the
masses mv.
Next
we
shall consider
equation
(3),
which
is
independent
of
the
hypo
thesis
that the
mass
depends
on
energy.
If
we
disregard the
dependence
of the
masses on
energy
and
denote the resultant of all
Xcomponents
of
the forces
acting
on
mv
by
Xv,
we
must
set
up
the
following
equation of motion
for
the
mass
mv:
[6]