88
DOC.
1
MANUSCRIPT ON
SPECIAL RELATIVITY
Here
we
have
set
\x
+
X
=
P_
c2
i-a!
c~
The
quantity
X
plays
the role of
the
density,
as
referred
to
the
coordinate
system,
while
|i
denotes the
"rest-density,"
i.e.,
the
density
from the
standpoint
of
the
comoving
observer.
K1
denotes the
momentum
imparted by
the
liquid per
unit
volume and unit time
in the direction of the
X-axis,
and thus
-K1
denotes the force exerted
on
the
liquid
from the outside
per
unit volume.
It
follows further from
(85)
and the
meaning
of
(Tmv)
that
icK4
is
equal
to
the work transferred
to
the
liquid by
the external forces
per
unit volume and unit time. Hence
we
have,
if
we
denote the volume force
exerted
on
the
liquid
from
the
outside
by f,
f*
=
-*i
(fa)
=
_^4
c2
c
If
we
rewrite the
first three
equations
of
(85a)
with the
help
of the
fourth,
we
obtain
the
equations
r
1
,c
X
j
f-^q(fq)
»gradp
+
W
+
^q^op1c
c
(fa)
dX^A-
ldp
C
(85b)
where
we
have
set
d d
d d d
Jt
~
in the
usual
manner.
The first of
equations
(85b)
corresponds
to
Euler's
equations,
and the last
one
to
the
continuity equation,
which
expresses
at
the
same
time the law
of the conservation of
energy.
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