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.