DOC.

45

241

the

body's

electrical

charges.1 The

energy AE

transferred

from

the field of

force

to

the

body

between

the times

t0

and

t1 is

given

by

the

expression

11

=

I

dlJvIfjdxdydz

`Ito

where

the

space integral

is

to be

extended

over

the

body

and

we

have

put

"

_

dX

A

dY

A

dZ

p

=

dx

+

dy

+

dz

.

Using

the transformation

equations given

in the

paper

cited

above2,

we

transform this

expression to

the

space-time

system

(E,

n,

c,

r),

which

corresponds

to

a

coordinate

system

that is

at rest

with

respect to

the

body

and

whose

axes

are

parallel

to

(x,y,z).

One

thus

obtains after

a

simple

calculation,

in

a

notation that

corresponds

exactly

to

that

used

in the

paper

quoted,

=

if

f3vP

ds~diy1Cdr

where,

as

there,

ß

denotes the

expression

1

1-(u/v)2.

Note

that

according

to

our

assumptions

the forces

X'

cannot be

arbitrary.

Rather,

at

all times

they

must

be such

that the

body

under consideration

does

not experience

any

acceleration.

The

necessary

(but

not

sufficient)

condition

for this,

according to

a

theorem

of

statics,

is

that,

observed

from

a

coordi-

nate

system

that

moves

together with the

body,

the

sum

of

the

X-components

of

the forces

acting

upon

the

body always

vanishes.

One

thus

has

for

each

r

J

X'pdtdr)d(

=

0

.

1We

introduce this

assumption

in

order

to be

able

to

assume

that the

acting

forces

are

not subjected

to

any

restricting

conditions

due to

the

way

they

are

produced.

[6]

2A.

Einstein,

Ann.

d.

Phys.

17

(1905),

§§

3

and

6.