308

THE

RELATIVITY

PRINCIPLE

quantities

p,

u, X,

L, £,

etc.,

if

we

limit ourselves

to

an

infinitesimally

short

period1

that is

infinitesimally

close

to

the time of relative

rest

of

S' and E.

Further,

we

have to replace

t'

by

the local time

a.

However,

we

must not

simply

put

d d

~5i

because

a

point which

is

at rest

relative

to

S,

and to which equations

transformed

to

S

should

refer,

changes

its

velocity

relative

to S'

during

the time element

dt'

=

da,

to which

change,

according

to

equations

(7a)

and

(7b),

there

corresponds

a

temporal

change

of the S-related field

component.

Hence

we

have to put

AT

dX

dV

31

=

Ta

IV

=

T

dY'

W

dY

Ta

+

2

N

dM dM

y

To

cZ7

dV

W

dN'

W

dN

To

+

1

Y

Hence

the

E-referred

electromagnetic

equations

are

1

c

+

If

dN

dM

Tj

1

c

dY

puV

+

To

+

c

%

N

dl

dN

^

M

1

c

dZ

[pu(

+

To

XCM

dM

dl

Tj

1

dl

c

T

dY

dZ

Tj

1

c

dM

y

7

To

c

Z

dZ

dX

1

c

dN

X

To

+

c

Yy

dX dY

Tj

M

1This

restriction

does

not

affect the

range

of

validity

of

our

results because

inherently

the

laws to be

derived

cannot

depend

on

the time.