THEORY OF RELATIVITY 
271 
time at B or ‘B-time’ and t' the time of arrival of the reflected 
A 
ray at A. The clocks then run synchronously if t^ — a””^b 
T his equation defines ‘time’ and ‘symultaneity of time’. The 
velocity of light is then also defined by : 
2AB 
( 2 ) 
We may now assume the uniform velocity of light in all di- 
rections, there being no reason for assuming the direction A-B 
as unique. 
Assuming now A as the source of light and at the same time 
the origin of a frame of reference K at rest. If r is the distance 
of the point B from A; after the time At the wave front will be 
the surface of sphere passing through B, for according to the 
second postulate the equation holds : 
Y^C.At 
If we express the difference of coordinates by Axv, d from 1 to 
3, and squaring we get 
:S(Ax^)2_c2At2=o (3) 
This equation evidently formulates the principle of the velocity 
of light relative to K which must be independent of the motion 
of the source A. 
We now take another frame of reference K' in uniform recti- 
linear motion relative to K. K and K' are then inertial systems. 
With respect to K' we have the equation : 
^(Axj^)“ — c^At“=^ (3a) 
Equations (3) and (3a) must be mutually consistent with each 
other with respect to a definite transformation, transforming K' 
into K. Again an interval has a physical meaning only if it is 
independent of the choice of coordinates which is true evidently 
also of the spherical surfaces represented by (3) and (3a). 
We will now develop the Lorentz transformation in an elemen- 
tary way. 
Let the Xj coordinate axis be parallel to the x\ axis and X 2 =x '2 
also X 3 =x '3 of K and K' respectively. Furthermore we must have : 
X“ cH=:X'2 — ct'^ . (4) 
and 
x'^=k(x — vt) ; x^ =k' (x' -f- vt' ) (5) 
