THE RESULTS of the last three sections show that
the apparent incompatibility of the law of propagation of light with the
principle of relativity (Section VII) has been derived by means of a consideration
which borrowed two unjustifiable hypotheses from classical mechanics; these
are as follows:

The timeinterval (time) between two events is independent of the condition
of motion of the body of reference.

The spaceinterval (distance) between two points of a rigid body is independent
of the condition of motion of the body of reference.

1 
If we drop these hypotheses, then the dilemma of Section VII disappears,
because the theorem of the addition of velocities derived in Section VI
becomes invalid. The possibility presents itself that the law of the propagation
of light in vacuo may be compatible with the principle of relativity,
and the question arises: How have we to modify the considerations of Section
VI in order to remove
the apparent disagreement between these two fundamental results of experience?
This question leads to a general one. In the discussion of Section VI we
have to do with places and times relative both to the train and to the embankment.
How are we to find the place and time of an event in relation to the train,
when we know the place and time of the event with respect to the railway
embankment? Is there a thinkable answer to this question of such a nature
that the law of transmission of light in vacuo does not contradict
the principle of relativity? In other words: Can we conceive of a relation
between place and time of the individual events relative to both referencebodies,
such that every ray of light possesses the velocity of transmission c
relative to the embankment and relative to the train? This question leads
to a quite definite positive answer, and to a perfectly definite transformation
law for the spacetime magnitudes of an event when changing over from one
body of reference to another. 
2 
Before we deal with this, we shall introduce the following incidental
consideration. Up to the present we have only considered events taking place
along the embankment, which had mathematically to assume the function of
a straight line. In the manner indicated in Section II we can imagine this
referencebody supplemented laterally and in a vertical direction by means
of a
framework of rods, so that an event which takes place anywhere can be localised
with reference to this framework. Similarly, we can imagine the train travelling
with the velocity v to be continued across the whole of space, so
that every event, no matter how far off it may be, could also be localised
with respect to the second framework. Without committing any fundamental
error, we can disregard the fact that in reality these frameworks would
continually interfere with each other, owing to the impenetrability of solid
bodies. In every such framework we imagine three surfaces perpendicular
to each other marked out, and designated as “coordinate planes” (“coordinate
system”). A coordinate system K then corresponds to the embankment,
and a coordinate system K' to the train. An event, wherever it may
have taken place, would be fixed in space with respect to K by the
three perpendiculars x, y, z on the coordinate planes, and with
regard to time by a timevalue t. Relative to K', the same event
would be fixed in respect of space and time by corresponding values x',
y', z', t', which of course are not identical with x, y, z, t.
It has already been set forth in detail how these magnitudes are to be regarded
as results of physical measurements. 
3 
Obviously our problem can be exactly formulated in the following
manner. What are the values x', y', z', t'
of an event with respect to K', when the magnitudes x, y, z,
t, of the same event with respect to K are given? The relations
must be so chosen that the law of the transmission of light in vacuo
is satisfied for one and the same ray of light (and of course for every
ray) with respect to K and K'. For the relative orientation
in space of the coordinate systems indicated in the diagram (Fig. 2),
this problem is solved by means of the equations:
This system of equations is known as the “Lorentz transformation.” 1
FIG. 2.

4 
If in place of the law of transmission of light we had taken
as our basis the tacit assumptions of the older mechanics as to the absolute
character of times and lengths, then instead
of the above we should have obtained the following equations:
x' = x  vt 
y' = y 
z' = z 
t' = t. 
This system of equations is often termed the “Galilei transformation.”
The Galilei transformation can be obtained from the Lorentz transformation
by substituting an infinitely large value for the velocity of light c in
the latter transformation. 
5 
Aided by the following illustration, we can readily see that,
in accordance with the Lorentz transformation, the law of the transmission
of light in vacuo is satisfied both for the referencebody K
and for the referencebody K'. A lightsignal is sent along the
positive xaxis, and this lightstimulus advances in accordance with the
equation
i.e. with the velocity c. According to the equations of the
Lorentz transformation, this simple relation between x and t
involves a relation between x' and t'. In point of fact,
if we substitute for x the value ct in the first and fourth
equations of the Lorentz transformation, we obtain:
from which, by division, the expression
immediately follows. If referred to the system K', the propagation
of light takes place according to this equation. We thus see that the velocity
of transmission relative to the referencebody K' is also equal
to c. The same result is obtained for rays of light advancing in
any other direction whatsoever. Of course this is not surprising, since
the equations of the Lorentz transformation were derived conformably to
this point of view. 
6 