and
image039
we obtain the equations
image040
We should thus have the solution of our problem, if the constants _a_
and _b_ were known. These result from the following discussion.
For the origin of _K _ we have permanently _x _ = 0, and hence
according to the first of the equations (5)
image041
If we call _v_ the velocity with which the origin of _K _ is moving
relative to _K_, we then have
image042
The same value _v_ can be obtained from equations (5), if we calculate
the velocity of another point of _K _ relative to _K_, or the velocity
(directed towards the negative _x_-axis) of a point of _K_ with respect
to _K _. In short, we can designate _v_ as the relative velocity of the
two systems.
Furthermore, the principle of relativity teaches us that, as judged
from K, the length of a unit measuring-rod which is at rest with
reference to _K _ must be exactly the same as the length, as judged
from _K _, of a unit measuring-rod which is at rest relative to _K_. In
order to see how the points of the _x _-axis appear as viewed from _K_,
we only require to take a snapshot of _K _ from _K_; this means that
we have to insert a particular value of _t_ (time of _K_), _e.g._ _t_ =
81