Page:SahaElectrodynamics.djvu/10

If we now introduce the condition that $\tau$ is a function of co-ordinates, and apply the principle of constancy of the velocity of light in the stationary system, we have

${\frac {1}{2}}\left[\tau (0,\ 0,\ 0,\ t)+\tau \left(0,\ 0,\ 0,\ \left\{t+{\frac {x'}{c-v}}+{\frac {x'}{c+v}}\right\}\right)\right]$

$=\tau \left(x',\ 0,\ 0,\ t+{\frac {x'}{c-v}}\right)$.

It is to be noticed that instead of the origin of coordinates, we could select some other point as the exit point for rays of light, and therefore the above equation holds for all values of (x', y, z, t).

A similar conception, being applied to the y- and z-axis gives us, when we take into consideration the fact that light when viewed from the stationary system, is always propagated along those axes with the velocity ${\sqrt {c^{2}-v^{2}}}$, we have the questions:

${\frac {\partial \tau }{\partial y}}=0,\ {\frac {\partial \tau }{\partial z}}=0$.

From these equations it follows that $\tau$ is a linear function of x' and t. From equations (1) we obtain

$\tau =a\left(t-{\frac {vx'}{c^{2}-v^{2}}}\right)$,

where $a$ is an unknown function of v.

With the help of these results it is easy to obtain the magnitudes (ξ, η, ζ), if we express by means of equations the fact that light, when measured in the moving system is always propagated with the constant velocity c (as the principle of constancy of light velocity in conjunction with the principle of relativity requires). For a