Page:SahaElectrodynamics.djvu/11

time $\tau =0$, if the ray is sent in the direction of increasing ξ, we have

$\xi =c\tau$, i.e. $\xi =ac\left(t-{\frac {vx'}{c^{2}-v^{2}}}\right)$.

Now the ray of light moves relative to the origin of k with a velocity c-v, measured in the stationary system ; therefore we have

${\frac {x'}{c-v}}=t$.

Substituting these values of t in the equation for ξ, we obtain

$\xi =a{\frac {c^{2}}{c^{2}-v^{2}}}x'$.

In an analogous manner, we obtain by considering the ray of light which moves along the y-axis,

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

where ${\frac {y}{\sqrt {c^{2}-v^{2}}}}=t,\ x'=0$.

Therefore $\eta =a{\frac {c}{\sqrt {c^{2}-v^{2}}}}y,\ \zeta =a{\frac {c}{\sqrt {c^{2}-v^{2}}}}z$.

If for x', we substitute its value x—tv, we obtain

$\tau =\phi \ (v)\cdot \beta \left(t-{\frac {vx}{c^{2}}}\right)$,

$\xi =\phi \ (v)\cdot \beta \left(x-vt\right)$,

$\eta =\phi \ (v)\ y$,

$\zeta =\phi \ (v)\ z$,

where $\beta ={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$, and $\phi (v)={\frac {\alpha c}{\sqrt {c^{2}-v^{2}}}}={\frac {\alpha }{\beta }}$ is a function of v.