Page:SahaElectrodynamics.djvu/25

the moving system. ${\frac {A'^{2}}{A^{2}}}$ would therefore denote the ratio between the energies of a definite light-complex "measured when moving" and "measured when stationary," the volumes of the light-complex measured in K and k being equal. Yet this is not the case. If $l,m,n$ are the direction-cosines of the wave-normal of light in the stationary system, then no energy passes through the surface elements of the spherical surface

$(x-clt)^{2}+(y-cmt)^{2}+(z-cnt)^{2}=R^{2}$,

which expands with the velocity of light. We can therefore say, that this surface always encloses the same light-complex. Let us now consider the quantity of energy, which this surface encloses, when regarded from the system k, i.e., the energy of the light-complex relative to the system k.

Regarded from the moving system, the spherical surface becomes an ellipsoidal surface, having, at the time $tau=0$, the equation :—

$\left(\beta \xi -l\beta {\frac {v}{c}}\xi \right)^{2}+\left(\eta -m\beta {\frac {v}{c}}\xi \right)^{2}+\left(\zeta -n\beta {\frac {v}{c}}\xi \right)^{2}=R^{2}$

If $S={\text{volume of the sphere}}$, $S'={\text{volume of this ellipsoid}}$, then a simple calculation shows that:

${\frac {S'}{S}}={\frac {\beta }{\sqrt {1-{\frac {v}{c}}\cos \Phi }}}$

If E denotes the quantity of light energy measured in the stationary system, E' the quantity measured in the