Page:The principle of relativity (1920).djvu/59

 the moving system. 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 τ = 0, the equation:—

(βξ - lβ(v/c)ξ)^2 + (η - mβ(v/c)ξ)^2 + (ζ - nβ(v/c)ξ)^2 = R^2

If S = volume of the sphere, S´ = volume of this ellipsoid, then a simple calculation shows that:

S´/S = β/[sqrt](1 - (v/c)cos Φ)

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