Page:The principle of relativity (1920).djvu/56

 It is further clear that the assymetry mentioned in the introduction which occurs when we treat of the current excited by the relative motion of a magnet and a conductor disappears. Also the question about the seat of electromagnetic energy is seen to be without any meaning.

§ 7. Theory of Döppler's Principle and Aberration.

In the system K, at a great distance from the origin of co-ordinates, let there be a source of electrodynamic waves, which is represented with sufficient approximation in a part of space not containing the origin, by the equations:—

X = X_{0} sin Φ } L = L_{0} sin Φ }

Y = Y_{0} sin Φ } M = M_{0} sin Φ } Φ = ω(t - (lx + my + nz)/c)

Z = Z_{0} sin Φ } N = N_{0} sin Φ }

Here (X_{0}, Y_{0}, Z_{0}) and (L_{0}, M_{0}, N_{0}) are the vectors which determine the amplitudes of the train of waves, (l, m, n) are the direction-cosines of the wave-normal.

Let us now ask ourselves about the composition of these waves, when they are investigated by an observer at rest in a moving medium k:—By applying the equations of transformation obtained in §6 for the electric and magnetic forces, and the equations of transformation obtained in § 3 for the co-ordinates, and time, we obtain immediately:—

X´ = X_{0} sin Φ´ L´= L_{0} sin Φ´

Y´ = β(Y_{0} - (v/c)N_{0})sin Φ´  M´ = β(M_{0} + (v/c)Z_{0})sin Φ´

Z´ = β(Z_{0} + (v/c)M_{0})sin Φ´  N´ = β(N_{0} - (v/c)Y_{0})sin Φ´,

Φ´ = ω´(τ - (l´ξ + m´η + n´ζ)/c),