Page:SahaElectrodynamics.djvu/22

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 :—

$\left.{\begin{array}{c}X=X_{0}\sin \Phi \\Y=Y_{0}\sin \Phi \\Z=Z_{0}\sin \Phi \end{array}}\right\}\left.{\begin{array}{c}L=L_{0}\sin \Phi \\M=M_{0}\sin \Phi \\N=N_{0}\sin \Phi \end{array}}\right\}\phi =\omega \left(t-{\frac {lx+my+nz}{c}}\right)$

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 :—

${\begin{array}{cc}X'=X_{0}\sin \Phi '&L'=L_{0}\sin \Phi '\\\\Y'=\beta \left(Y_{0}-{\frac {v}{c}}N_{0}\right)\sin \Phi '&M'=\beta \left(M_{0}+{\frac {v}{c}}Z_{0}\right)\sin \Phi '\\\\Z'=\beta \left(Z_{0}+{\frac {v}{c}}M_{0}\right)\sin \Phi '&N'=\beta \left(N_{0}-{\frac {v}{c}}Y_{0}\right)\sin \Phi '\end{array}}$ $\Phi '=\omega '\left(\tau -{\frac {l'\xi +m'\eta +n'\zeta }{c}}\right)$,