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Rh therefore to examine the ray reflected from a moving mirror by an interference method simpler than that on which the action of the diffraction depends, as has been said above.

Before expounding this method it is well to recall that many theoretical researches have been made on the influence of the motion of the mirror upon the reflected luminous wave, amongst them those of Abraham, Brown, Edser, Harnack, Larmor, Planck. These researches make of the problem either a simply geometrical investigation, or an application of the electromagnetic theory of light. But without discussing the result of these researches we may accept the conclusion of Harnack respecting the frequency of the vibrations reflected by a mirror in uniform motion. Let v be the velocity of the latter, normal to its plane, reckoned as positive towards the source; c the velocity of the luminous ray in vacua which makes the angle of incidence θ with the mirror; n, n' the frequencies of the ray before and after reflexion, the source and observer being at rest. If we put β = v/c we shall have

$$n'=n\frac{1+2\beta\ \cos\theta+\beta^{2}}{1-\beta^{2}}$$

which, neglecting the terms in β², reduces to

This relation is the same as that of Ketteler, which was employed by Belopolski in his investigation of the Doppler effect, and follows simply from the consideration that the image of the source moves with the velocity 2v in the direction of the normal to the mirror and, consequently, the component of this velocity in the direction of the reflected ray is 2v cos θ.

If now we suppose that the ray is, by suitable arrangements, reflected with the incidence θ, k times from several mirrors in motion with the velocity v, we shall have

Therefore, according to the hypothesis of constant velocity of light, neglecting the terms in β² we shall have

If, on the contrary, we suppose that the velocity of the