37° ROY J. KENNEDY The theory of the arrangement is as follows: The interference phenomena will be the same as if the mirror M_{2} were replaced by its image in M_{z}. Under the conditions of the experiment, where the paths are nearly equal, M_{x} is perpendicular to the beam incident on it, and the reflected beams are brought nearly to parallelism, the image of M_{2} will be nearly parallel and coincident with the face of M_{x}. Elementary theory shows that the resulting interference pattern then practically coincides with M_{x}. It would needlessly complicate this discussion to develop the general theory of interference for all inclinations of the mirrors; the experimentally realized case of near parallelism alone is necessary. Let Figure 10 represent a greatly exaggerated cross-section of M_{x} and the image of M_{2}, normal to their planes and to the dividing line in M_{2}. M_{x} lies in the plane x = o, and the levels of M_{2} are at equal distances on opposite sides of a parallel plane at the distance x from M_{x}. Let a monochromatic wave, in which the displacement is given by fall on M_{x} and M_{2} from the left. At the surface of M_{x} the displacement in the reflected wave is then given by if we ignore the loss through imperfect reflection. The displacement in the plane of M_{x} in the wave reflected from the upper part of M_{2} is £_{x} — a cos co(/+e) The square of the resultant displacement is then ^{=} O'^{2} j cos GO(t“j^{-}é) “f"cos co |~f-€---—-—- This can be reduced to the form | CONFERENCE ON MICHELSON-MORLEY EXPERIMENT 371 Similarly, the square of the resultant displacement in the interfering beams below the dividing line is found to be 2#^{2} i-|-cos ^ (x+a)J cos^{2} co(/ — 8) . The intensities, being proportional to the squares of the amplitudes, can be represented by J^Æa^i+cos ^ (%— a)j and /_{2} = &a^{2}£i+cos ^ (#+a)J . Now co = 27tv where v = frequency of the light. Hence co/c=27r/X. Therefore and /i = &a^{2}|^i+cos ^ (x— a) j J_{2} = &a^{2}£i+cos —■ (#+a)j . For values oi x = n\/4, where n is an integer, 7i = Æa^{2}^i±cos > the sign being positive for even values of n and negative for odd values. The same expression holds for Z_{2}; hence, under these conditions, Jx = /_{a}. To the observer, then, the field of view is equally intense on both sides of the dividing line when x = n\/4. We have now to determine the least change in x from this value which will produce a perceptible difference in illumination in the two sides of the field. If x is given the variation dx while a is kept constant, the difference in intensity will be |