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Rh. XLVI.—Interference Phenomena in a new form of Refractometer; by

an experiment undertaken with a view to detecting the relative motion of the earth and the luminiferous ether (Am. Journal of Science, No. 128, vol. xxii,) it was necessary to produce interference of two pencils of light, which had traversed paths at right angles with each other. This was accomplished as follows: The light from a lamp at $$a$$, fig. 1, was separated into two pencils at right angles, $$bc$$, $$bd$$, by the plane-parallel glass $$b$$, and these two pencils were returned to $$b$$ by the mirrors $$c$$ and $$d$$, whence they coincided along $$be$$, where they were viewed by the eye, or by a small telescope at $$e$$.

It is evident that, so far as the interference is concerned, the apparatus may be replaced by a film of air whose thickness is $$bc-cd$$, and whose angle is that formed by the image of $$d$$ in $$b$$, with $$c$$.

The problem of interference in thin films has been studied by Feussner, but his equations do not appear to give the explanation of the phenomena observed. In particular, in the "Annalen der Physik und Chemie," No. 12, 1881, on page 558, Feussner draws the conclusion that the interference fringes are straight lines, whereas, in the above described apparatus they are in general curves: and there is but one case—that of the central fringe in white light—which is straight.

I have therefore thought it worth while to attempt the solution of the problem for a film of air, for small angles of incidence and neglecting successive reflections; and though the solution is not perhaps adapted to the general problem, it accounts for all the phenomena observed in the special case.

Let $$Om_{1}$$, $$Om_{2}$$, fig. 2, be two plane mirrors whose intersection is projected at $$O$$, and whose mutual inclination is $$\varphi$$. The illumination at any point, $$P$$ (not necessarily in the plane of the figure), will depend on the mean difference of phase of all the pairs of rays starting from the source and reaching $$P$$, after