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 and then decompose these residual aberrations into terms depending on the squares and on the cubes of the ratio of velocities. To a thousandth part of the residual aberrations, their difference is represented by the equation

$\delta'-\delta\ =\ \frac{v^2}{\mathrm{V}^2}\cos 2\alpha+\frac{v^3}{\mathrm{V}^3}(\sqrt{0.5}\sin{2\alpha}+\tfrac{1}{6}\sin{4\alpha}+\cos\alpha)$.

For the velocity-ratio 10,000 the agreement would be much closer. Fig. 9 (Pl. IX.) A shows the laws of the variation in the residual aberrations of the two rays I and II, coming from the mirrors I and II. The unit for A and C is $$\sin^{-1}{\frac{v^2}{\mathrm{V}^2}}$$, and for B and D, $$\sin^{-1}{\frac{v^3}{\mathrm{V}^3}}$$. The curves A are nearly represented by $$\frac{v^2}{\mathrm{V}^2}(\sin{2\alpha}\pm\tfrac{1}{2}\cos{2\alpha})$$. Subtracting the ordinates given by this expression from the actual ordinates, we get the residuals shown (after multiplication by the reciprocal of the velocity-ratio) at B. C shows the difference of the curves I. and II. of A, and thus gives directly the angle of divergence of the emergent wave-fronts which is the object of our study. D gives the difference between this curve and the sine curve $$\delta'-\delta\ =\ \frac{v^2}{\mathrm{V}^2}\cos{2\alpha}$$. The latter curve shows that the difference of the aberrations of I and II is at an undisturbed maximum at 90° and at 270°; at 0°, it is less than the undisturbed maximum by the quantity $$\frac{v^3}{\mathrm{V}^3}\cos\alpha$$, or $$\frac{v^3}{\mathrm{V}^3}$$; at 180° it is greater by the same quantity.

It may be thought that the adjustment of the angles between the mirrors which has been assumed will limit too narrowly the use of the apparatus. We may simply say that experience with mirrors as nearly plane as those used by us has shown us that the method of observation supposed would suffice for angles of aberration at least twenty-five times that expected if the velocity-ratio is 10,000.

Since the experiment gives a null result, it is not worth the space to prove that what is true of this adjustment is true with sufficient approximation for an adjustment which differs from the assumed adjustment only by the rotation of mirror I by an angle of ten seconds around a perpendicular axis passing through its surface. Instead, we may compare the results here obtained with those of Dr. Hicks.

In the first place, he declares that the position of the fringes is displaced by aberration. This point is eliminated from our discussion by the fact that the fringes are infinitely