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 wide. We simply remark that, if we understand rightly his statement, this aberration is annulled by the motion of the telescope. Also, his discussion contains a term expressing the fact that the waves of one system gain upon those of the other while passing towards the observer. We have shown that, in the conditions assumed (and realized), this effect is nil for central rays in the eight principal azimuths, and is small in all others. At its maximum, for central rays, it is $$0.3\frac{v^4}{\mathrm{V}^4}$$. With our present large apparatus, whose length is $$54\times10^6\lambda$$, the gain of one wave-front over the other in the whole length is much less than $$10^{-6}\lambda$$.

In the theory of 1887, powers of the velocity-ratio higher than the second were expressly regarded as negligible. Dr. Hicks virtually supplies one such term. He writes, displacement of fringes $$=\ \frac{\frac{1}{2}\xi^2\mathrm{L}\cos{2\alpha}}{\sin(\mathrm{B-A})-\frac{1}{2}\xi^{2}\cos{2\alpha}}$$, where is the velocity-ratio, L is the length of path in the apparatus, from D to I, fig. 5, and B—A is the difference between the angles DB I and DB II. Without the small term in the denominator, this gives precisely the same value as the expression in the paper of 1887, as a simple numerical computation shows. The effect of the small term is the following:—the value of the denominator is decreased or increased by $$\tfrac{1}{2}\frac{v^2}{\mathrm{V}^2}$$ at alternate quadrants, and the value of the fraction is therefore increased or decreased at alternate quadrants. But, according to the present solution, the expression should have a mean value at 90° and 270°, and have, further, a maximum at 180° and a minimum at 0°. At three quadrants we agree, but at the fourth we differ by twice the term in question. The difference is easily explained and is negligible, especially in view of the null result of experiment.

It should be noted that, when there is aberration of the wave-front, there are four closely related magnitudes. One is the distance travelled by the wave-front in the period; a second is the perpendicular distance between consecutive wave-fronts, called in Dr. Hicks's paper; a third is the distance between wave-fronts, resolved parallel to some line dictated by the geometric conditions of the case; and the fourth is the distance between wave-fronts in the line of sight, which is the true wave-length. The perpendicular distance between wave-fronts is used rightly, as we conceive, in establishing the conditions of the network of intersecting wave-fronts in Dr. Hicks's admirable paper. But in one