Scientific Papers of Josiah Willard Gibbs, Volume 2/Chapter XVII

by Josiah Willard Gibbs

XVII.

ON THE VELOCITY OF LIGHT AS DETERMINED BY FOUCAULT'S REVOLVING MIRROR.

[Nature, vol. xxxiii. p. 582, April 22, 1886.]

It has been shown by Lord Rayleigh and others that the velocity ($U$) with which a group of waves is propagated in any mediam may be calculated by the formula

$U=V\left(1-{\frac {d\log V}{d\log \lambda }}\right),$

where $V$ is the wave-velocity, and $\lambda$ the wave-length. It has also been observed by Lord Rayleigh that the fronts of the waves reflected by the revolving mirror in Foucault's experiment are inclined one to another, and in consequence must rotate with an angular velocity

${\frac {dV}{d\lambda }}\alpha ,$

where $\alpha$ is the angle between two successive wave-planes of similar phase. When $dV/d\lambda$ is positive (the usual case), the direction of rotation is such that the following wave-plane rotates towards the position of the preceding (see Nature, vol. xxv. p. 52). But I am not aware that attention has been called to the important fact, that while the individual wave rotates the wave-normal of the group remains unchanged, or, in other words, that if we fix our attention on a point moving with the group, therefore with the velocity $U$, the successive wave-planes, as they pass through that point, have all the same orientation. This follows immediately from the two formulæ quoted above. For the interval of time between the arrival of two successive wave-planes of similar phase at the moving point is evidently $\lambda /(V-U)$, which reduces by the first formula to $d\lambda /dV$. In this time the second of the wave-planes, having the angular velocity $\alpha dV/d\lambda$, will rotate through an angle a towards the position of the first wave-plane. But $\alpha$ is the angle between the two planes. The second plane, therefore, in passing the moving point, will have exactly the same orientation which the first had. To get a picture of the phenomenon, we may imagine that we are able to see a few inches of the top of a moving carriage-wheel. The individual spokes rotate, while the group maintains a vertical direction.

This consideration greatly simplifies the theory of Foucanlt's experiment, and makes it evident, I think, that the results of all such experiments depend upon the value of $U,$ and not upon that of $V.$

The discussion of the experiment by following a single wave, and taking account of its rotation, is a complicated process, and one in which it is very easy to leave out of account some of the elements of the problem. The principal objection to it, however, is its unreality. If the dispersion is considerable, no wave which leaves the revolving mirror will return to it. The individual disappears, only the group has permanence. Prof. Schuster, in his communication of March 11 (p. 439), has nevertheless obtained by this method, as the quantity determined by "the experiments hitherto performed," $V^{2}/(2V-U),$ which, as he observes, is nearly equal to $U.$ He would, I think, have obtained $U$ precisely, if for the angle between two successive wave-planes of similar phase, instead of $2\omega \lambda /V,$ he had used the more exact value $2\omega \lambda /U.$

By the kindness of Prof. Michelson, I am informed with respect to his recent experiments on the velocity of light in bisulphide of carbon that he would be inclined to place the maximum brilliancy of the light between the spectral lines ${\text{D}}$ and ${\text{E}},$ but nearer to ${\text{D}}.$ If we take the mean between ${\text{D}}$ and ${\text{E}},$ we have

${\frac {K}{U}}=1.745,$${\frac {K(2V-U)}{V^{2}}}=1.737,$

$K$ denoting the velocity in vacuo (see p. 249 of this volume). The number observed was 1.76, "with an uncertainty of two units in the second place of decimals." This agrees best with the first formula. The same would be true if we used values nearer to the line ${\text{D}}.$

J. Willard Gibbs.

New Haven, Connecticut, April 1. [1886.]