Page:1860 Fizeau en.djvu/8

 the preceding formula becomes

$\Delta=4L\frac{u}{v}m^{2}$

and the numerical calculation being performed, we find

$\Delta$ = 0.0002418 millim.

Such is the difference of path which, under the present hypothesis, ought to exist between the two rays.

Strictly speaking, this number has reference to a vacuum, and ought to be divided by the index of refraction for air; but this index differs so little from unity, that, for the sake of simplicity, the correction, which would not alter the last figure by a unit, may be neglected.

The above quantity being divided by the length of an undulation, will give the displacement of the bands in terms of the breadth of one of them. In fact, for a difference of path amounting to 1, 2,. . . $$m$$ undulations, the system of bands suffer a displacement equal to the breadth of 1, 2,. . . $$m$$ bands.

For the ray E the length of an undulation is $$\lambda$$ = 0.000526, and the rays about it appear to preserve the greatest intensity after the light has traversed a rather considerable thickness of water. Selecting this ray, then, we find for the displacement the value

$\frac{\Delta}{\lambda}=0.4597.$

Had, therefore, the æther participated fully in the motion of the water, in accordance with the hypothesis under consideration, a displacement of 0.46 of a band would have been observed in the foregoing experiments. But the mean of our observations gave only 0.23; and on examining the greatest particular values, it will be found that none approached the number 0.46. I may even remark that the latter number ought to be still greater, in consequence of a small error committed in the determination of the velocity of the water; an error whose tendency is known, although, as will soon be seen, it was impossible to correct it perfectly.

I conclude, then, that this hypothesis does not agree with experiment. We shall next see that, on the contrary, the third, or Fresnel's hypothesis, leads to a value of the displacement which differs very little from the result of observation.

We know that the ordinary phænomena of refraction are due to the fact that light is propagated with less velocity in the interior of a body than in a vacuum. Fresnel supposes that this change of velocity occurs because the density of the æther within a body is greater than that in a vacuum. Now for two media