# Page:1860 Fizeau en.djvu/7

rapidly. Had it been possible to maintain the velocity of the current of water constant for a greater length of time, the measurements would have been more precise; but this did not appear to be possible without considerably altering the apparatus, and such alterations would have retarded the prosecution of my research until the season was no longer favourable for experiments requiring solar light.

I proceed to compare the observed displacement with those which would result from the first and third hypotheses before alluded to. As to the second hypothesis, it may be at once rejected; for the very existence of displacements produced by the motion of water is incompatible with the supposition of an æther perfectly free and independent of the motion of bodies.

In order to calculate the displacement of the bands under the supposition that the æther is united to the molecules of bodies in such a manner as to partake of their movements, let

$v$ be the velocity of light in a vacuum,
$v'$ the velocity of light in water when at rest,
$u$ the velocity of the water supposed to be moving in a direction parallel to that of the light. It follows that
$v'+u$ is the velocity of light when the ray and the water move in the same direction, and
$v'-u$ when they move in opposite directions.

If $\Delta$ be the required retardation and E the length of the column of water traversed by each ray, we have, according to the principles proved in the theory of the interference of light,

$\Delta=E\left(\frac{v}{v'-u}-\frac{v}{v'+u}\right),$

or

$\Delta=2E\frac{u}{v}\frac{v^{2}}{v'^{2}-u^{2}}.$

Since $u$ is only the thirty-three millionth part of $v$, this expression may, without appreciable error, be reduced to

$\Delta=2E\frac{u}{v}\cdot\frac{v^{2}}{v'^{2}}$

If $m=\tfrac{v}{v'}$ be the index of refraction of water, we have the approximate formula

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

Since each ray traverses the tubes twice, the length E is double the real length of the tubes. Calling the latter L = 1.4875 metre,