Page:Michelson1904.djvu/2

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717
Relative Motion of Earth and Æther.

measurement of the difference of time required for the two pencils to traverse the circuit would furnish a quantitative test of the entrainement.

But it is not necessary that the path should encircle the globe, for there would still be a difference in time for any position of the circuit.

This difference is given by the formula

where is the velocity of light, the velocity of the earth's surface at the element of path , and the angle between and .

If the circuit be horizontal, and and denote distances east and west and north and south respectively, and the latitude of the origin, and the radius of the earth, then for small values of we have approximately

The integral being taken round the circuit the first, term vanishes, and if area of the circuit,

The corresponding difference of path for equal times expressed in light-waves of length is

Thus, for latitude 45° ; the velocity of light is in the same units, and the length of a light-wave is : which approximate values substituted in the preceding formula give

Thus if the circuit be one kilometre square

The system of interference-fringes produced by the superposition of the two pencils—one of which has traversed the circuit clockwise, and the other counterclockwise—would be shifted through seven-tenths of the distance between the fringes, in the direction corresponding to a retardation of

Phil. Mag. S. 6. Vol. 8. No. 48. Dec. 1904.
3 D