Page:Michelson1904.djvu/2

Rh 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

$$\mathrm{T}=\frac{2}{\mathrm{V}^{2}}\int v\ \cos\ \theta\ ds$$

where $$\mathrm{V}$$ is the velocity of light, $$v$$ the velocity of the earth's surface at the element of path $$ds$$, and $$\theta$$ the angle between $$v$$ and $$ds$$.

If the circuit be horizontal, and $$x$$ and $$y$$ denote distances east and west and north and south respectively, and $$\phi$$ the latitude of the origin, and $$\mathrm{R}$$ the radius of the earth, then for small values of $$y/\mathrm{R}$$ we have approximately

$$\mathrm{T}=\frac{2v_{0}}{\mathrm{V}^{2}}\int\left(\cos\phi-\frac{y}{\mathrm{R}}\sin\phi\right)dx.$$

The integral being taken round the circuit the first, term vanishes, and if $$\mathrm{A}={\textstyle\int} ydx=$$ area of the circuit,

$$T=\frac{2v_{0}\mathrm{A}}{\mathrm{V}^{2}\mathrm{R}}\sin\phi.$$

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

$$\Delta=\frac{2v_{0}\mathrm{A}}{\mathrm{VR}\lambda}\sin\phi.$$

Thus, for latitude 45° $$\sin\phi=\sqrt{1/2},\ \frac{v_{0}}{\mathrm{R}}=\frac{2\pi}{\mathrm{T}}$$; the velocity of light is $$3\times10^{8}$$ in the same units, and the length of a light-wave is $$5\times10^{-7}$$: which approximate values substituted in the preceding formula give

$$\Delta=7\times10^{-7}\mathrm{A}.$$

Thus if the circuit be one kilometre square

$$\Delta=0.7.$$

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.