Page:On the Relative Motion of the Earth and the Luminiferous Ether.djvu/4

tions and distances traversed by the rays will be altered thus:— The ray sa is reflected along ab, fig. 2; the angle bab, being equal to the aberration =a, is returned along ba_/, (aba_/ =2a), and goes to the focus of the telescope, whose direction is unaltered. The transmitted ray goes along ac, is returned along ca_/, and is reflected at a_/, making ca_/e equal 90—a, and therefore still coinciding with the first ray. It may be remarked that the rays ba_/ and ca_/, do not now meet exactly in the same point a_/, though the difference is of the second order; this does not affect the validity of the reasoning. Let it now be required to find the difference in the two paths aba_/, and aca_/.

Let V= velocity of light.

v= velocity of the earth in its orbit,

D=distance ab or ac, fig. 1.

T=time light occupies to pass from a to c.

T =time light occupies to return from c to a_/, (fig. 2.)

Then $\scriptstyle {T={\frac {D}{V-v}}}$, $\scriptstyle {T_{/}={\frac {D}{V+v}}}$. The whole time of going and coming is $\scriptstyle {T+T_{/}=2D{\frac {V}{V^{2}-v^{2}}}}$, and the distance traveled in this time is $\scriptstyle {2D{\frac {V^{2}}{V^{2}-v^{2}}}=2D\left(1+{\frac {v^{2}}{V^{2}}}\right)}$, neglecting terms of the fourth order. The length of the other path is evidently $\scriptstyle {2D{\sqrt {(1+{\frac {v^{2}}{V^{2}}}}}}$ or to the same degree of accuracy, $\scriptstyle {2D\left(1+{\frac {v^{2}}{2V^{2}}}\right)}$. The difference is therefore $\scriptstyle {D{\frac {v^{2}}{V^{2}}}}$. If now the whole apparatus be turned through 90°, the difference will be in the opposite direction, hence the displacement of the interference fringes should be $\scriptstyle {2D{\frac {v^{2}}{V^{2}}}}$. Considering only the velocity of the earth in its orbit, this would be $\scriptstyle {2D\times 10^{-8}}$. If, as was the case in the first experiment, $\scriptstyle {D=2\times 10^{6}}$ waves of yellow light, the displacement to be expected would be 0.04 of the distance between the interference fringes.

In the first experiment one of the principal difficulties encountered was that of revolving the apparatus without producing distortion; and another was its extreme sensitiveness to vibration. This was so great that it was impossible to see the interference fringes except at brief intervals when working in the city, even at two o'clock in the morning. Finally, as before remarked, the quantity to be observed, namely, a displacement of something less than a twentieth of the distance between the interference fringes may have been too small to be detected when masked by experimental errors.