Page:A history of the theories of aether and electricity. Whittacker E.T. (1910).pdf/140

and therefore (neglecting second-order terms in w/c)

${\frac {\sin A_{1}{\hat {B}}_{1}D_{1}}{\sin i}}={\frac {c-w\cos {\delta _{1}}}{c_{1}-w{\frac {c_{1}^{2}}{c^{2}}}}}={\frac {c}{c_{1}}}+{\frac {w}{c}}-{\frac {w}{c_{1}}}\cos {\delta _{1}}$.

Denoting by δ the value of δ₁, when w is zero, we have

${\frac {\sin {(i-\delta )}}{\sin {\delta }}}={\frac {c}{c_{1}}}$.

Subtracting this equation from the preceding, we have

${\frac {\delta -\delta _{1}}{\sin {\delta }}}={\frac {w}{c}}$.

Now the telescope by which the emergent wave-front B₁D is received is itself being carried forward by the earth's motion; and we must therefore apply the usual correction for aberration in order to find the apparent direction of the emergent ray. But this correction is w sin δ/c, and precisely counteracts the effect which has been calculated as due to the motion of the prison. So finally we see that the motion of the earth has no first-order influence on the refraction of light from the stars.

Fresnel inferred from his formula that if observations were made with a telescope filled with water, the aberration would be unaffected by the presence of the water—a result which was verified by Airy^[1] in 1871. He showed, moreover, that the apparent positions of terrestrial objects, carried along with the observer, are not displaced by the earth's motion; that experiments in refraction and interference are not influenced by any motion which is common to the source, apparatus, and observer; and that light travels between given points of a moving material system by the path of least time. These predictions have also been confirmed by observation: Respighi^[2] in 1861, and Hoek^[3] in 1868, experimenting with a telescope filled with water and a terrestrial source of light, found that no effect was produced on the phenomena of reflexion and refraction by altering the orienta-

1. Proc. Roy. Soc., xx, p. 35.
2. Mem. Accad. Sci. Bologna, 71, p. 279.
3. Ast. Nach, lxxiii, p. 193.