Page:Philosophical Transactions of the Royal Society A - Volume 184.djvu/848

752 space, it is necessary* that the difference of potential between two points $$\mathrm{A}$$ and $$\mathrm{B}$$ should be the same whether the space between is filled with dense matter or not (or, say, whether the ray-path is taken through or outside a portion of dense medium); in other words (calling $$\phi$$ the outside and $$\phi^\prime$$ the inside potential-function), in order to secure that $$\mathrm{T}^\prime$$ shall not differ from $$\mu\mathrm{T}$$ by anything depending on the first power of motion, it is necessary that $$\phi^\prime_\mathrm{B} - \phi^\prime_\mathrm{A}$$ shall equal $$\phi_\mathrm{B} - \phi_\mathrm{A}$$, i.e., that the potential inside and outside matter shall be the same up to a constant, or that $$\mu^2 v^\prime\cos\theta^\prime = v\,\cos\theta$$; which for the case of drift along a ray is precisely hypothesis.

Another way of putting the matter is to say that to the first power of drift velocity

$\mathrm{T}^\prime = \mu\mathrm{T} - \int \left(\mu^2 v^\prime\cos\theta - v\cos\theta \right) ds/\mathrm{V}^2$,

and that the second or disturbing term must vanish.

29. Hence hypothesis as to the behaviour of ether inside matter is equivalent to the assumption that a potential-function, $$\mu^2v\cos\theta\;ds$$, exists throughout all transparent space, so far as motion of ether alone is concerned.

Given that condition, no first-order interference effect clue to drift can be obtained from stationary matter by sending rays round any kind of closed contour, nor can the path of a ray be altered by ethereal drift through any stationary matter.

As soon as matter is locally moved, however, its motion may readily produce an effect, for it has no potential conditions to satisfy; it may easily be moved in a closed contour. Suppose it moves with velocity $$u$$, always with the light, the relative drift of ether thereby caused in it must, as above, be $$u/\mu^2$$, and so it may be said to virtually carry the ether inside it forward with velocity $$u - u/\mu^2$$; for that is the amount by which it affects the time of journey of a ray. This does not mean that it carries with it any ether of space; in fact, it definitely means that it does not appreciably disturb the ether of space (cf. § 3, b).

The equation to a ray in moving matter, subject to an independent ether drift, is

$\int \frac{ds}{\mathrm{V}/\mu\cos\epsilon + v/\mu^2 \cos\theta + u\left[1 - \left(1/\mu^2 \right)\right]\cos\phi} =$ const.

30. It is noteworthy that almost all the observations which have been made with negative results as to the effect of the Earth's orbital motion on the ether are equally consistent with complete connexion and complete independence between ether and * [The argument has here been slightly expanded since the MS. was sent in to meet a suggestion of inadequacy made by Dr., to whom I am also indebted for an objection to the term “velocity-potential” at first applied to this function $\phi^\prime$. As Professor has observed, it is more general than a velocity-potential, though it reduces to that when the medium is homogeneous, or when $\mu = 1$. The text has been altered accordingly.—July, 1893.]