168 Dr. L. Silberstein on the recent Eclipse Results
answer is: On experience. For, clearly, we cannot deduce <i relation, which is essentially electro-mechanical, from mechanical principles alone, or from electromagnetism alone. Nor can we imitate the usual dispersion theory (which makes use of both kinds of principles), for we are interested in those portions of the aether in which there are no atoms and no electrons.
In short, as was announced in section 3, let us write down the required relation by utilizing the observational result obtained by the Eclipse Expedition. In other words, let us see what that relation must be like in order to give the observed effect.
Now, if we disregard the small discrepancies (which may be either due to accidental errors or, perhaps, due to a superposed slight ordinary refraction), the observed total deflexions of the rays passing near the Sun are represented hj Einstein's formula (quite apart from his theory)
where r0 is the minimum distance of the (undeflected) ray from the Sun's centre, and it can easily be shown that such will be the case* if the refractive index n — c/c at any 'distance r>R from the Sun’s centre be determined by
or, denoting the potential by XI, and generalizing to any distribution of gravitational matter,
......(9)
[This, in fact, is the formula which would follow at once from Einstein's approximate line-element
for a “ static ” field.]
In order to obtain the required relation, that is to say the assumption to be made on the optical behaviour of the condensed aether, it is enough to combine equation (9) with our last equation (8), which gives
.... (10)
* Approximately, that is, for small A6, and consequently for a refractive index but little differing from unity.
and Stokes-Planck’s JEther.
161)
Such, then, would be the required refractivity of the •condensed aether, obeying aiiy law ^=/(/o). In particular, if it obeys Boyle's law, we have
.... (10 a)
which is of a surprisingly simple form, and reads: n2 —1 ^qual to four time'» the logarithm of condensation multiplied ly the squared ratio of the two velocities of propagation characterizing the cether.
Notwithstanding this temptingly simple form of the relation, I shall not try to “deduce” it from things more familiar. I prefer to regard it as an assumption, dictated by observation.
If the reader so desires, he can write n2 — l = 4w/c2, where w is the work, per unit mass of: matter, done by the gravitational field in condensing the aether. The small fraction w2 — 1 being known from the Eclipse results (for any r),the numerical value of this work is determined without any further assumptions. If we agree to the lowest estimate of log 5 at the Sun’s surface, as required by the aberration theory, we can also evaluate separately the ratio v/c, as already mentioned. This, however, is only a secondary matter.
7. Some details and further implications of the Stokes-Planck aether theory, supplemented by assumption (10), must be postponed to a later opportunity. Here it will be enough to add only a few more general remarks. It will be kept in mind that the proposed theory would account not only for the observed astronomical aberration and for the older terrestrial optical nil-effects, but manifestly also for the nil--effect of the Michelson-Morley experiment. The bending of rays round the more massive celestial bodies would be only a by-product of the theory. Again, in view of the exceedingly small condensation of the aether round single atoms or corpuscles there will be no difficulty in working out a satisfactory 'electromagnetic theory of ponderable media. The proposed theory would also have the advantage of not predicting th<* •obstinately absent gravitational shift of the spectrum lines. It might also react, in part at least, upon the 1905 relativity, depriving it of its indispensability in most cases, but by no means banishing it from the whole domain of physico-mathematical investigations. Finally, the just objections raised by the advocates of the physical principle of causality against the fixed and homogeneous aether of Fresnel-Lorentz would not apply to Stokes’s modified aether. For this