and Stokes-Planck's sEther.
167
,9=1 (or log s = 0) to the actual value of the condensation. Thus the condensation formula (6) becomes
where ft has been written for the total gravitational potential at the place under consideration. For constant V (Boyle's law), and for a single spherical body, the previous formula (3) reappears.
It will be kept in mind that although the aether is assumed to behave in this way (say, like a gas) with respect to slow processes, it can still propagate rapid transversal light— disturbances as if it were an elastic solid (like the famous cobbler’s wax of Lord Kelvin) ; but it will be best to think of light as of electromagnetic disturbances. The normal velocity c of propagating them is another property of the aether, independent of I hat which is represented by v, and subjected only to slight variations with condensation, as will appear presently. The ratio of V to c wTill be of importance, but as to the longitudinal waves themselves, they are of no physical interest for the present and, on the other hand, are not likely to become a nuisance. For it is not in our power to produce them to any relevant extent, and even if they are generated and maintained by some gigantic natural processes, their only effect would be to alter very slightly, here and there, the normal velocity of light-propagation.
If we wish to form an idea of the numerical value of t?, or at least of its upper limit, for the case of Boyle’s law, say, it is enough to take the value of a given above for the Sun, and to remember that Afjcl = 1*5 km., and, in round figures,
R = 1 . 105km. Then the result will be=8’2 .10~6, that is
to say, V equal to about 2*5 km. per second *. This is quoted by the way only. But the ratio of these two velocities will be seen to acquire a particular interest in connexion with the recent astronomical discovery.
6. Let c, as before, stand for the propagation velocity of light in uncondensed aether, i. e. in absence of, or far away from, gravitating masses, and let c' be the light velocity at a place where the aether has undergone a condensation s. The question is : How are we to correlate c' with s ? In other words : On what are we to base the optical behaviour of the aether modified by a condensation ? The only reasonable
* If so, then the condensational disturbances due to the Earth and other planets, whose velocities exceed will be confined to conical regions as in Mach’s famous experiments.
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.