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166 Dr. L. Silberstein on the recent Eclipse Results
Michelson-Morley experiment a pronounced positive effect. But enough has now been said in illustration of the formula* for the condensation and for the slip.
5. Before passing to consider the Eclipse result it may be well to generalize the condensation formula (3) for the case in which Boyle’s law is replaced by any relation between the pressure and the density of the aether. The corresponding generalization of the slip-formula (2), not required for our present purposes, may be postponed to a later opportunity.
Let the pressure p be any function of the density p aloner and let there be any distribution of gravitating masses. Introduce the function, familiar from hydrodynamics,
Then, in the state of equilibrium, and with dm written for any mass-element in astronomical units,
where r is the distance of the contemplated point from dmr and the integral, representing the total gravitational potential, extends over all material bodies. <£> being a known function of p, formula (6) gives the required relation. It will be seen from the definition (5) that the dimensions of <I> (work per unit mass of aether) are those of a squared velocity. In order to bring this into evidence, let, us recall that
is the velocity of propagation of longitudinal waves in any compressible non-viscous fluid *. This velocity is, in general,, a function of p, and becomes a constantfor the special case of Boyle’s law, namely, our previousUsing (7) and
writing, as before,we have
. . . . (5 a)
the required form. The integral is to be extended from
* This result, known as the formula of Laplace, holds also for the most characteristic kind of waves—to wit, for a wave of longitudinal discontinuity (Hugoniot, Hadamard), for which it follows directly, without integration, from the hydrodynamical equations of motion. Sec, for instance, my ' Vectorial Mechanics/ p. 169.
and Stokes-Planck's sEther.
,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.