and Stokes- Planck9s JEther.
165
aberration, i. e. <r= log5= 10*2, we shall have at the surface of the Sun, as already mentioned ina footnote,
which means, no doubt, an enormous condensation *. The corresponding relative velocity ot* slipping v/vw will, by (2), be almost evanescent; the drag will be almost complete.
On the other hand, at the surface of a hydrogen atom, assumed for the moment to be a homogeneous sphere (and the only existing body), we shall have log 5=1*7 . 10“34, that as to say,
indistinguishable from unity. Notice that for small cr the denominator in (2) reduces to Jo*3 + 4- higher terms, so that the relative slip becomes
For such bodies, therefore, as a hydrogen atom, or in fact any other atom, the ratio in question will be exceedingly nearly equal its limiting value 3/2, wThich is well known to be the maximum relative slipping for a sphere moving in an incompressible liquid. In short, for such small bodies there will be practically no drag at all. The more so for electrons, if one wished to attribute to them gravitational properties. This behaviour will be important in connexion writh some such eleptrodynamic theories of ponderable media, as is that proposed by Lorentz, which require a complete slip. But •even a sphere of the mass of 1. kg. and the radius of 10 cm., for which a—109 .10_16, will practically have no “ grip upon the aether.” This will readily be seen to account, among •other things, for the negative results of Sir Oliver Lodge’s ingenious experiments with the Ether machine, even if its whirling part were made much more massive. As a mere curiosity notice that even the Moon would have only a partial, weak grip upon our rehabilitated aether. In fact, at the Moon’s surface we should have= 10*2 x 0*094 = 0*96, and
therefore, by (2),which differs only by 0*35 from
the full slip. Thusthe Selenites would obtain with a
* Such fantastically large condensations need not frighten us. They can be reduced if Hoyle’s law is replaced by some other appropriate form of relation between pressure and density. Boyle’s law, which is by no means necessary, is here used only, as the simplest one, for the sake of illustration.
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.