1. Any estimate of the rigidity of the Earth must be based partly on some observations from which a deformation of the Earth’s surface can be inferred, and partly on some hypothesis as to the internal constitution of the Earth. The observations may be concerned with tides of long period, variations of the vertical, variations of latitude, and so on. The hypothesis must relate to the arrangement of the matter as regards density in different parts, and to the state of the parts in respect of solidity, compressibility, and so on. In the simplest hypothesis, the one on which Lord Kelvin’s well-known, estimate was based, the Earth is treated as absolutely incompressible and of uniform density and rigidity. This hypothesis was adopted to simplify the problem, not because it is a true one. No matter is absolutely incompressible, and, the Earth is not a body of uniform density. It cannot be held to be probable that it is a body of uniform rigidity. But when any part of the hypothesis,
e. g
., the assumption of uniform density, is discarded, the estimate of rigidity is affected. Different estimates are obtained when different laws of density are assumed. Again, whatever hypothesis we adopt as regards the arrangement of the matter, so long as we consider the Earth to be absolutely incompressible and of uniform rigidity, different estimates of this rigidity are obtained by using observations of different phenomena. Variations of the vertical may give one value, variations of latitude a notably different value. It follows that “the rigidity of the Earth” is not a definite physical constant. But there are two determinate constant numbers related to the methods that have been used for obtaining estimates of the rigidity of the Earth. One of these numbers specifies the amount by which the surface of the Earth yields to forces of the type of the tide-generating attractions of the Sun and Moon. The other number specifies the amount by which the potential of the Earth is altered through the rearrangement of the matter within it when this matter is displaced by the deforming influence of the Sun and Moon. If we adopt the ordinarily-accepted theory of the Figure of the Earth, the so-called theory of “fluid equilibrium,” and if we make the very probable assumption that the physical constants of the matter within the Earth, such as the density or the incompressibility, are nearly uniform over any spherical surface having its centre at the Earth’s centre, we can determine both these numbers without introducing any additional hypothesis as to the law of density or the state of the matter. We shall find, in fact, that observations of variations of latitude lead to a determination of the number related to the inequality of potential, and that, when this number is known, observations of variations of the vertical lead to a determination of the number related to the inequality of figure. [
Note added
, December 15, 1908.—This statement needs, perhaps, some additional qualification. It is assumed that, in calculating the two numbers from the two kinds of observations, we may adopt an equilibrium theory of the deformations produced in the Earth by the corresponding forces. If the constitution of the Earth is really such that an equilibrium theory of the effects produced in it by these forces is inadequate, we should expect a marked discordance of phase between the inequality of figure produced and the force producing it. Now Hecker’s observations, cited in § 6 below, show that, in the case of the semidiurnal term in the variation of the vertical due to the lunar deflexion of gravity, the agreement of phase is close. If, however, an equilibrium theory is adequate, as it appears to be, for the semidiurnal corporeal tide, a similar theory must be adequate for the corporeal tides of long period and for the variations of latitude.]
The Restricted 2+2 Body Problem: Parametric Variation of the Equilibrium States of the Minor Bodies and Their Attracting Regions
