38 PRECESSION OF THE EQUINOXES AND NUTATION 



oscillations given by the dynamic theory approximate rapidly to those of the 

 "equilibrium theory," with which, when the depth is very great, or the spheroid 

 wholly fluid, they are essentially identical. Moreover, he shows (p. 219, Vol. I) 

 that the vertical motions of the particles, when the depth is small, may be disre- 

 garded. When the spheroid is wholly fluid, all the relative motions of the particles 

 are of the same order as the vertical ones and exceedingly minute ; and the forces 

 of inertia thereby developed are insensible compared with those we have been con- 

 sidering. 1 



By parity of reasoning the truth of Sir W. Thomson's propositions concerning a 

 solid but yielding spheroid is made evident ; for exactly in the same ratio to the 

 tides of a fluid spheroid that the solid tidal elevations are produced (the actual 

 ellipticity of the earth being nearly that of equilibrium with the centrifugal forces), 

 will the processional couple due to the tide-producing attraction be neutralized by 

 their centrifugal action. 2 That a thin solid crust, such as geologists generally 

 assume, would yield and exhibit tidal elongations, seems without calculation very 

 probable ; but if Sir W. Thomson is correct as to the rigidity required in even a 

 wholly solid earth, the hypothesis of a thin crust 'must be abandoned, and it would 

 seem indeed that rigidity several times as great as the actual rigidity of iron 

 throughout 2000 or more miles thickness of crust would be incompatible with a 

 very high internal temperature. 



Without having recourse to Sir W. Thomson's profound analysis, the necessity, 

 in order that there shall be no sensible solid tidal wave, of a very high rigidity 



1 The foregoing demonstration does not conflict with Laplace's theorem that ocean tides do not 

 affect the precession ; for his theorem applies only to a shallow ocean over a rigid nucleus, of 

 which oeean the precessional couple, by altered attractions, pressures, and centrifugal forces due to 

 generation of living forces in the fluid, is transferred to the nucleus. I have already alluded to 

 the minuteness of the motions of the particles of a fluid spheroid. The remarks apply, d fortiori, 

 to those of an elastic solid. Vibratory motions, properly speaking, cannot exist, for the elastic 

 forces extremely minute are always held (sensibly) in equilibrium by the distorting forces. The 

 solid surface would oscillate in the same sense that the ocean tides oscillate, i. e., by a "forced'' 

 tide-wave. 



2 " It is interesting to remark," say Thomson and Tait ( 848, " Treatise, &c."), " that the popular 

 geological hypothesis of a thin shell of solid material, having a hollow space within it filled with 

 liquid, involves two effects of deviation from perfect rigidity which would influence in opposite ways 

 the amount of precession. The comparatively easy yielding of the shell must render the effective 

 moving couple due to sun and moon much smaller than it would be if the whole interior were solid, 

 and, on this account, must tend to diminish the amount of precession and nutation. But the effective 

 moment of inertia of a thin solid shell, containing fluid in its interior, would be much less than that 

 of the whole mass if solid throughout; and the tendency would be to much greater amounts of pre- 

 cession and nutation on this account." 



The co-efficient of precession of the "thin solid shell" would be (p. 34) the same, nearly, as that of 

 the spheroid of which the homogeneous strata have the same ellipticity. Its precession-resisting 

 couple (48) due to tidal distortion would be just what is necessary to develop its proportional influ- 

 ence upon the precession of that shell, upon which the fluid contents can exert influence only through 

 their pressure. This is identically Prof. Hopkins' problem. The thin shell of popular geological 

 hypothesis would, however, be subject to tidal distortions scarcely inferior in magnitude to those of 

 a wholly fluid spheroid; by which, as we have seen, the sun and moon's "moving-couple" is wholly 

 neutralized throughout the whole spheroid. 



