8 INTERNAL STRUCTURE OF THE EARTH 



'3 V 15 



cos V = ~ {C — A) sin v cos v, [since e = — (C — A) ], which is identically the ex- 



pression for the tilting conple exerted on the earth by the sun's attraction — tlie 

 couple which causes 2> recession ; and to which, also, tlie above distortion {if it Uke- 

 ■wise take i>lace) is due. Hence this tilting of the planes is exactly the distortion 

 the solar couple is capable of producing, and the proper exponent of the intensity 

 of the sun's effort. 



But, if the tidal protuberance — the "small elliptic deviation" — be thus the result 

 of a sliglit tilting of the planes of rotation, it follows tliat tlie centrifugal forces of 

 the fluid particles moving in these planes viust shift direction with the planes 

 themselves; in other words, the planes of repulsive action of the "infinite repelling 

 line" must be regarded as diverted from perpendicularity to the repelling line.. A 

 couple, neutralizing the solar couple, will not be generated, hence my demonstration 

 founded upon it fails. In fact if we refer to the actual phenomena as I now exhibit 

 them (see Fig. 2),' we find that each circle of revolution of the normal fiijure is still 

 a inrfeet circle (or very nearly so) in the distorted figure. The centrifugal forces, 

 exactly balanced before distortion, are no less so after distortion. 



The planes, then, in which lie the repulsive force produced by rotation, neither 

 remain stationari/, as the rejielling line paraphrase of authoritative usage leaves 

 them; nor do they shift correspondingly with the shifting of th<e figure ; but they 

 actually (lohile the rejycUin'j line itself remains stationary) undergo a much slighter 

 change of direction by which, as eff'ectually as if they shifted equally with the 

 shifting of the figure while continuing normal to its axis, they frustrate the 

 development of a precession-neutralizing "couple."' 



Having, in what precedes, shown the plausible basis on which my demonstra- 

 tion of the absence of precession in a non-rigid spheroid was founded, and having 

 now explained precisely where the defect of the demonstration lies, I need scarcely 

 say that I find myself compelled to Avithdraw my conclusion as regards a "solid 

 but yielding splicroid," viz., that "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 tliat of equilibrium with the centrifugal forces), will the preces- 

 sional couple due to the tide-producing attraction be neutralized by their centrifugal 

 action;" a withdrawal which requires an "expunging" of considerable portions of 

 article cited at head of this paper, or at least a disclaimer of their virtue as demon- 

 strative of the above dictum. 



' In Fig. 2 the full periphery is the meridional section of the rotating fluid earth : P P the axis 

 of rotation (answering to the repelling-line, which continues slill the axis of rotation after distor- 

 tion) ; 6; 6, (parallel to equator E E) the normal planes of rotation (only one of which is drawn); 

 b\b\, etc., the same planes as tilted (about a, a, a, etc.) through the minute angle 2('y, causing 

 a motion of the axex of fgure, though the whole angle y, from E to E' , and from P to P', and a 

 distortion of figure (or tidal development), represented by the broken peripherj^ 



In the analytical expressions above C and A are the moments of inertia with reference to polar 

 and equatorial axes, respectively. 



^ The above demonstration refers to the homogeneous fluid spheroid; but we may infer, I think, 

 for lietcrogeneousness whether fluid, or solid but imperfectly rigid, that, however the internal motion 

 of their distortion be thereby modified, the above will hold true. 



