44 PRECESSION OF THE EQtTINOXES AND NUTATION 



form it appears in the analysis which, otherwise, gives to the internal fluid mass a 

 precession identical with that of the enveloping shell ; an important fact to be noted. 

 Prof. Hopkins confines his analysis for the case of homogeneousness to equal 

 ellipticities for the bounding surfaces of the shell. Excepting the case of sphericity 

 for the inner surface, the result would be the same viz., an unchanged precession, 

 however the ellipticities might differ. 



I now return to the formula (a) and remark, that it is an approximate expres- 

 sion for the difference (P l P 1 ) due to the fact that the spheroid is heterogeneous 

 that it is not capable of being made a test of internal fluidity, or a measure of 

 thickness of crust. 



I have already shown that for homogeneousness the couple due to pressure on 

 the inner surface of the shell is identical with the sun-couple upon the fluid mass 

 a result approximately true if the density of the fluid strata vary, the methods by 

 which pressures are obtained in the latter case only differing by the introduction 

 of the variable density into the differentials to be integrated. Hence, if we take 

 the sum of the sun-couples exerted on a shell of interior and exterior ellipticities, 

 e and e,, ariti upon the fluid nucleus considered as solid, and divide by the moment 

 of inertia of the entire mass and by o, we shall have the rate of precession of the 

 entire mass considered as a solid. 



Referring to Prof. Hopkins' analysis and symbolism, the quotient will be 



d(a 6 ^ 



3 n . S a * da' (i) 



2 A -^ ' 



Denote the moment of inertia of the entire spheroid by I, 

 " " " " " " shell " I 



Then (oj = A fa (aja (a)\ 



and the above expression will reduce to 



nntational movements. In case the precessional force were augmented by so large a ratio as _ ^ 



would be for a thin shell, these nutational movements would surpass in magnitude those necessary 

 to generate the required counteracting pressures. 



1 This implies, of course, that the second term of the numerator should be equivalent to the symbolic 



expression \ a f ^ at 'da, which would be used for the couple due to the contents of the shell regarded 

 J o da 



as solid. The statement of the text is, however, demonstrably true. The difference between the 

 couple due to pressure (used above from Prof. Hopkins' determination) and that denoted by the 

 symbol just written, can only be due to the portion of the sun-force expended in producing internal 

 motions in the fluid, proved in the analysis to be, for the case in hand, omissible. Hence, the 

 analytical discrepancy depends upon such neglected quantities : but, more generally, the effect of 

 such motions would be practically nil in a rotating body. 



