IN Hi: I. A TION TO THE EAHTII'S INTKKNAL STIIUCTUUK. 45 



Prof'. Hopkins gets for the precession of the same spheroid considered as fluid 

 within the >hell 



In this last expression (y,) and (y,) denote coefficients of gyration which one and 

 the same couple (i.e. the centrifugal force, by pressure on the shell and by reaction 

 on the fluid mass tin .///y///// l>< imj mail' that the latter, having its proportionate 

 force on each particle, gyrates as a solid) produce upon the shell and fluid mass re- 

 spectively. They are therefore inversely proportional to the respective moments of 

 inertia of the shell and nucleus; 1 hence, the expressions (x) and (y) are identical. 

 The precession is therefore the same whether the entire spheroid be solid, or whe- 

 ther it be a shell with a fluid nucleus.* 



If confirmation be needed it will be found in the fact that the analysis itself 

 gives, as in the case of homogeneousness, the same precession to the internal 

 fluid spheroid as to the shell. 



The two masses retaining their configurations, dispositions of matter, and rotary 

 velocities, essentially unaltered by the hypothesis of internal fluidity, it would be 

 a violation of fundamental mechanical principles were the resulting precession not 

 identical with that due to the entire mass considered as a solid.* 



I am aware that the author's ratio, rr^v is not identical with the above but it evidently should 



be. The discrepancy is of the same character as that treated in the note to p. 44 concerning the 

 value of h for the sun-couple; its origin being, unquestionably, in the assumption upon which the 

 yalnc of (y,) is independently calculated, viz. : that the planes of rotation are, by the angular separa- 

 tion of the axes of the shell and fluid, deflected in perfect planes. This assumption (which docs 

 not at all affect the calculated value of (y,) ), is, as we have seen (p. 43 and note), sustained for the 

 case of homogeneousness. It is not so in the other case. The internal denser fluid rings undergo 

 less deflection than the lighter ones. It roust be borne in mind that the angle t, of these deflections, 

 uniform or not, is very minute, even with reference to the angular separation of the azes, always to 

 be regarded as extremely small ; hence of the second order of minuteness. 



1 An exception must be made for perfect, or nearly perfect, sphericity of interior. In the expres- 

 sion (y), s becomes infinite and A zero for this case. 



* The oneness of the precession of the shell and fluid contents, resulting from the analysis, may 

 be put in a stronger light by the statement that for the sun or moon, separately, a suitable determi- 

 nation of arbitrary constants will eliminate even the minute oscillation about each other of the axes 

 (pp. 43, 44), of shell and fluid, which depends on (y,) and (y,); this unison, I think, could not really 

 obtain ; but it is quite consistent with the convenient but imperfect view of gyration which makes it 

 only a motion perpendicular to the plane of the generating couple. 



I do not concur with Sir William Thomson in the opinions quoted in note, p. 38, from Thomson 

 and Tait, and expressed in his letter to Mr. Q. Poulett Scrope, so far as regards fluidity, or imperfect 

 rigidity, within an infinitely rigid envelope. I do not think the rate of precession would be affected. 



That no increase arises from fluidity I have endeavored to show; and it is, unquestionably, a 

 corollary of Prof. Hopkins' investigations. As regards imperfect rigidity, Sir William Thomson 

 bases his argument upon the assumption that "the whole would not rotate as a rigid body round 

 one ' instantaneous axis' at each instant, but the rotation would take place internally, round axes 

 deviating from the axis of external figure, by angles to be measured in the plane through it and the 

 line perpendicular to the ecliptic in the direction towards the latter line. These angular deviations 

 would be greater and greater the more near we come to the earth's centre. ***** Hence the 



