^^ NEW ADDENDUM. 



couple which the sun would exert on the fluid mass considered as a solid.^ It would 

 increase the precessioual force of the sJiell in the ratio ^^ of the analysis. By 



virtue of this pressure tlie fluid tends to transform its own precession into an 

 augmented precession of tiie shell. 



It requires, however, but an extremely minute angular separation of the axes of 

 the shell and fluid to generate counter-pressures equivalent to those which caused 

 the separation.^ The divergence cannot, therefore, be progressive, but is simply a 

 minute oscillation of tlie two axes, or a rotation around each other. In the latter 

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

 precession identical vi/h tltat of the enveloi>iiHj shell. 



Prof. Hopkins conflnes his analysis for tlie case of homogeneousness to equal 

 ellipticitics for the bounding surfaces of the sliell. Excepting the case of sphe- 

 ricity for the inner surf\ice, the result would be the same — viz., an unchanged pre- 

 cession, however the ellipticities might differ. 



I now return to the formula («) and remark, that it is an inaccurate expression 

 for a sliglit difterence (Pj — P') 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 thick- 

 ness of cnist. 



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

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

 solidified, a result ajiproxiinately true (as will be shown hereafter) if the density 

 of tlie fluid strata vary. Hence, if we take the sum of the sun-couple exerted on 

 a shell of interior and exterior ellipticities, e and fj, and of the pressure-couple 

 developed in the fluid,^ and divide by the moment of inertia of the entire mass and 

 by (J, we shall have the rate of gyration of the entire mass considered as a solid. 



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



(-) ^ J"^ sin 2A -^^^ ^^^ ^"^ 



a (a,) = 1 p' - da 



' The lever arm is also 2e sin x cos %. Multiply the above by this arm, by g, by the elementary 

 surface rf ^ </ o, and, again, by cos b, and we get the elementary component tending to tilt the shell. 

 The integral, with proper substitutions, is equivalent again to (19) or (48). 



' There is anollic:r [.rocess which may take effect in neutralizing internal pressure. I have remarked 

 (last par. p. 6), that, considered as a perfectly rigid body, the precessional motions of the earth 

 cannot be preciseb/ those assumed. In fact, our imperfect integrals of the conditional differential 

 equaUons present the anomaly of a varied motion in which the generating force does no work ; no 

 yielding to the tilting couple having place. There are necessarily some, too minute to be detected, 



nutational movements. In case the precessional force were augmented by so large a ratio as 'f 



q" — 1 

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

 to generate the required counteracting pressures. 



' I use provisionally Prof. Uopkins' computations for this, involving /^ f'a'da'; its erroneous- 

 ncss will appear hereafter. 



* "^hc symbols ^, a, <o, correspond to S, 9, n, of p. 7 ; p' is the density of stratum, solid or fluid, 

 ■•r "inch . ,s the ellipticity, and a' the polar radius ; a, is the external, and a the interna pclar 

 radius of the shell. 



