44 NEW AD DEN BUM. 



couple which the sun would exert on the fluid mass considered as a solid. 1 It would 



Q^ 



increase the processional force of the shell in the ratio --^- of the analysis. By 



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

 augmented precession of the 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. 2 The divergence cannot, therefore, be progressive, but is simply a 

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

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

 precession identical with that of the enveloping shell. 



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

 ellipticities for the bounding surfaces of the shell. Excepting the case of sphe- 

 ricity for the inner surface, the result would be the same viz., an unchanged pre- 

 cession, however the ellipticities might differ. 



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

 for a slight difference (P^ 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 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 

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

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

 a shell of interior and exterior ellipticities, s and c l5 and of the pressure-couple 

 developed in the fluid, 3 and divide by the moment of inertia of the entire mass and 

 by M, 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 4 



j p' ^') da' + 2* e f a o'a'da! 



() 



- 



fa, , da b 



o ? 'da' 



_ 



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

 surface d/ide, and, again, by cos e, and we get the elementary component tending to tilt the shell. 

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



1 There is another process which may take effect in neutralizing internal pressure. I have remarked 

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

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

 equations 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 processional force were augmented by so large a ratio as ^ 



<? 5 1 



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

 to generate the required counteracting pressures. 



1 I use provisionally Prof. Hopkins' computations for this, involving /J f'a'da'; its erroneous- 

 ness will appear hereafter. 



' The symbols M , A , , correspond to S, 9, n, of p. 1 f > is the density of stratum, solid or fluid, 

 for ^ which , IB the ellipticity, and a' the polar radius; a; is the external, and a the interntf wlur 

 radius of the shell. 



