42 NEW ADDENDUM. 



NEW ADDENDUM. 1 



A FEW words are in place here concerning the results of the late Prof. Hop- 

 kins' investigation (against which M. Delaunay's objections, (note, page 38,) are 

 especially directed), briefly stated, pages 34, 35. They are as follows: First FOR 

 HOMOGENEOUSNE^S. "Supposing the earth to consist of a homogeneous spheroidal 

 shell (the ellipticities of the outer and inner surfaces being the same) filled with a 

 fluid mass of the same uniform density as the shell;" then, "the precession will 

 be the same, whatever be the thickness of the shell, as if the whole earth were 

 homogeneous and solid." 



SECOND, FOR HETEROGENEOUSNESS, his result may be thus expressed : 



(a) Pi-P = 



" where P l denotes the precession of a solid homogeneous spheroid of which the 

 ellipticity =e,, that of the earth's exterior surface, and P' the precession of the 

 earth, supposing it to consist of an interior heterogeneous fluid contained in a 

 heterogeneous spheroidal shell, of which the interior and exterior ellipticities are 

 respectively s and e,, the transition being immediate from the entire solidity of the 

 shell to the perfect fluidity of the interior mass." 



In the multiplier of -, second member of (a), q is the ratio of external to internal 



polar radius of the shell ; s depends on the varying ellipticity and density of the 

 strata of equal density of the shell ; h depends on the density of the fluid interior. 

 For a thin crust the coefficient in question is unity nearly; for a thick one it will 

 be somewhat greater if e be less than EJ. 



It cannot fail to be observed that, under the conditions just before expressed for 

 homogeneousness i. e., equality of external and internal ellipticities we get from 

 the formula (s becoming zero) the same result, i. e , P = P,, as for that case. 



In accordance with rational hypothesis as to the internal condition of the earth, 

 equalities of ellipticities for the surfaces of a thin crust (and corresponding equality 

 of densities), or closely approximate equalities would be expected. The necessity 

 for a thick crust arises, therefore, from the alleged discrepancy between the observed 

 and calculated annual precessions (50 seconds and 57 seconds), which, according 



P - P 1 

 to Prof. Hopkins, makes - L_ _ _ 1, nearly, assuming the moon's mass Y V, and 



*t 



the earth's ellipticity ^1^. (The real discrepancy is probably very much less. See 

 page 34, et sequential) 



The original ADDENDUM, hurriedly written while the work was in the printer's hands, has been, 

 in what follows, somewhat modified and amplified. 



