42 PRECESSION OF THE EQUINOXES AND NUTATION 



Second, FOR HETEROGENEOUSNESS, his result is thus expressed: 



1 </ I > 



" 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 1 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 e and e lt the transition being immediate from the entire solidity of the 

 shell to the perfect fluidity of the interior mass." 



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

 polar radius of the shell ; n 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 factor in question is unity nearly ; for a thick one it may be 

 considerably less. 



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 the same result, i. e., P 1 =P 1 , as for that case. 



In our ignorance of 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 illicit, crust arises, 

 therefore, from the discrepancy between the observed and calculated annual pre- 

 cessions (50 seconds and 57 seconds), which, according to Prof. Hopkins, makes 



*~ =|, nearly, assuming the moon's mass ^\ and the earth's ellipticity 



(The real discrepancy is probably less. See note, page 34.) 



If, in applying the expression (a), the second factor, for a first approximation, is 

 omitted, we have =! This value of e will be in excess ; hence, the thickness of 

 crust deduced from it will err the other way, and a determination on this basis 

 will give a thickness which must, in fact, be exceeded. 



The limit of solidity, proceeding inwards, may and probably does depend upon 

 both temperature and pressure. Isothermal surfaces Prof. Hopkins finds to have 

 increasing ellipticities. Surfaces of equal pressure, deduced from the hypothetical 



law of density 2L-, have diminishing ellipticities, and if qb 1 ^\50 the above law 



agrees sufficiently well with the actual ellipticity and ratio of surface to mean 

 density of the earth. This law for E=ffi demands a thickness of crust of ^ the 

 radius, or 1000 miles. This is a minimum, since the actual surface of solidifica- 

 tion (lying between this and the corresponding isothermal surface) would have 

 greater (and hence too great) ellipticity. 



Before commenting upon this application, and upon the real meaning of the 

 formula, I return to the case of homogeneousness. Some of the results arrived at 

 by the analysis of Prof. Hopkins may be illustrated by the following considera- 





