NEWADDENDUM. • 47 



Q 



integrals from zero to a^ and multiplying by - - 7t, the last becomes (Thomson and 

 Tait, § 825), 



2 



^ Ma^" (ej — i m) ==CS — 3(, for the earth as actually constituted. 

 o 



And the first (deduced from liopkins, Phil. Trans., 1840, pp. 203 and 204), is 



(nearly) 



- Ifa^^ ei= C — A, for same spheroid with unifonn internal cUipticitics. 

 o 



(il/= mass of the earth). 



[The value of the first, using the constants of density of Archdeacon Pratt, 

 "Figure of the Earth," 4th ed., p. 113, is somewhat less than this last expression.] 

 Now ??2 = 2 Itj (ratio of centrifugal force to gravity) and hence 2 (fj — j/h) is very 

 little less than Cj. For a fluid nucleus, the inequality would be still less. 



Hence it appears that both (x) and (y) (when corrected) express very nearly 

 the precession of the solidified earth; and, moreover, that tlie effect upon preces- 

 sion due to the variation of internal cllipticity is very small, the precession of the 

 earth considered as rigid being, essentially, that corresponding to uniform cllipticity; 

 or what is the same thing, that of a homogeneous sjfheroid of its exicrncd J'orm. 



Tliis also appears in the comparison of the value of , as established from 



A 



observation, and the resulting calculated cllipticity. The first is .00327 and tlie 



second ^ir (Thomson and Tait, § 828). Now a homogeneous spheroid of the 



(J 4 



latter ellipticity would have for — ^^— a value (e — 4e-) of .00332 ; a difference 



of about Jg. Variations in the constants which enter into the expressions for 

 internal density give rise to variations in the calculated ellipticity — and, of course, 

 in the resulting precession ; but if the external cllipticity is defined by a rigid shell, 

 the effect of internal variation is, in the case in liand, almost nil. Hence, had 

 the hypothetical consolidation of the earth, of p. 43, been carried to tlie very 

 centre, no material approximation to the desired correction of ^ in the calculated 

 precession would have been found. ^ In fact, the problem for heterogencousness 



subjected to a foreign attraction, and a condition of static equilibrium assumed, tlie pressure-couple 

 exerted by the fluid on the shell cannot difler from that which the attraction would exert on the 

 solidified fluid. 



In the case of homogeneousncss, I have arrived (p. 43: the results, though based on an cllipticity 

 corresponding to fluid equilibrium, hold good for any sntall ellipticity) at the exact expression for the 

 pressure-couple, from the function expressing the tidal protuberance due to the foreign attraction. 

 The tidal conflguration of the heterogeneous earth, wholly liquefied, would result from the transcen- 

 dental analysis of Hopkins, pp. 203, 204, or of Thomson and Tait, § 822-824, and the maximum 

 height would be one foot, very nearly; but the pressure function cannot be readily deduced. It 

 would depend on gravity and [§ 825] "the value of C — A may be determined solely from a 

 knowledge of surface or external gravity, or from the figure of the sea level without any data regard- 

 ing the internal distribution of density." 



* It is curious, to say the least, that there should be ground for the remark that the expressions (a) 



C 1 ( s ) 



and (y), which latter, with its author's valuation of -'' , raav be writcn -J ^ "f" h ( P, g'^e 



