NEW ADDENDUM. 45 



S 

 Denote the moment of inertia of the entire spheroid by /j ^ -^7ta(«.) 



" shell " /' =^7t[a(«,)-o(a)] 



Q 



nucleus " /" = _ _ 71 a (a) 



Then a (aj = - L / a (oj _ a ((()) 



and the above expression, reduced to precession, will become 



Prof Hopkins gets for the precession of tjie same spheroid considered as fluid 

 witliin the shell, (his symbolic abbreviations used in both cases) 



(yi) + (72) \ ' 2" — 1/ 



In this last expression (^^i) and (^o) denote coefficients of gyration wliich one 

 and the same couple (i. e. the centrifugal force, by pressure on the shell and by 

 reaction on the fluid mass — the assumj^dou Icing made that the latter, having its 

 proportionate force on each particle, gyrates as a solid) produce upon the shell and 

 fluid mass respectively. They should be therefore inversely proportional to the 

 respective moments of inertia of the shell and nucleus, rendering the expressions (.r) 

 and (//) identical. 



But this apparent identity is brought about by assuming that Prof Hopkins' 

 expression for the pressure-couple on the shell arising from the sun's attraction on 

 the fluid to be identical (or at least approximately so) with that which would be 

 exerted on the same heterogeneous fluid solidified ; by Avliicli assumption I introduce 



in {x), -J — -, Avhich is Prof. Hopkins' symbolic abbreviation of 



— ^-^- ) : instead of 

 -a (a) I 



da 

 <7(«i)— (7(«) '' ""''""""" t[CT(«i)a(a)] 

 wliich latter expression belongs to the case just specified, of the solidified fluid. 



Now in the case of nature — /'. e. the earth with the received hypothetical laws 

 of density and ellipticity, the two expressions difl'er in a ratio (about 4 : 3) so 

 greatly exceeding unity, as to forbid the assumption of approximate equality. The 

 error of the first expression will be better appreciated by referring to the quan- 



' The interpretation of (,r) and (ij) is obvious. P is the coefficient of precession for a homoqe- 

 neous shell of uniform ellipticity f ; instead thereof let the shell be heU'rorjeneous with same internal 

 surface, but of an external ellipticity t,. P for such a shell will have a fractional increment denoted 

 by the ratio s. By the pressure of the internal fluid the precessional coefficient of this shell will be 



still further increased by a ratio denoted (a result of the analysis) by ^' . But the shell is con- 



cf — ] 

 strained to carry along and to take up a common precession with the nucleus, and the coefficient will 



be thereby diminished in a ratio of (.r) or (according to Prof Hopkins) the corresponding ex- 

 pression of (y). Since P—P^ ^, expression {a) is readily deducible from {ij). 



