268 The Rev. Samuel Haughton on the 



These are the equations which correspond, on the supposition of homoge- 

 neity, to the equations (17). Equating the values of e,, we obtain the following 

 equation to determine : 



2^= + 50^ = 3 5^^^ . (26) 



^ ^ 7e — bin ^ ' 



Substituting in this equation for m and e their values in tlie case of the Earth, 

 viz., -^ and ^, we find, 



2(^^ + 50- = 13'57743. (27) 



Applying Stuem's theorem to this equation, it is easy to prove that it has only 

 one real root, which lies between = 1 and = 2. The numerical value of 

 this root is 



— = 0= 1-2407. 

 a. 



Hence, since a = 3958 miles ; a, = 3190 miles, and 



a - a, = 768 miles. (28) 



This is the thickness of the earth's crust, on the hypothesis that both the crust 

 and nucleus are homogeneous, and the surfaces of both perpendicular to gravity. 



I shall now prove that this thickness of crust is a major limit to the depth 

 to which the density of the rocks at the surface can extend into the interior ; 

 the density being supposed heterogeneous. 



The difference of the moments of inertia of the nucleus with respect to its 

 polar and equatorial axis may be expressed as follows : 



C-A- — p— y— = — p-a-, (29) 



15 J; da \f)^ a ' ^ ' 



a denoting an unknown number, depending on the structure of the nucleus, and 

 which, if the nucleus be supposed fluid, is greater than unity. 

 Substituting from (29) in equations (14) and (15) we find 



and, 



I [Po (^«^^- ^.) + - \P«+ (^ - Po) <t>'\]=l^ (2^ - '«) 4>'- (31) 



