214 ON THE PHYSICAL STRUCTURE OF THE EARTH. 



where b and a are the outer semiaxes of the shell composed of all the 

 strata of equal density. Bat 



2 V. «27 c 



C-A G^ + C,+ . . . +(7,-(Ai4-A2+ . . . +AJ 



C'i+6\+...+(7„ 



This is the symbolical form of the proposition just stated. 



In a homogeneous solid of revolution the general expression for the 

 moment of inertia is 



Ttfy^xdx ; 



and from the ordinary treatises on mechanics it readily appears that 

 from a sj)heroid, 



15 ' io ^ '' 



wbere h is the semi-polar and a the semi-equatorial axis. Hence we have 



G-A 2a'h-a'h-a?¥ a^h-a'V {a^-¥) ^^ a'-b^ 1^^ 6^ 

 C~~ 2a' b - ^a'b ~ 2a'b 



and 



2 (C-A) / 62 

 G -\^~a' 





In a spheroidal shell for whose inner surface the semi-axes are &i aud 

 «i, we have the moments of inertia with respect to the axes by taking 

 the moments for the inner spheroid bounded by bi and ai from those of 

 the outer spheroid. 



Calling the former Ci and Ai, we have as before, 



8 4 



Gi= r-Ttai^bi, Ai= j^ ;rfl,^&i(ai2+&j2). 



Calling Gi and Ai the moments of inertia of the shell, we have there- 

 fore, 



C,=~^7r(«^&-oV/>i), A, = -^^7r[a'b{a^+b^)-a,'b,{a,'J^b,')]; 



and hence 



/ b^ \ / b ^\ 



0,-Ai a^b{a'-b')-aiHi{a,-'b-bi') "'\^~dO ~^'' ^' V~a7 V. 

 Gi ~ 2{a*b—a^'bi) ~ 2{a'b—ai'bx) 



If e and Ci be the outer and inner ellipticities of the shell, 

 e = 1 - ^, ., = 1 - ^^, and If . = c, ^^ = -. 



In this case — ^t 2(a^b^^ = i (^1 - o^j 



2 h 



Ci-A, _ C-A_ 

 Gi - G ' 



