214 ON Tin: physical structure of the earth. 



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

 strata of equal density. Bat 



T/' h'^\_C-A_C:-\-G,+ . . . +6',.-( ^i + ^+ . • . - \-A„) 



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 



and from the ordinary treatises on mechanics it readily appears that 

 from a spheroid, 



8 4 



where b is the semi-polar and a the semi-equatorial axis. Hence we have 



a^b: ' '' 



G ~ 2a* b ~ 2a*b ~ 2a'b 



and 



2 (G-A) /_¥ 



G ~\ a 



In a spheroidal shell for whose inner surface the semi-axes are 6i and 

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

 the moments for the inner spheroid bounded by ^i and fli from those of 

 the outer spheroid. 



Calling the former Gx and ^i, we have as before, 



8 4 



Ci= -.rrTTfiiH)], Ax= , - 7rai'^/>i(ai2-f/V). 



Calling Gi and ^li the moments of inertia of the shell, wo have there- 

 fore, 



and hence 



C'l "" " 2{n:'h-a,U)x) " 2{a'b-ax'bx) 



If e and e^ be the outer and inner ellipticities of the shell 



\ b . ^1 1 •. ^> ^> 



e = l-^, e, = l-^^,andif. = .„^^=^^ 



In this case 



/ b' 



C, — Ai ^^ ' " V «-' _ 1 / -, ^ 



■ Gx 2 {a*b - ax*bx) ' - -^ \^ a? I' 



Gx-Ax G-A 



Cr - G 



