UH 



inner boundaries two 



:ir Ami similarly situated ellipsoids; required the 

 i'avity o! lith pjut of it included between 



planes. Let a, / - scmi*axcs of the exterior 



ra, rb, re those of the inner ellipsoid, r being a 



semi-axes of an ellipsoid; the volume of 

 ;i part is JITCI/K% and the co-ordinates of its centre of 

 gravity are fa, |6, and \c (see Art 129). Hence 



|a . lirabc - jra . lwr*abc + x (\vabc - }irr j a&c) ; 



Ifjre suppose the shell indcfmitrly tliin. we must put r l f 

 and thru x - Ja. Similar results may be found for y a: 



! ellipsoid is composed of an infinite number of in- 

 thin shelU; each shell has for its outer and 

 boundaries two similar an 1 ilipxoids ; the 



;' each shell is constv 'y Taries from 



occonling to a given law : 



of gravity of t ii part of the ellipsoid included between 



three principal planes. 



7, t represent the three semi-axes of an ellipsoid; 

 then the volume of the ellipsoid :. Suppose that 



jf wuj and t - fix, where m and * are constants, then the 



i ^^ 



ne becomes - x*, and if there be a similar i-llipsoid 



having 2 + Ax for the semi-axis corresponding to the s 



first ellipsoid, the volume of the second ellipsoid 



will be - (x-f Ax}". Hence the volume of a shell bounded 







by two similar and similarly situated ellipsoids may be dc- 



by {(aj + Aa:)"-**}, and therefore by 4irm*< 



the thickness is i miuishcd. Let (x) de- 



shell, then its matt is 4ir?n/t$ (x) - 



