214 FIGURE OF THE EARTH. [11 



The attraction in the direction MO is 



dV 4 16 /3.. 



from which r' has disappeared, so that the attraction is independent of 

 the dimensions of the spheroid, and varies directly as the distance MO. 



It will be convenient to eliminate r', which varies with X, from the 

 expression for V. We have seen in 6, 



"- where PN = c ( 1 - X 2 ) 4 , 



_ 



C 



or if we introduce a quantity K, the radius of a sphere of volume equal 



(2 \ 

 1 - e I , then 

 3 / 



Substitute above for r', and collect the terms in e ; we find 



The potential at an external point expressed in a parallel manner is 



4 K 3 8 K S /3,, 1> 



We have supposed the density uniform and have taken it as unit. If 

 it be called p, these expressions must be multiplied throughout by p. 



10. Let us now consider the case of a heterogeneous spheroid, the 

 surfaces of equal density being also spheroidal, concentric and coaxial with 

 the external surface, but not necessarily similar to it. 



Take a shell of homogeneous matter of density p contained between 

 two spheroidal surfaces; let the semiaxes of the interior surface be c, c(l+e), 

 and the semiaxes of the exterior surface c', c'(H-e'); also let /c, /c' be the 

 radii of the spherical surfaces enclosing volumes equal to the spheroidal. 

 Then for an attracted particle inside this shell 



and for an attracted particle outside 



These expressions hold whether the shell is indefinitely thin or of finite 

 thickness. 



