Shell revolving freely within a Hollow Spheroid. 429 



it is easy to see that it can only have one positive root, because 

 the first term must be negative. 



Let us particularly consider the case in which the outer 

 bounding surface of the spheroid is spherical, and the inner sur- 

 face is oblate. Then 



N=M=^/, 



M'=2^/ ( i±^(arctanX- j^), 



N'=47rp/i-±^( X- arc tan X), 

 where p denotes the density of the spheroid. Putting 



(10) 



t«r 



Zvp'f 



equations (3) and (9) become 

 E = m ( 3 -^_3(1±^ 



g _ m+l 

 l+k~~ 2 



= E, and £ 



r 



X 3 



3 (1 + X 2 ) 



arc tan X 



)■ 



(11) 



2 



X 3 



(A,-arctan\)=^(l-E). (12) 



Equations (11) and (12) determine the relations which must 

 exist between X, e, m } and k, in order that equilibrium may result. 

 If the right-hand member of (12) be denoted by n } we get 



TTk =n ' 



This equation, if equilibrium be possible, must give a positive 

 fraction as the value of k. If resolved in regard to this quantity, 

 we get, observing that n must necessarily be positive, 



*=tv?+»> 



which gives as the condition for k < 1, 



n < 1, or 1 - ^ * t^ (X- arc tan X) < 0. . . (13) 



This condition is always fulfilled when X > 0. In order that 

 n may be positive, 



m(l-?^i X 5(X-arctanX))+1^0,ore<:l. (14) 



This inequality determines the greatest value m can have when 



