430 Form assumed by a Fluid Shell revolving in a Hollow Spheroid. 



\ has a given value. Hence it appears that when X is given, we 

 can always find such positive values for m, that the condition 

 given by the inequality may be fulfilled. Equation (11) in that 

 case always gives positive values for e when X and m are deter- 

 mined. 



Suppose, for example, \=1, then the condition (14) will be 



-0-2876m+ 1 > 0, or m < ^^. 



Suppose, for instance, m = l, or that the densities of the solid 

 spheroid and of the fluid sphere were equal, then we should get 



,— 7 =0-3562, whence k =0-801. 



1 + a: 



The inner radius of the hollow sphere ought therefore to be 

 about four-fifths of the outer one. Lastly, we find from (11) 

 that 



€=0-2876. 



There is another class of forms of equilibrium which may be 

 assumed by fluids revolving in a hollow spheroid, which we will 

 only mention here, as being of comparatively of less interest. A 

 heavy homogeneous fluid, revolving in the cavity of the hollow 

 spheroid, may assume a spheroidal figure of equilibrium. In 

 fact, when M, M', N, and N' have the same signification as 

 before, and — M"#, — M"y, and — Wz denote the component 

 parts of the attraction of the fluid spheroid, then the equation of 

 equilibrium will be 



(--M + M'-M , '4-^ 2 )^ + ^//) + (--N + N , --N , 0-^=O,(15) 



where M, M', M", N, N', N", w are independent of x, y } z. If 

 the equation of the fluid surface be supposed to be 



* 2 + ?/ 2 , _£!__-, 

 a 2 i "a 2 (l+\ 2 )~ i ' 



then the condition of equilibrium will be 



-M + M'-Mff + uf 

 1+X " -N + N'-N" ■• ' * < 16 ) 

 By putting the values for M, M ; , M", N, N', N" in equa- 

 tion (16), we find that equilibrium is possible under an infinite 

 number of different conditions with regard to the form and 

 density of the spheroids, and the velocity of rotation of the fluid 

 mass. 



Gothenburgh, October 24, 1860. 



