f 



124 On Figures of Equilibrium for Revolving Fluids, 



these conditions are expressed by 



/ 2 r*\ 



V P+ 3?M2 > * • ' .• • ( 15) 

 which, after integration gives 



!#*&**&? < 16 ) 



If this condition is fulfilled, it is clear that the force which acts 

 upon a particle on the inner surface of the fluid, must also be 

 directed outward from the cavity. 



Let us now, lastly, consider the special case when ?w=l, i. e. 

 when the density of the hollow spheroid is equal to that of the 

 fluid. The equation (11) gives then 



P=-2 



1+X 2 



X 3 



[mx +v /i + x*)- 7! =]. - tm 



If the smaller semiaxis of the inner spheroid be equal to the 



r 1 



radius of the surface of the sphere, we set - = . . and the 



condition (16) is then expressed by 



+ 5OTS " 8 > 



The equation of condition (10) will in that case be 



E= ^_3vTO* M W_ 2) _ . . (19) 



When \ is given, we can define by (18) the greatest value of 

 X' which is compatible with the conditions of equilibrium. The 

 equation (19) then gives the corresponding value of E. If, for 

 instance, \=0, then 



|=-|, X<=0, E = 0; 



if \=1, then 

 P 



= -0-2464, X'=0-699, E = 0-2608; 

 if\=2*5, then 



| = -0-22169, V = 1732, E= 0-628. 



Every value of \' smaller than those here calculated must, there- 

 fore, also satisfy the conditions of equilibrium. 

 Gothenburgh, June 12, 1860. 



