of Equilibrium for Revolving Fluids. 121 



Differentiating this equation, we get 



aedx + ydy+ 1+ ^//a = ° ( 4 ) 



In order that the equations (2) and (4) may both exist at the 

 same time, it is necessary that 



The equation (5) shows the relation that must exist between 

 the several quantities in order that the inner bounding sur- 

 face may form a spheroid, whereby it is supposed, as before 

 mentioned, that the pressure on the surface is either counter- 

 balanced by the pressure of a body of gas, or is =0 in conse- 

 quence of attraction and centrifugal force. The general exami- 

 nation of the circumstances which occur in this case is par- 

 ticularly complicated, as the equation (2) cannot be integrated ; 

 we shall therefore confine ourselves to the single case in which 

 an integration can be effected, when the inner bounding surface 

 of the fluid becomes a sphere, and both the outer spheroids are 

 oblong. 



Let p denote the density of the solid part of the ellipsoid, and 

 p' that of the fluid ; / the attraction between two unit masses at 

 the unit of distance ; and let X and V have the same significa- 

 tion for the two outer spheroids as \" has in the equation (3). 

 Further, let r be the radius of the spherical cavity. Then 



M=27r P r l + x *[WT+x>-i. (\+vT+i\?) % 



M'=27r/)/^^~ 2 [xVlTx^-/. (V + x/T+V*)], 



N^W/^^f^v+yiT^-^^^ 



3 x/{x 2 + y* + z 2 ) 3 

 Phil. Mag. S. 4. Vol. 20. No. 131. Aug. 1860. 



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