HOMOGENEOUS FLUID AT LIBERTY. 507 



rection of R : then ^J —f-\ being a function of two dimensions, in which x, y, z are 

 the only variables, ^3 X \j ~~r\ will be a quantity of no dimensions ; it will, there- 

 fore, be a function of ^, ^, -j^, or of «, 5, c ; so that we shall have 



[/^]=R2xF(«,6,c), (3.) 



F being the mark of a function. The same value may be expressed by means of the 

 coordinates, viz. 



If the value just found be substituted in equation (2.), the result will be, 



C = (x2+3,2+.^) X F (;7FTFTT" 7?7FT?> ;7?TFTP) + i X (^' + -') = 



which proves that the forces in action at the surface of the fluid are not sufficient to 

 determine the equilibrium of the mass. For the equation of the figure of the fluid at 

 which we have arrived, containing an arbitrary function, is indeterminate ; and, on 

 examination, it will be found to comprehend the elhpsoid and innumerable other 

 figures*. 



If for x, y, z we substitute their values R «, R ^, Re, the equation of the surface 

 will assume this form, 



C = R2 X {f {a, ^,c) + -^ {b^ + c^)]. 



The equation of a level surface is deduced from the equation of the upper surface by 

 changing the constant, and substituting the coordinates of the level surface for those 

 of the upper surface : now, supposing that r, in the same straight line with R, is a 

 radius of a level surface, the coordinates of the point in that surface at the extremity 

 of r will he r a, r b, r c, because r and R have the same direction : wherefore, by sub- 

 stituting 7' a,rh,rc for x, y, z in the equation of the upper surface, and denoting the 

 new constant by C, the equation of the level surface of which r is the radius will be 



C = r2x|F(«^,&,c) + -|-(i2^.e2)j. 



The comparison of this equation with that of the upper surface of the fluid leads to 



this result, 



r^ _ a 

 R^ — C » 



* In a particular examination of Clairaut's theory that occurs in the sequel of this Paper, it is proved from 

 diflFerent principles, that the equation of the figure of the fluid deduced from the forces in action at the surface 

 is indeterminate, and admits of innumerable solutions. 



