20 Mr green, ON THE LAWS OF THE EQUILIBRIUM OF FLUIDS. 



<p being the characteristic of any function whatever; and we afterwards 

 form the quantity 



where dv represents an element of the sphere's volume, and g the dis- 

 tance between dv and any particle p under consideration, the resulting 

 value of V wiU always be of the form 



V^^ being what I^'*" becomes by changing 9^, nr , the polar co-ordinates 

 of the element dv into Q, w, the co-ordinates of the point p; and R 

 being a function of r, the remaining co-ordinate of p, only. 



4. Having thus demonstrated a very general property of functions 

 of the form P"*'', let us now endeavour to determine the value of F" 

 for a sphere whose radius is unity, and containing fluid of which the 

 density is every where represented by 



p = {l-x''-y"-zyf{x',y',z'); 



on', y , z' being the rectangular co-ordinates of dv, an element of the 

 sphere's volume, and Jl the characteristic of any rational and entire 

 function whatever. 



For this purpose we will substitute in the place of the co-ordinates 

 x', y , z' their values 



x = r cos &\ y = r sin & cos w'. z = r' sin ff sin -bt' ; 



and afterwards expand the function/(a;', y', s) by Laplace's simple method, 

 {Mec. Cel. Liv. iii. No. 16.). Thus, 



(7) /{x, y, z) =/<«>+/'" +/'<^> + &c +/'«; 



s being the degree of the function /{x, y', z'). 



It is likewise easy to perceive that any term /'■''' of this expansion 

 may be again developed thus, 



/'(•■) =/;(•■>/* +/'«/-= +^'<V^+^ + &c.; 



and as every coefficient of the last developement is of the form [/'", 

 (Mec. Cel. Liv. iii.), it is easy to see that the general term y'''V'+^' may 

 always be reduced to a rational and entire function of the original 

 co-ordinates x, y', »'. 



