Mr green, on THE LAWS OF THE EQUHJBRIUM OF FLUIDS. 57 

 we shall readily deduce 



A«= (1 - ef^e^^^ + ^e.%+ ""'^'""l^ e.% + &c., 



and ii,^ being thus given, B'-'^ and consequently the second line of the 

 expansion (25) are also given. 



From what has preceded, it is clear that when V is given equal to 

 any rational and entire function whatever of x and y, the value of 

 f{x', y') entering into the expression 



p={l-r'-^)-^.f{x',y'), 

 will immediately be determined by means of the most simple formulas. 



The preceding results being quite independent of the degree s of 

 the function f(x', y) will be equally applicable when s is infinite, or 

 wherever this function can be expanded in a series of the entire powers 

 of x, y', and the various products of these powers. 



We will now endeavour to determine the manner in which one fluid 

 will distribute itself on the circular conducting plane A when acted 

 upon by fluid distributed in any way in its own plane. 



For this purpose, let us in the first place conceive a quantity q of 

 fluid concentrated in a point P, where /• = « and 6 = 0, to act upon a 

 conducting plate whose radius is unity. Then the value of V due to this 

 fluid will evidently be 



g V' 



((^ — 9,ar cos Q + r^)~^ 



and consequently the equation of equilibrium analogous to the one marked 

 (20) Art. 10., will be 



(27) const. = ^ ^+ F; 



(«'-2«rcos e + r^)~ 



V being due to the fluid on the conducting plate only. 



If now we expand the value of V deduced from this equation, and 

 Vol. V. Part I. H 



