THE EQUILIBRIUM OF FLUIDS. 177 



we shall readily deduce 



and b being thus given, 'B w 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 off(x, y'} entering into the expression 



will immediately be determined by means of the most simple 

 formulae. 



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 quan- 

 tity q of fluid concentrated in a point P, where r = a 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 



V' 



(a 2 2ar cos 6 + r a ) * 



and consequently the equation of equilibrium analogous to the 

 one marked (20) Art. -10, will be 



const. = - 2 ^ + F (27) ; 



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



12 



