THE EQUILIBRIUM OF FLUIDS. 179 



and consequently, 



A* = K (i) + <V 2 + 2 <V 4 + &c.} r fi 



fn-l 

 m( 



2sm 



, 2 , 4 



7T 



the particular value ^ (0) being one half only of what would 

 result from making t'=0 in this general formula. 



But e ( S = evidently gives jE' (i) = 0, and therefore the expan- 

 sion of/(o;', y] before given becomes 



f (V, y') = A (0) + A {1] cos ^ + A cos 2(9' + ^ (3) cos 3(9' + &c. 



/TO 1 \ 



2sm^-7r) ^ 



\ 7 . / 2 i\ 2 /_2 '2\-l 



+ ^ cos 2^' + &c.i 

 or by summing the series included between the braces, 



_ gar' cos (9'+ r' 2 

 n-l 



R being the distance between P, the point in which the quan- 

 tity of fluid q is concentrated, and that to which the density p is 

 supposed to belong. 



122 



