ACTION OF TWO CIRCULAR CURRENTS. 157 



When the potential of the circuit A, at a point (x, y) of the 

 surface S', can be advantageously expanded as a function of in- 

 creasing powers of y, by one of the expressions found above, the 

 value of f will be expressed by an analogous series which is also 

 convergent. 



If the circuit A is a circumference of radius a, we have, from 

 equation (29) of (736), 



27r l3~^ 2 ~r^ + ~, \5 3*7 *") 



V* 3 2.2* dx* 3(2. 4) 2 ^ 7 / 



Putting S = Ira 2 , we thus find : 

 by equations (32) of (736), as a function of u = *Jju 2 + x 2 , 



/"OP*/ '9 *> O 



6SS / 5 a 2 4^ - 3 J 



....), 



2.2 u 

 (62) 



3.5 # ' 4 5 6x* 



by equation (37) of (740), which is especially suited for small values 

 of x, 



6SS' f 5Jc 2 5.7 jc* 5.7.9* 



= -^ i--^ + -,- -7- Q + 



a b (_ 2 a" 2. 4 a^ 2.4.6 a" 



i a 

 (63) + ~2~^~a 



