MAXWELL'S METHOD. 149 



779. Let us now suppose that the planes of the two circles are 

 at a very small distance x from each other, their radii being a and 

 a+y. The flow of force from the second current, and which 

 traverses the first of radius a, has no longer the same value as 

 before ; but if x is very small, we shall have a very close approxi- 

 mation if we replace the shortest distance c of the two arcs by its 

 new value r = ^x 2 +y 2 . We might then represent the coefficient M 

 by an expression of the form 



(53) 



the coefficients A and B being functions of x and y, whose approxi- 

 mate values are respectively a and - 2#, and which we have now to 

 determine. 



Let us observe, in the first case, that these two functions ought not 

 to change their value, when x is changed to -x; hence their expan- 

 sion in series only comprise uneven powers of x. We may put then 



V 2 X 2 V 3 VX 2 



B= - 2a + E iy + 3./- + E',- + ^ + y/ 



It follows, moreover, from the fundamental property of the 

 coefficient M (341), that its value does not alter when the two 

 circles are interchanged that is to say, when a is replaced by a +y, 

 and y by -y. 



On the other hand, the function M should satisfy the general 

 equation 



<) 2 M ^M i 3M 

 ty 2 Djc 2 a+y ty 



These two conditions furnish the number of equations necessary 

 for determining the coefficients ; we find thus 



Iy y + 3^ y + 37^ 



d I H ---- 1 -- : ---- - -- f- . . . , 



L 2 a i6a 2 32# 3 J 



2 a 



