Field of Circular Currents. 381 



of practical importance, we shall confine our attention to 

 the case o£ small values of the coordinates £, rj referred 

 to the mean point as origin, so that 



£ = *;■ v = y-f; (9) 



where x> y are referred to axes with the origin at the centre 

 of the first circle. The magnetic force at f, rj is the sum of 

 the forces arising from the first and the second circle. 

 Evidently the expressions (8) can be used in calculating 

 the component forces; as to the modulus of fhe elliptic 

 integrals K and E, we notice that 



*i 2 =■ ~^ whera n 2 = + 1 )'-' + (j + vj ■ • (9') 



for the first coil, and for the second 



k ^=~l w tere r./ = (« + $)* +(^-vf, ■ ■ (9") 



The expressions for X and Y due to the circle carrying 

 unit current are 



2 



' a (l + V ) ($ + r >) 



X = - 3 - ~ ■ ' E - ^-^ (K - E) 



>■ (10) 



T = 2 f 2a(a-f) 



L {- ,? V E + (K-E)). 



This may be conveniently written under the form 



~ 2^«p U' 2 * J <Mi ' 



2y/rtlk'*\S ) dky K 



