ACTION OF TWO COILS. 159 



The principal term is then --- , so that the reciprocal action is 



00 



inversely as the distance of the two circles. This action arises prin- 

 cipally from the parallel portions of the two currents (480). 



789. In a more general manner, if the coefficient of mutual 

 induction of two parallel circular currents, having the same axis, 

 is expressed by the aid of the polynomials X n by equation (47), 

 we have, making use of equation 



and the second of the equations (27) 



3M_ 2 SST i 3a - 



~t X - ' U* \ ^ 2-2U* 





n+ i 2.4 ... 2^ 



_j 



790. ACTION OF Two COILS. For two coils with rectangular 

 channels, we might use the value of M, expressed in elliptic func- 

 tions either by Maxwell's formula (764) or by Lord Rayleigh's 

 expression (765) ; but the calculations are generally very compli- 

 cated, and it is better to have recourse to expansion in series. 



In order to replace the circuit A' of the preceding paragraph by a 

 coil with a rectangular channel, of dimensions a', zb ', zc', n' lt and n', 

 one of the expressions found is multiplied by ri^dxdy, and the double 

 integration made between the ordinary limits a c' and a' + <:', x b' 

 and x + b'. 



When the ratios - and are very small, we may calculate the 

 a a 



value F m (#i) of the mean action on the mean circle of radius a( 

 (727), and we have sensibly 



(68) 



791. ACTION ON A LONG COIL. In order to calculate the 

 action of a system of currents on a long coil, it will often be 

 simpler to estimate directly the action exerted on equivalent mag- 

 netic surfaces. We shall consider a particular case as an example. 



