82 PROPERTIES OF CIRCULAR CURRENTS. 



or that the thickness of the insulator is proportional to the radius of 

 the wire that is to say, that 8 is constant, and all the windings are in 

 contact. For if we replace the original wire of the coil by another 

 wire of m times smaller diameter, so as still to fill up the volume of 

 the channel, the values of R and of G become 



which gives, for the action at the centre, 



EG 



F 



R, + R' 



The factor m being here the only variable, the condition of 

 maximum is 



whence 



Hence the resistance of the coil must be equal to the resistance 

 of the external circuit. 



This condition is independent of the form of the coil. It would 

 still be true if the diameter of the wire, instead of being uniform, varied 

 from one winding to another, provided that the ratio of the diameter 

 of the two coils compared are the same at all points. 



If the thickness z of the insulator is constant instead of being 

 proportional to the diameter of the wire, we must take the general 

 expression 



R pl 



= ' 



as the windings are in contact, and the coiling is supposed to be 

 uniform, we have 



(19) u = 



If g is % a constant, we may put 



(20) t G =/, 



for it is clear that in all cases the total magnetic action for a given 



