84 PROPERTIES OF CIRCULAR CURRENTS. 



In the second case the equations 



give 





The former equation may be written 



.To 



The approximate value of ^'represents the radius of the wire, if 

 the thickness z of the insulator were equal to zero. This equation 

 would be solved by successive approximation, by giving to y under 

 the root an arbitrary value, y Q for instance. 



733. BEST FORM OF THE CHANNEL. We have hitherto sup- 

 posed that the section of the channel is rectangular, but it is readily 

 seen that this shape is not the most advantageous. As the action of 

 a winding, other things being equal, is inversely as its radius, it is 

 manifestly advantageous to multiply the number of those which 

 correspond to the smallest radii that is, those whose effect pre- 

 dominates. 



We are not able to diminish indefinitely the radius of the first 

 windings ; in the first place there must be a core to support the wires, 

 and in coils which are to be used in galvanometers we must more- 

 over arrange about the axis the place necessary for the movement of 

 the magnet. 



Reasoning similar to that given above (730) shows that the most 

 advantageous form for the contour of the channel is that in which all 

 the windings on the surface have the same specific action on the 

 magnet, for otherwise, it would be better to displace some of the 



