458 THE ELECTROMAGNETIC FIELD. [PT. III. CH. XI. 



considering the element in which it occurs as straight the integral 



fds 



J 



becomes logarithmically infinite. We may see the reason for the 

 self-inductance becoming infinite in another way by considering 

 Biot and Savart's Law, for as we approach a linear conductor 

 the force is inversely proportional to the distance from the 

 conductor. The flux therefore increases like the logarithm of 

 this distance, and is not finite when we approach the linear 

 conductor indefinitely. We may avoid this difficulty by con- 

 sidering conductors of finite cross-section, for in that case the 

 corresponding element of the integral, in which the integrand 



becomes infinite, I is not infinite, as was proved for an ordinary 

 potential, 76. 



We shall now consider currents flowing in three-dimensional 

 conductors in the form of cylinders of infinite length whose 

 generators are all parallel. We might treat the problem by the 

 application of the law of Biot and Savart to each infinitesimal 

 tube of flow, but we shall prefer to make use of the general 

 equations 222 (2), and 228 (2). It is evident that the lines 

 of force are in planes perpendicular to the conducting cylinders, 

 which we shall take for the XF-plane, so that N0 and the field 

 is independent of the coordinate z. The problem is accordingly 

 a two-dimensional problem, and all the quantities concerned are 

 independent of z. Since u = v = we have F= G = so that our 

 equations are 



dM dL 



-= -- ~ , 

 das dy 



from which results, if ^ is constant, 



&H 



(4) -47^ = ^ 



But this is Poisson's equation for the logarithmic potential, 91 (10), 



