316 ELECTROSTATICS. [PT. II. CH. VII. 



so that 



_lj / ^T7^ 7 / 0?+^ ) 



(4) 



<fy = 2 



1 ,_ , ( dX 2 



(5) ds 2 : 



C^yLt 2 



For a conformal relation we must have 



Consequently if we put 



du = 

 (6) 



the functions u, v will give us a conformal relation in which the 

 straight lines u = const., v = const, in the ^JT^-plane correspond to 

 the ellipses and hyperbolas X, /u,, in the JTF-plane. 



Integrating the differential equations (6) 



u = log {Va 2 + X + V& 2 + X}, 

 (7) 1 



Taking the antilogarithm and its reciprocal, of the first equation, 



_ tt _ 



Solving these for +X and V 



J (e w + (a 2 - 6 



ftM^ = J- [e u - (a 2 - 

 From the integral for v we get 



1 + cos 2v a 2 



" * 



1 - COS 2 



