264 Methods for the Solution of Special Problems [CH. vni 



It may be noticed that the transformation 



gives the transformation appropriate to a line-charge at z = a. 



Also we notice that 



TJT , z a 

 W = log - 

 6 z + a 



gives a field equivalent to the superposition of the fields given by 

 W=\og(z-a) and W = - log (z + a). 



This transformation is accordingly that appropriate to two equal and oppo- 

 site line-charges along the parallel lines z = a and z = a. 



This last transformation gives 7 = when y = 0, so that it gives the 

 transformation for a line-charge in front of a parallel infinite plane. 



GENERAL METHODS. 

 I. Unicursal Curves. 



319. Suppose that the coordinates of a point on a conductor can be 

 expressed as real functions of a real parameter, which varies as the point 

 moves over the conductor, in such a way that the whole range of variation 

 of the parameter just corresponds to motion over the whole conductor. In 

 other words, suppose that the coordinates x, y can be expressed in the form 



and that all real values of p give points on the conductor, while, conversely, 

 all points on the conductor correspond to real values of p. 



Then the transformation 



z = f(W) + iF(W) ........................... (235) 



will give F=0 over the conductor. For on putting F=0 in equation (235) 

 we obtain 



*+iy-f(Uy+iF(U\ 



so that x=f(U), y = F(U), 



and by hypothesis the elimination of U will lead to the equation of the 

 conductor. 



320. For example, consider the parabola (referred to its focus as origin), 



y 2 - = 4a (x + a). 



We can write the coordinates of any point on this parabola in the form 



x 4- a am 2 , y = 2am, 



