308 ELECTROSTATICS. [PT. II. CH. VII. 



electrification of very long cylindrical conductors at a distance 

 from the ends, the potential is independent of the coordinate whose 

 axis is parallel to the given direction, and Laplace's equation 

 reduces to two terms. 



3 2 F 



(I) -5-i + TT 



2 z 



Such problems are called two-dimensional, or uniplanar, since 

 the distribution is the same in all planes parallel to a single one. 

 For the solution of two-dimensional problems the method of the 

 conformal representation by means of functions of a complex 

 variable gives a powerful method. Let V be a function of x, y, 

 holomorphic in a certain region of the plane. Let us make a 

 conformal transformation of the plane by means of two conjugate 

 functions u, v. V then becomes a function of u t v in the trans- 

 formed plane. That is, u, v are to be taken as rectangular 

 coordinates in the transformed plane. Then to every point P in 

 the original plane, having the coordinates x, y, there correspond 

 definite values of the functions u, v, and the point P' in the 

 transformed plane having the coordinates u, v corresponds to the 

 point P. To corresponding values points P and P' belong the 

 same values of V. Thus level lines of V in the JTF-plane corre- 

 spond to level lines of different form in the Z7F-plane. We shall 

 first show that a certain function of the derivatives of V remains 

 unchanged if we replace in it one set of rectangular coordinates x, y 

 by the other u, v. 



Considering F as a function of u, v 



dV_dVdu dVdv dV^dVdu dV dv 

 fa ~~ "du dx + dv dx ' dy ~~ du dy + dv 'by ' 



__uV d*Vdudv 

 da? ~ ~du? (da;) dvdu dx dx du 



dudv dx dx 9tf 2 dx 



dvdu dy dy du dy* 



du dv 9 2 F / 



