323-328] Conjugate Functions 269 



transformation =/(,&) transform the real axis of f into a surface S, and the 

 point f = a into the point s = , so that a =f(z ). Then the transformation 



gives the solution when a line-charge is placed at s = s in the presence of 

 the surface S. In this transformation it must be remembered that U, and 

 not V, is the potential (cf. 318). 



326. Conductors at different potentials. Let us suppose that the trans- 

 formation =<(/) transforms a conductor into the real axis of f. The 

 further transformation TF=(7-f Dlogf ( 318) will give the solution when 

 the two parts of this plane on different sides of the origin are raised to 

 different potentials G and C + nrD. 



Thus the transformation obtained by elimination of f, namely 



F = (7 + D log ((*), 



will transform two parts of the same conductor into two parallel planes, and 

 so will give the solution of a problem in which two parts of the same con- 

 ductor are raised to different potentials. 



EXAMPLES OF THE USE OF CONJUGATE FUNCTIONS. 



327. Two examples of practical importance will now be given to illus- 

 trate the use of the methods of conjugate functions. 



Example I. Parallel Plate Condenser. 



328. The transformation 



* = -(- log f-MV), 



has been found to transform the two plates in fig. 90 into the positive and 

 negative parts of the real axis of f. The further transformation W=log f 

 gives the solution when these two parts of the real axis of f are at potentials 

 and TT respectively ( 326). 



Thus the transformation obtained by the elimination of f, namely 



7T 



(243), 



will transform the two planes of fig. 90 one infinite and one semi-infinite 

 into two infinite parallel planes. Thus equation (243) gives the transfor- 

 mation suitable to the case of a semi-infinite plane at distance h from a 

 parallel infinite plane, the potential difference being TT. 



By the principle of images it is obvious that the distribution on the 

 upper plate is the same as it would be if the lower plate were a semi-infinite 



