268 Methods for the Solution of Special Problems [OH. vm 



coinciding. Let us take A to be W = oo , B or C to be W = 0, D to be 

 W = 1 and E to be W = + oc . The angles of the polygon are zero at (BC) 

 and 2-7T at D. Thus the transformation is 



dz _ r W-l 

 dW~ ~W~' 

 giving upon integration 



2=C{W-\og W+D] (239), 



where C, D are constants of integration which may be obtained from the 



pr >- - - E > W = +) 



w="+r" -<,-- 



lw-o 



A.--..V- .... ...-B- 



W=-oo 



FIG. 90. 



condition that the two planes are to be, say, y = and y = h. From these 

 conditions we obtain C = - , D = ITT, so that the transformation is 



7T 



* = -{F-log W + iir] (240). 



On replacing z, W by z, W, the transformation assumes the simpler form 



k 



z 



= -(W + log W) (241). 



7T 



III. Successive Transformations. 



324. If =<t>(z), TF=/(f) are any two transformations, then by elimi- 

 nation of a relation 



W=F(z) .............................. (242) 



is obtained, which may be regarded as a new transformation. 



We may regard the relation f = < (z) as expressing a transformation from 

 the ^-plane into a f-plane, while the second relation W=/() expresses a 

 further transformation from the f-plane into a W -plane. Thus the final 

 transformation (242) may be regarded as the result of two successive trans- 

 formations. 



Two uses of successive transformations are of particular importance. 



325. Conductor influenced by line-charge. The transformation 



gives, as we have seen ( 318) the solution when a line-charge is placed at 

 f = a in front of the plane represented by the real axis of Let the further 



