158,159] CONFORMAL REPRESENTATION. 311 



If "SP" be the conjugate function to V, we have for the charge 

 upon any conductor V=C between points A and B, 



e = I ads = -j I ;r ds 

 - J A 4f7rJ A dn e 



or since cos (nx) ds = dy, cos (ny) ds = dx, 



i ( B fiv , dv, \ 



e=.-~ \-5-dy -^- da) 

 4f7rJ A \dx ' d J 



so that the flux of force of any tube of force is measured by 1/4-n- 

 times the difference of the values at its two sides of the conjugate 

 function to the potential, as in 103. 



159. Examples. Eccentric Cylinders. Let us transform 

 by means of the function w = log z giving ( 45), 



1 i ) u = log r = log 



A pair of parallel planes u=V 1) u=V Zt transforms into a pair 

 of concentric circular cylinders r = r l} r=r z . To the potential 

 function V u we have the conjugate function 'W = v so that 

 for the charge of the cylinders we have between < = and 2ir, 



( 2 ) e = -: {^ 27r - ^0} = T'- . 2?r=4, 



4?r l 4?r 2 



and the capacity is 



(3) K = + V 6 _ v } = Z = 1/2 log ^ , as in 144. 



log n 



The use of the fractional linear function w = (2 + )/(& ), 

 gives us an important new result. Replacing i in 



(4) + 



by i, as a reference to the theory of the complex variable shows 

 is always possible, gives 



,. . x iy+a 



(5) u-iv= - , 



x l a 



