321-323] Conjugate Functions 267 



Now arg f jij measures the inclination of the axis F=0 to the edge of 



the polygon at any point, so that if the polygon is to be rectilinear, this 

 must remain constant as we pass along any edge. It follows that there must 

 be no change in arg F as we pass along any side of the polygon. 



This condition can be satisfied by supposing F to be a pure numerical 

 constant. Taking it to be real, we have, from equation (236), 



W2 )+ ......... (237). 



On passing through the angular point at which W = u z , the quantities 

 W u lt Wu 3) etc. remain of the same sign, while the single quantity 

 Wu 2 changes sign. Thus arg ( W u 2 ) increases by TT, whence, by equa- 



/ / \ 



tion (237), ar gTw increases by \ 2 7r. 



The axis F=0 does not turn in the TF-plane on passing through the 



value W=u 2 , while ar g(;rTfr) measures the inclination of the element of 



\ct w j 



the polygon in the ^-plane to the corresponding element of the axis V in 

 the TF-plane. 



Hence, on passing through the value W=u 2 , the perimeter of the 

 polygon in the 2-plane must turn through an angle equal to the increase in 



arg -TT , namely X 2 7r, the direction of turning being from Ox to Oy. Thus 



X!TT, X 2 7r, must be the exterior angles of the polygon, these being positive 

 when the polygon is convex to the axis Ox. Or, if a l} 2 , ... are the interior 

 angles, reckoned positive when the polygon is concave to the axis of x, we 

 must have 



\ 1 = ^-1, etc. 



7T 



Thus the transformation required for a polygon having internal angles 

 !, 2 , ... is 



7 a, cuj 

 a /-y/Trr \ * / TI7 \ 1 /000\ 



- =0(W u^ n (W - u 2 )" (238), 



d W 



where u lt u 2 , ... are real quantities, which give the values of U at the angular 

 points. 



323. As an illustration of the use of Schwarz's transformation, suppose 

 the conducting system to consist of a semi-infinite plane placed parallel to an 

 infinite plane. 



In fig. 90, let the conductor be supposed to be a polygon ABODE, which 

 is described by following the dotted line in the direction of the arrows. The 

 points A, B, 0, E are all supposed to be at infinity, the points B and 



