182 Mr Love, On the Theory of Discontinuous [May 4, 



The point u = c will correspond to <£ = — oo , and u= cc to (f> = oo . 

 The figure in the w plane is 



-l 



where the lower line is the axis of real quantities ^ = 0, and 

 the upper is ^ = 7r. The strip between these lines can be con- 

 formably represented upon a half-plane u by means of the trans- 

 formation 



dw 1 



du u — c 

 and the boundary in the u plane is 



.(1), 



— co — 1 e a x 1 oo 



4. To find the boundary in the Q. plane. 



Let a r be the value of u that corresponds to any angle a. 

 of the polygon. The stream-line yjr = has for initial direction 

 6 = — \tt. At the first angle a x the velocity vanishes, at oo it 

 has a definite value. Thus log q' 1 increases from a certain value 

 to + oo as u moves from c to a x , and 6 remains constant and 

 equal to — \tt, and the corresponding part of the boundary in 

 the H plane is therefore a straight line 6 = — \ir from a certain 

 point for which the real part of XI is positive to oo . 



As u increases from a t to a 2 , log*/" 1 diminishes from oo to 

 a certain minimum and then increases to oo , 6 remains positive 

 and equal to — \ir + (tt — aj. Thus we have the two sides of 

 a second line starting from an unknown point (say u = c 1 ) and 

 running to oo in the O plane in the direction of O real. 



We shall obtain a line in the same way for each side of the 

 vessel until we come to the side that contains the hole. On 

 this line 6 = 0, and log q' 1 diminishes from oo to zero, its value 

 at the edge of the hole where the velocity is unity. 



The next part of the boundary consists of the line log q' 1 = 

 drawn from the origin downwards (because 6 becomes negative 

 on the jet), to some value corresponding to w=». This corre- 

 sponds to the free part of the stream-line aJt = 0. Passing through 

 this point the line can be continued downwards to = — ir the 

 value at the other edge of the hole corresponding to u = — 1, 

 this part of the line corresponds to the free part of the stream- 

 line ^r = 7r. 



Proceeding from this point we have to trace the line 6 = — ir 

 to oo , then both sides of the other lines 6 = const, corresponding 



