104 MOTION OF A LIQUID IN TWO DIMENSIONS. [CHAP. IV 



z, f, w, and of two subsidiary variables z 1} z. 2 *. A reference to 

 the diagram on p. 77 will shew that the relation 



*i = a?! + tyi = log? (1) 



transforms the boundaries in the plane of f into the axis of x l 

 from (oo , 0) to the origin, the axis of y x from the origin to (0, - 2?r), 

 and the line y x = - 2?r from (0, - 2?r) to (oo , - 2?r), respectively. 

 If we now put 



z z = x 2 + iy^. = cosh ^ (2), 



these boundaries become the portions of the axis of x 2 for which 

 # 2 > 1, 1 > # 2 > 1, and x 2 < 1, respectively ; see Art. 66, 1. It 

 remains to transform the figure so that the positive and negative 

 portions of the axis of x 2 shall correspond respectively to the two 

 bounding stream-lines, and that the point z 2 = (marked / in the 

 figure) shall correspond to w = oo . All these conditions are satis- 

 fied by the assumption 



w = \ogz 2 (3), 



(see Art. 62), provided the two bounding stream-lines be taken 

 to be ^ = 0, ty = 7r respectively. In other words the final 

 breadth of the stream (where q - 1) is taken to be equal to TT. 

 This is equivalent to imposing a further relation between the units 

 of length and time, in addition to that already adopted in Art. 74, 

 so that these units are now, in any given case, determinate. An 

 arbitrary constant might be added to (3) ; the equation, as it 

 stands, makes the edge A of the canal correspond to w = 0. 



Eliminating z lt z. 2) we get f* 4- ~* = 2e w , whence, finally, 



f = -l + 2#' + 2^(^-1)* (4). 



The free portion of the stream-line >/r=0 is that for which f is complex 

 and therefore < < 0. To trace its form we remark that along it we 

 have - dcfr/ds q=l, and therefore (f> = s, the arc being measured 

 from the edge of the canal. Also f = dx/ds + idyjds. Hence 



dx/ds = - 1 + 2e- 2 , dyjds = - 2e~ s (1 - e~- s )? (5), 



or, integrating, 



x = 1 - s - e~-\ y = - \nr + e~ s (1 - e~- s )^ + sin- 1 e~ s . .. (6), 

 the constants of integration being so chosen as to make the origin 

 of (x, y) coincide with the point A of the first figure. For s = oo , 



* The heavy lines represent rigid boundaries, and the fine continuous lines the 

 free surfaces. Corresponding points in the various figures are indicated by the 

 same letters. 



