Flow of Liquid past a Wedge. 805 



case in which the left-hand free stream-line is caught for 

 a portion of its length by a second plane set anyhow with 

 respect to the first. The four quantities to be disposed of, 

 a, d, f, g, are determined by the angle 7 between the planes, 

 the angle of impact a. on the first plane and the two co- 

 ordinates which fix H, the outer edge of the second plane, 

 with respect to DB. If arbitrary values are assigned to dg, 

 /is determined by 7 and then a by a by the relations : 



1 {u— /)(1— u 2 )~i(u— d)~i(u— g)~* du — 



(V 



*J a 



(5) 



f)(l-u 2 )~*(u-d)-i(u-g)-idu = a. . . (6) 



In other words, we can ensure that the planes shall make 

 an assigned angle with each other, and that the stream shall 

 meet them in an assigned direction, but the breadth and 

 position of the second plane relative to the first can only be 

 found by trial. Thus, to obtain solutions for a wedge, we 

 could keep, say d, fixed, and work out the configuration of 

 the ^r-plane for different values of g, until we found the 

 value which made the second plane pass through the edge 

 of the first, repeating the process with other d's until a range 

 of wedge-cases, with varying-breadth ratio, could be ob- 

 tained. This lengthy procedure is considerably shortened 

 when the wedge is right-angled, for then, taking the origin 

 of z at D, the required condition is the vanishing of x at the 

 point G. 



Reverting to the general case, let Q' be the velocity over 

 the free stream-line bounding the pocket DFG. Then 



log ,y = HG on the Q, diagram 



= 1 ' (u-f)tu 2 -l)-*(u-d)-*(u-g)-Mu.. . (7) 



"9 



Along DFG the real part of O is log /^ , its imaginary 

 part is ifo say. where * 



% = 7r _ ^\u-f)(u*-l)-*{u-d)-K9-u)-Uu, . (8) 



Jd 



X is minimum for u=f corresponding to the inflexion F and 

 then rises to 7 at u=g. 



