These equations can easily be solved, either for (ft and i/i, or for x and y. The signs of (f> and 

 xp may be inferred either from physical considerations or from a detailed study of Equation 

 [85b]. 



On the y-axis; i// = 0, = f^ \Ja?- + y-^; u ^ C, v = - Uy yja^ + y^. 



On the a;-axis where |a;|< a: li = 0, = i C/ \/ a - x , also v = Q, u =- Vx/\ja - x , 

 where the upper sign refers to the front face and the lower sign to the rear face. Thus 

 q= \u\ = |f/| at \x\ = a/^J^. 



On the a!-axis where \x\> a: cfi = Q, ip = + V yjx^ - a^, where the sign + is opposite to 

 the sign of x. 



In this case, where the y-axis is a streamline and may represent an infinite rigid sur- 

 face, half of the flow may represent a stream flowing past a straight boundary carrying a 

 straight rigid stiffener of width a and negligible thickness, perpendicular both to the boundary 

 and to the stream. 



Two cases for F = 0, with a = 45 deg and a = 90 deg, respectively, are shown in 

 Figures 136 and 137. In the first figure, the a;-axis is rotated into a convenient direction. 

 The points at which u = - V are shown by short marks. 



Figure 136 — Flow past a plane lamina 

 -in a direction inclined at 45° 

 to the lamina. 



Rotation of the stream and lamina through -90 deg, so that U (if positive) is directed 

 toward negative x and the lamina lies along the y-axis, gives, perhaps by using Equation 

 [25k] with a = - 90 deg, A; = 1 and A = 0, w = V {z'^ + a})'^ and 



if,"- - if/ = U^ {x"- + a^ - y^), (fjifj = U^ xy. 



In steady motion, the lift on the lamina is in any case pVU, as on the cylinder, and the 

 torque on it, from Equation [83t], is 



A' = n p a?' V^ sin 2 a 



2 ^ 



[85e] 



where a is the angle between its direction of motion and the plane of its faces. The torque 

 tends to set the lamina at right angles to the stream. 



205 



