treatment of Section 111 requires only certain changes in order to fit the present case. The 

 point C is a stagnation point, from which the fluid flows away toward both sides. Hence the 

 parts CA and C^'of the lamina are represented by segments of the real axis of ^, as shown 

 in Figure 183, while the free streamlines follow the unit circle below the real axis as before. 

 The geometrical boundary on the ^-plane is thus the same as in Section 111, and the same 

 transformation from i^ to (5 can be used: 



dz „ 1 /2 



C = - — --t-(t^ -1) , 



aw 



t = 



1-7 



[113a, b] 



For the amplitudes, see Section 107. 



The flow on the <-plane is again along the real axis toward the origin, but in the pre- 

 sent case i/f has the same value on the two halves of the axis, which represent on the z-plane 

 parts of a single divided streamline. Hence there cannot be a source or sink at the origin; 

 the fluid must flow away along the imaginary axis. A simple flow of this type is that of 

 Section 38, whose complex potential may be written 



M = - — 

 ^2 



[113c] 



In this flow the axial streamlines continue to the origin; on the i;,"-plane, therefore, the corre- 

 sponding curves continue to the point I or t^ = - i. It follows that the free streamlines become 

 parallel at infinity. 



From Equation [113a], dz = - i^dw. After substituting from Equations [113a] and [113c] 

 integrating with the help of the substitution t = 1/u and choosing c and the constant of inte- 

 gration so as to make z = - 1/2 at A and /I' or ^ = + 1, it is found that 



n- + 4 



[113d] 



21 



77 + 4 



1 1 „ 1/2 1 ,1 



— + — (r - 1) + — sin ^ — 

 t 2t' 2 t 



[113e] 



Here, for real t, and |<| > 1, - 77/2 < sin ^ — £ n/2. 



The particle velocity at any point is obtainable from the equation 



dw 1 2 1/2 



- u + iv = = = t - (t -1) 



dz C 



[113f] 



282 



