I sin 



4 + 77- sina 



sine sin 0" {1 - cos a cos 6") 1 - cos {a+ 0") 

 - In 



Note that 



(cos 6" - cos a ) 



[114k] 



1 + cos (6 +a) cos 6 + cosi 



cos (( 



-a) I 



The free streamlines for a = 55° are plotted to scale in Figure 186. 



The force F^ on the lamina per unit of its length can be found, as in Equations [113j,k], 

 using dx = dz from Equation [114b], c from Equation [114c], and evaluating /"-^ and /°° 



-<xi 1 



separately. When the velocity at infinity and on the free streamlines is V instead of unity, 



TT plU sin a 



F. = — ^ . [1141] 



4 + 77 Sin a 



The force acts perpendicularly on the lamina. 



The center of pressure, at which F. may be supposed to act, is displaced from the 

 center to the position x = x where 



3 I cos a 



[114m] 



4 4 + 77 sin a 



The deduction of x is given in Section 77 of Lamb's Hydrodynamics, where Equation [15] 

 is equivalent to Equation [114b] here; a table of values of F^, x and x for various values of 

 a is also given. 



(For notation and method; see Section 34; Reference 1, Article 77; Reference 2, 

 Section 12.3). 



115. INFINITE STREAM ON A V-SHAPED LAMINA. 



Let a steady stream impinge symmetrically on a lamina whose section consists of 

 two equal straight lines meeting at an angle 2 a as measured on the downstream side. 



The solution in Section 113 can be modified to suit this problem by interposing a 

 transformation 



^^ g-i(l/2 7T -a) ^2a/77^ [115a] 



Since only values of i^ and t^^ on or below the real axis are involved, let all amplitudes 6 



be taken in the range - n ^ < n. Then the positive real axis of ^^, on which = 0, becomes 



on the ^-plane a radius inclined downward at the angle - I — 77 - a j ; whereas the negative 



288 



