dz ~ dx + i dy - ds (sin a - i cos a ) = ds e 



-.(tH 



whence 



ds = e 



(2 --«) 



■2a/7T 



dt 





from Equations [115a, b,c]. Hence, if b is the half-width of the lamina. 



dt 



b = lds = -2c /-I (- t + y^2 _ ^^2a/n _ 



The integral can be evaluated numerically, and the equation then fixes c in terms of b. 

 Also, on the right-hand half of the lamina 



?^ = 



c' 



= C1- 



■4a/77 



since /^^ is real and positive. Hence, the force on the entire lamina per unit of its length, in 

 the direction of the stream, is 



-1 



F^=jp{s-ina)f{l-q^)ds = -2cp{sinci) f {C^''''' - C;^''/") 



-1 p 



/r-5 2a/n r-T. la, 



(_^ + x/r-l) - (-t-^Jt - 1) 



dt 

 *3 



= - 2 cp(sir 



[1156 



The evaluation of the integrals is discussed in Article 78 of Lamb's Hydrodynamics; 

 the last integral there written is evaluated by complex integration on page 363 of Wilson's 

 Advanced Calculus. In the location first cited is given a table of values of the force, 



there called "pressure." 



(For nonsymmetrical cases; see Reference 2, Section 12.50 and Morton;^ ^^ see 

 Reference 1, Article 78; Reference 2, Section 12.52.) 



290 



