Hence substituting also w/U for g', a complex potential w defined by 



mw - cU I 2 - c 



mu, + cV \ z + c 



[89a] 



will represent flow past the given cylinder, with a uniform velocity U toward negative x at 

 infinity. Then, in terms of ^, X, and fi as defined by Equations [88d,e,f, k] or [881, m,n], 



cU c 

 — coth 



m 2m 



cU , k cU , n 



sinh — , i// = sin — 



2mG m 2mG m 



G = sinh — + sin = — cosh — - cos — 1 



2m 2m 2 \ m m j 



dw dw I dz^ 

 dz d^ \ d^ I 



sinh — / sinh — 



„2 V 2 2m 



[89b] 



[89c, d] 



[89e] 



[89f] 



w 



X 



sinh — + sin — = 

 2n \ 2 



\U\ 



(cosh A - cos /x). [89g] 



m^G \ ■^ ^ I 2m^G 



On the a!-axis where la;] > c and ii = 6^ - 0^- 0, ^ = X, s = ar, g' = lw| and 



U cosh X - 1 X + c 



M = — - — — , A = In . 



^2 cosh (X/ot) -1 X - c 



On the y-axis outside of the cylinder, X = 0, C, = ~ ip., q= \u\ and 



V 1 - cos 



fi _i y 

 ) ^ = 2 cot — 



^2 1 - cos {yi/m) 



On the cylinder itself ^ = - mn and, from Equation [89g], 



• |f| cosh X - cos {mn) 



2 cosh (X/ot) + 1 



[89h,i] 



[89j,k] 



[891] 



214 



