If convention (a) is adopted for the values of ^ and t], with f > and -TT<r]<n, the ring 

 between two ellipses, or the interior of any ellipse, is mapped onto a rectangle with sides at 

 jj = i 77 and with ends at the proper values of ^. The ring may be supposed to be cut along 

 the negative ar-axis and straightened out. Using convention (b), the area between two hyper- 

 bolas is mapped onto an infinite strip parallel to the ^-axis. 



(See Reference 1, Article 71; Reference 1, Section 6.32.) 



83. FLOW PAST AN ELLIPTIC CYLINDER 



By means of the transformation discussed in Section 81, or s = 2"+ c /z", the flow 

 around a stationary circular cylinder can be transformed into that around an elliptic cylinder. 

 The appropriate complex potential w can be obtained by replacing s - 2^ by 2 "in Equation 

 [69j] and then substituting for 2" in terms of 2. The result will be written down in a modi- 

 fied notation and verified. It is 



w =V{a+b) cosh iC-^Q-ia) + ~ {C- Q, [83a] 



Zn 



2 = X + iy = c cosh ^, <^ = ^ + i 77, ^ ^ 0, [83b, c] 



where a and b are positive real constants, a and U are real constants, 

 and 



/-^ 5- f - a + b a + b 



c a - b 



Here ^, tj are the elliptic coordinates of Section 82. 

 Writing cosh t^ = (e'= + e~^)/2, it is found that 



2 / c 2z 



/ i;2 / 1 1 / c^ \ 22 



,-<. 



2c f 1 



c^ 

 2 2 \l ~ 2 I 22 



[83d, e] 



sinh ^0 = — , cosh ^0 = — ^ [83f,g] 



