«, = - c csch — i , /?. 



1 ^ 2 



c csch — ^ 



c coth — - , x^ = - c coth — = 

 4 ' 2 ^ 



[42h,i] 

 [42j,k] 

 [42l,m] 



[42n] 



x^ =yc2 +/?2, ^2 = - \/c2 + /e 2 



Then D = x^ - x , and, after eliminating all radicals by squaring twice, 

 4 c2 ^2 = [^£)2 _ (/^^ + fi^f\ \n2 _(/^^ -/?2) J 



This formula fixes c when D, /?p and ^2 ^""S given, and the values of x^ and x„ then locate 

 the origin of coordinates. 



The singular points (i c,0) lie inside the cylinders. Hence a valid representation is 

 obtained of purely circulatory flow between and around two parallel cylinders. The difference 

 lA, -'/'i represents the volume of fluid that passes between them per second, per unit of their 

 length. 



B. Line Vortex Outside a Circular Cylinder 



If cylinder number 1 is omitted and the formulas are continued down to the point (c,0), 

 the ideal flow is represented due to a line vortex outside a rigid cylindrical boundary of 

 circular cross-section. The vortex, located at (c,0), is at a distance h from the axis of the 

 cylinder, which is located at (rs^jO), where h = c - a;.. If R is written in place of /?„ for the 

 radius of the cylinder and ih as before for Uie valye of \b on it, using [42i] and [42m] 

 (Figure 58), 



"^9 ro 9 A2_^2 



ft = c csch -^ , h = c+\lc^+R^, c = 



A 2h 



h^ + R^ 

 X = ; — [42o,p,q,r] 



^ 2A 



The last equations serve to fix c and x when A and R are given, and A - r/2ff, where V is the 

 assumed circulation around the vortex. The circulation around a curve encircling the cylinder 

 once in a positive direction is —V. 



Figure 58 — Illustration of a line vortex of circulational strength T near a 

 circular cylinder with circulation F'- F around it; see Section 42B. 



93 



