y 4>. 



Figure 93 - Confocal ellipses and 

 hyperbolas; the flow new on the 

 2-plane for 2 = c cosh (0 + iijj). 



In the hydrodynamical applications, the double-valued character of dw/dz makes it 

 necessary to insert boundaries so as not only to exclude the singular points but also to pre- 

 vent the fluid from circulating around just one of them. When this has been done, a choice 

 can be made for the values of and such that they vary continuously throughout the fluid 

 and such that their derivatives represent a single-valued velocity. The singular points can- 

 not be interpreted as representing a source and a sink, either simple or compound; mathemat- 

 ically, they are not poles but branch points. 



Flow between Hyperbolic Cylinders 



If <^ is taken as the potential, boundaries may be inserted along any two of the i/( hy- 

 perbolas. The formulas then represent the flow between two hyperbolic cylinders. Conven- 

 ient ranges for the variables are: 



< ./. < 



- 00 < (;6 < , 



Since dw./dz = - u + iv, from [61a— f] 

 1 



O^lP ^7T. 



u (sinh (f> cos i/»), v ■= cosh <f) sin 0. 



cG cG 



[61k,l] 



On a boundary defined by = "A,, from [61c, d] and [61f], 



q =_ (sin-: 0j + 



sin^ 



[61m] 



On the y-axis, = n/2 and q = 1/ ^c"^ + y^ ■ In the plane of the opening (j> = and the veloc- 

 ity is from Equations [61d,f,b], 



138 



