Eliminating A and ^ , 



w = — CO (a + b) e ^, 



[106n] 



= — (a + bf e~^^ sin 2 77, i// = — (a + 6)^ e ^^ cos 2 tj. 

 4 4 



[106o,p] 



From Equations [82s, t,u] the components of velocity, in directions given by Equations 

 [82o, p], are 



c^ (a + b) T <:: CO (a + b) _„£ 



q^=—- e-^^ sin 2 Ti, g^ ^ - — e'^^ cos 2 r?, [106q,r] 



G = (sinh^ ^ + sin"^ 77) 



2 „^l/2 



cG 



— (cosh 2 ^ - cos 2 17) 



1/2 



[106s 



At large distances from the origin, ^ is large and cosh f = sinh f = eV2, nearly, so 

 that x^ + y^ = c^ e ^/4. Thus q vanishes in proportion to (a; + y )~^'^^. 



On the a;-axis, cos r] = -l,x = -c cosh ^ and u = - qt = 0, while t) has the opposite 

 sign to that of x and is, since e~'^^ = (cosh ^ + sinh C)~ > 



t; = ± fl =+— (a- 6)(a+ bf [y/x^ - c^ (\x\ + yjx^ - c^) ] 



[106t] 



On the y-axis, sin -q =- 1, y = - c sinh ^, and v -- qt= Q, while u has a sign opposite to 

 that of y and is 



= + ?^=+-^(«-^')(«+ ^)'[n/2/^ +c2(|y| + v/y^ + c^) ] 



[106u] 



The formulas represent the flow around an elliptical cylinder of cross-sectional 

 semiaxes a and 6, rotating about its axis at angular velocity di, in fluid that is at rest at 

 infinity. The origin lies on the axis of rotation, and the axes rotate with the cylinder. The 

 velocities as given refer, however, to fixed axes; it may be supposed that the axes are 

 momentarily stationary in their instantaneous positions. 



The streamlines for 1// = correspond to 77 = i 45° or - 135° and are easily seen to be 

 asymptotic to the radii y = - x. These streamlines separate those that cross the a;-axis at 

 their outer extremities from those that cross the y-axis. 



259 



