On the cylinder ^ = ^q ^^'^^ ^'"°'^ Equations [1061, m, o, p], 



— (a^ - b^) sin 2 7?, i// = — {a? - b^) cos 2 r?. 



[106v,w] 



The flow net is the same for all confocal cylinders; the values of (f) and i// at corre- 

 sponding points are proportional to {a + b) . For, the values of ^ and ?/ at a given point in 

 space depend only on the locations of the foci, and in particular upon c. 



If 6 -♦ 0, the ellipsoid becomes a lamina of width 2c = 2a, rotating about its median 

 axis. On its surface ^=0, y=0, a; = a cos rj and 



w = + « 



CO '2.x'- - a"- 



2 Al 2 

 \ia - X 



qt = oji 



[106x,y] 



Here the upper sign refers to the face toward positive y, the lower sign to the other face, and 

 &j is positive as usual for rotation from x toward y. Thus m = at a; = i aj\j2. 



Streamlines for equally spaced i// are shown around the elliptical cylinder ab in 

 Figure 173. The curves inside the ellipse are to be disregarded in this connection. The 

 fluid is at rest relative to the cylinder at points o and i. The curve a '6 'represents a con- 

 focal cylinder that would give rise to the same streamlines at external points. The flow net 

 for a lamina is illustrated in Figure 174. The streamlines differ in appearance in the two 

 cases only because different spacings of i/i were chosen. 



Figure 174 — Flow net produced by a plane lamina rotating about its median line. 

 (Copied from Reference 10.) 



260 



