Substituting for if/ in Equation [99c], 



This can be writien 



A{x^ -y')- - {x^ + y^) = C. 



2 2 



^ y , 2 

 + =1, a = 



^2 z2 



a 



2C 



2C 



-2A 



+ 2 A 



[105a, b,c] 



These equations represent a real ellipse with semiaxes a and b provided C < 0, |^1 < (y/2. 

 Substituting for A in terms of oj, a , and b , 



1 ,2 a^ -b"- 

 - u>k xy, 4' = — (^k (x - y ), k = 



2 ^ .2,2 



[105d,e,f] 



These formulas represent the flow inside a cylindrical shell whose cross section is an 

 ellipse of semiaxes a and 6, when it is rotating at angular velocity co about its axis, on which 

 the origin of coordinates has been taken. The coordinate axes must be allowed to rotate with 

 the shell, but all velocities are referred to a fixed frame of reference. The streamlines are 

 rectangular hyperbolas. They are illustrated in the interior of the heavy ellipse drawn in 

 Figure 173, which may be taken to represent half of the symmetrical shell. 



Figure 173 - The heavy ellipse may represent half of the contour of an elliptic cylinder that 

 is rotating about its axis and producing streamlines represented by the solid curves, or 

 half of an elliptic shell rotating similarly and producing streamlines in the contained 

 fluid represented by the broken curves. See Section 106. 

 (Copied from Reference 174.) 



256 



