If the fluid rotated as a rigid body, T^ would be rrpab (a^ + 6^) aj^/8. 

 For the elliptic shell; see Reference 1, Articles 71, 72. A shell whose cross section 

 is an elliptic quadrant was studied by Sen,^^^ one composed of arcs and lines, by Ghose.^^^ 

 (For notation and method; see Section 34.) 



106. ROTATION OF ELLIPTIC CYLINDER OR LAMINA. 



A transformation that yields a quadratic stream function with the fluid at rest at infinity 

 is that of Section 84 with F = or, after a slight change of notation, 



w = i A e~-^^, z = c cosh Ci [106a, b] 



where s = x + iy, t^ = ^ + irj and the elliptic coordinates f and 7y on the 3-plane are described 

 in Section 82. Here ^ > and 77 is multiple valued like an angle. The ellipse for a given 

 ^ has semiaxes a', 6 'such that 



a' = c cosh ^, b' = c sinh ^, a'+ b' = ce'=, c = \i a' - b' . [106c,d,e,f] 



a;= ccosh ^cos 77 = c'cos r], y = c s'lrih ^ s\n r] = b' s\n r). [106g, h] 



From w - <p + i\]j ■' 



= ^e~2^sin 2r?, i// = ^ e"^^ cos 2 rj. [106i,j] 



From Equation [106g,h] and a hyperbolic formula in Section 32, x + y - {c /2) (cosh 2^ + 

 cos 277); substitution for x + y in Equation [99c] gives 



A e~^^ cos 2 Ti - — CO (cosh 2 f + cos 2 7/) = C: [106k] 



This equation is satisfied for any value of 77 provided .f = f = constant and 



C = -— 6jcosh2^„, Ae °= — co. 

 4 4 



Thus for this particular value of C the curve defined by Equation [106k] degenerates into the 

 ellipse defined by ,f = ^ . Its semiaxes a, b are such that, by Equation [106e,f], 



a+ b = ce °, c = x/a^ - b^. ' , [1061, m] 



258 



