Equation [137b"] with sin a> omitted. The coefficient 2 e^h^/{l - e^) becomes 2, as for a 

 cylinder, as e h. 1, and 3/2, as for a sphere, as e ^ 0. 



Case 5. Rotation of a Prolate Spheroid about an Equatorial Axis. Let the spheroid 

 represented by Equation [137u] rotate at angular velocity fi about the y-axis, in fluid at rest 

 at infinity. 



At any instant, the velocity potential can be expressed in terms of ellipsoidal co- 

 ordinates whose axes coincide with those of the ellipsoid, and the component of the fluid 

 velocity that is normal to the surface of the ellipsoid at any point will then be given by Equa 

 tion [137o]. The direction cosines of the normal to the surface, on the other hand, are, from 

 Equations [136b, c, d]. 



8C dx 



8^dy_ 

 8s ^ dC 



SC dz 



Ss 



^ K 



Substitution of these values for q ^ I, m, n, and of w = i 



gives as the boundary condition at the surface of the rotating ellipsoid 



0, CO = n, in Equation [135d] 



dcj) I dx dz 



— = -0 3 — -X — 



[137c 



From Equations [137a, c] 



^ = v,-^ = kae - 1)- '/' (1 - ^')'^' sin 



The following potential will be found to satisfy both the boundary condition and the 

 Laplace Equation [137r]: 



1 



4> = A,ie-l)'^^il-,y/^^Jln^-i- 



[137d"] 



A = k^n 



Cn + 1 



-(2^^-l)/n- -Ho 



4>o -L 



-^0 



<l-y 



[137€ 



355 



