72] ELLIPTIC CYLINDER. 99 



The above formulae therefore give the motion of an infinite 

 mass of liquid, otherwise at rest, produced by the rotation of 

 an elliptic cylinder about its axis with angular velocity &>*. The 

 diagram shews the stream-lines both inside and outside a rigid 

 elliptical cylindrical case rotating about its axis. 



The kinetic energy of the external fluid is given by 



.(o 2 ........................... (16). 



It is remarkable that this is the same for all confocal elliptic 

 forms of the section of the cylinder. 



Combining these results with those of Arts. 66, 71 we find that if an elliptic 

 cylinder be moving with velocities u, v parallel to the principal axes of its 

 cross-section, and rotating with angular velocity o>, and if (further) the fluid 

 be circulating irrotationally round it, the cyclic constant being K, then the 

 stream-function relative to the aforesaid axes is 



= - /( a -~\ *~* (u& sin n - 



* Quart. Journ. Math., t. xiv. (1875) ; see also Beltrami, 1. c. ante p. 93. 



72 



