282 Prof. M. J. M. Hill. Motion of Fluid, part of [Feb. 7, 



Example 2. Kirchhoff's Elliptic Yortex Cylinder. 

 x', y' are rectangular co-ordinates referred to axes rotating with 

 uniform angular velocity w about their origin. 

 /, a, b are constants. 



u', v' the components of the velocity parallel to the rotating axes. 



£=i(^— — )= const, in this case (u, v are velocities parallel fixed 



\dx dijj 



rectangular axes in space). 



It is shown that u'=y'(^L — o^j ; v' = — x'^~ — u?j satisfy the equa- 

 tions of motion. 



The surfaces containing the same particles are the elliptic cylinders 



x' 2 v' 2 



'—-f^- = const. 



a 2 D* 



If u = — (which is= ^ a '^ \ then the motion may be supposed 

 to be given by the above values of u', v' inside the cylinder 



a £ b z 



and therefore to be rotational ; but outside this it may be supposed 

 irrotational, and its velocity potential 



x y { 1 — — l—Vab sin 1 



a 2 -b 2 J \ 2s/ {a 2 + e)(b , + e) J 



X 



x '1 y'2 



where e is the root of — — -f -4 — = 1, which lies between — b 2 and oo , 

 a 2 + e b 2 + e 



a being >b. 



Example 3. With the same values for vf, v' as in the last example, 

 except, however, the relation between / and w, it is shown that an 

 irrotational motion continuous with the rotational motion inside the 



cylinder — =1 can exist between this and a confocal elliptic 



a 2 b 2 



smooth rigid cylinder surrounding it ; provided that confocal elliptic 

 cylinder be made to rotate with the same angular velocity w. 

 The eof the confocal cylinder is — 



[a 2 +b 2 w\ 

 Xf) a 2 W 



w 2 

 where — oo <-< — — -. 



/ ab 



