Certain Problems of Two-Dimensional Physics. 767 



Similarly, in the neighbourhood of the extremities of the 

 major axis, put 



sinh £= tan \v sin 77, 

 Then at these points, 



B 1 , N B 



u=± 7 _ (1+ cosv), *= — =-(v— Bin-y), 



where, again, — tt-^v^tt, but v is otherwise arbitrary. 

 And in each case the upper sign corresponds to the positive 

 end of the axis, the lower to the negative. 



The velocity is thus indeterminate, and it is easy to verify 

 that the vorticity is infinite, at the extremities of the axes. 

 In other words, eddies are formed at these points. The eddies 

 do not disappear when the ellipse becomes a circle. And 

 this appears to be the true explanation of the result obtained 

 by Stokes in attempting to find a slow steady motion of an 

 infinite cylinder in viscous fluid — namely, that such a motion 

 is impossible. 



If the cylinder is moving, with a velocity whose com- 

 ponents parallel to the axes are U and V, in fluid at rest at 

 infinity, we have 



^ =tty-V.<e+ -(6U-aV)f 



. 2 V i/sinhfx 2U , ./sinhgx 

 H x tan M ) — — y tan M — — * . 



7T \ COS 7] / 7T \ Sin 7] / 



In the case of a circular cylinder moving with velocity V 

 parallel to the axis of y this becomes 



\ it 2a,v ) it G a 2 



6. For a cylinder slowly rotating with angular velocity co 

 in viscous fluid at rest at infinity, we have 



In the case of the ellipse it is easy to verify that the stream 

 function is 



^=-io)c 2 {2fcosh2\ + ^- 2 ^ +x )+ cos 2 V ], 



which, in the case of the circle, reduces to the obvious result, 



yjr— —^ooa 2 log r. 



