582 



Mb. stokes, on THE CRITICAL VALUES OF 



55. When a is greater than ir, it»will be observed that the expression for the velocity which 

 is obtained from (119) becomes infinite when r vanishes. Of course the velocity cannot really 

 become infinite, but the expression (119) fails for points very near the axis. In fact, in obtaining 

 this expression it has been assumed that the motion of the fluid is continuous, and that a fluid 

 particle at the axis may be considered to belong to either of the plane faces indifterently, so that 

 its velocity in a direction normal to either of the faces is zero. The velocity obtained from (119) 

 satisfies this latter condition so long as a is not greater than tt. For when a <ir the velocity 

 vanishes with r, and when a = w the velocity is finite when r vanishes, and is directed along the 

 single plane face which is made up of the two plane faces before considered. 



But when a is greater than tt the motion which takes place appears to be as follows. Let 

 OABC be a section of the cylindrical sector made by a plane perpendicular to the axis, and 

 cutting it in O. Suppose the cylinder to be turning round O in the direction indicated by the 

 arrow at B. Then the fluid in contact with OA and near O will be flowing, relatively to OA, 



towards 0, as indicated by the arrow a. When it gets to O it will shoot past the face OC \ so 

 that there will be formed a surface of discontinuity Oe extending some way into the fluid, the fluid 



