155] SPECIAL CASES. 247 



where c is the radius of the circle. If P be any point on the circle, we have 



AP AE _AD_ 

 ~BP~BE~^D~ ; 



so that the circle occupies the position of a stream-line due to a pair of 

 vortices, whose strengths are equal and opposite in sign, situated at J, E in 

 an unlimited mass of fluid. Since the motion of the vortex A would then be 

 perpendicular to AB, it is plain that all the conditions of the problem will be 

 satisfied if we suppose A to describe a circle about the axis of the cylinder 

 with the constant velocity 



m tn . CA 



where m denotes the strength of A. 



In the same way a single vortex of strength m, situated inside a fixed 

 circular cylinder, say at B, would describe a circle with constant velocity 



m.CB 



It is to be noticed, however*, that in the case of the external vortex the 

 motion is not completely determinate unless, in addition to the strength 

 m of the vortex, the value of the circulation in a circuit embracing the 

 cylinder (but not the vortex) is prescribed. In the above solution, this 

 circulation is that due to the vortex-image at B and is -2m. This may 

 be annulled by the superposition of an additional vortex + m at (7, in which 

 case we have, for the velocity of ,4, 



m . CA m me 2 



7T (CA 2 - C 2 } 7T.CA 7T.CA (CA 2 - C 2 ) 



For a prescribed circulation K we must add to this the term */2ir . CA. 



3. If we have four parallel rectilinear vortices whose centres 

 form a rectangle ABB A , the strengths being m for the vortices 

 A , B, and m for the vortices A, B , it is evident that the 

 centres will always form a rectangle. Further, the various rota 

 tions having the directions indicated in the figure, we see that 



* See F. A. Tarleton, &quot;On a Problem in Vortex Motion,&quot; Proc. R. L A., 

 December 12, 1892. 



