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 



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

 vortices, whose strengths are equal and opposite iu sign, situated at A, B 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 m. 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 A, 



m . CA m me 2 



For a prescribed circulation K we must add to this the term K/2n . 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, "On a Problem in Vortex Motion," Proc. R. L A., 

 December 12, 1892, 



