166 VORTEX MOTION. [CHAP. VI. 



of the vortex (supposed in this case external), and let B be the 

 image* of A with respect to the circle DPE, viz. C being the 

 centre, let 



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

 circle, we have 



AP_ AE_AD _ 



BP~ BE~ BD~ 



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 A t 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 problems are satisfied if we suppose 

 A to describe a circle about the axis of the cylinder with uniform 

 velocity 



m m . CA 



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

 a fixed circular cylinder, say at B, would describe a circle with 



. f T . . m.CB 



uniform velocity ^ 77- 



77 (C L&amp;gt;JJ ) 



(c) 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, if the various 

 rotations have the directions indicated in the figure, we see that 



Fig. 12. 



the effect of the presence of the pair A , B on A, B is to separate 

 them, and at the same time to diminish their velocity perpen 

 dicular to the line joining them. The planes which bisect AB, 



