510 Prof. Helmholtz on Integrals expressing Vortex-motion, 



the outer side of the ring is in the direction of z positive, on the 

 inner side in the direction of z negative; K, h, and R are from 

 their nature necessarily positive. 



Hence in a circular vortex -filament of very small section in an in- 

 definitely extended fluid, the centre of gravity of the section has, from 

 the commencement, an approximately constant and very great velocity 

 parallel to the axis of the vortex -ring, and this is directed towards the 

 side to which the fluid flows through the ring. Indefinitely thin 

 vortex-filaments of finite radius would have indefinitely great velo- 

 city of translation. But if the radius be indefinitely great, of the 



order -, then R 2 is indefinitely great compared with K, and / 



becomes constant. The vortex-filament, which has now become 

 rectilinear, becomes stationary, as we have already proved for 

 the case of such filaments. 



We can now see generally how two ring-formed vortex-filaments 

 having the same axis would mutually affect each other, since 

 each, in addition to its proper motion, has that of its elements 

 of fluid as produced by the other. If they have the same direc- 

 tion of rotation, they travel in the same direction ; the foremost 

 widens and travels more slowly, the pursuer shrinks and travels 

 faster, till finally, if their velocities are not too different, it over- 

 takes the first and penetrates it. Then the same game goes on 

 in the opposite order, so that the rings pass through each other 

 alternately. 



If they have equal radii and equal and opposite angular velo- 

 cities, they will approach each other and widen one another; 

 so that finally, when they are very near each other, their velo- 

 city of approach becomes smaller and smaller, and their rate of 

 widening faster and faster. If they are perfectly symmetrical, 

 the velocity of fluid elements midway between them parallel 

 to f ^the axis is zero. Here, then, we might imagine a rigid 

 plane to be inserted, which would not disturb the motion, and 

 so obtain the case of a vortex-ring which encounters a fixed 

 plane. 



In addition it may be noticed that it is easy in nature to 

 study these motions of circular vortex-rings, by drawing rapidly 

 for a short space along the surface of a fluid a half-immersed 

 circular disk, or the nearly semicircular point of a spoon, and 

 quickly withdrawing it. There remain in the fluid half vortex- 

 rings whose axis is in the free surface. The free surface forms a 

 bounding plane of the fluid through the axis, and thus there is 

 no essential change in the motion. These vortex-rings travel 

 on, widen when they come to a wall, and are widened or con- 

 tracted by other vortex-rings, exactly as we have deduced from 

 theory. 



