162-163] MUTUAL INFLUENCE OF VORTEX-RINGS. 261 



the rear will in turn overtake and pass through the other, and 

 so on, the rings alternately passing one through the other*. 



If the rotations in the two rings be opposite, and such that 

 the rings approach one another, the mutual influence will be to 

 enlarge the radius of each ring. If the two rings be moreover 

 equal in size and strength, the velocity of approach will con- 

 tinually diminish. In this case the motion at all points of the 

 plane which is parallel to the two rings, and half-way between 

 them, is tangential to this plane. We may therefore, if we 

 please, regard this plane as a fixed boundary to the fluid on 

 either side of it, and so obtain the case of a single vortex-ring 

 moving directly towards a fixed rigid wall. 



The foregoing remarks are taken from von Helmholtz' paper. 

 He adds, in conclusion, that the mutual influence of vortex-rings 

 may easily be studied experimentally in the case of the (roughly) 

 semicircular rings produced by drawing rapidly the point of a 

 spoon for a short space through the surface of a liquid, the spots 

 where the vortex-filaments meet the surface being marked by 

 dimples. (Cf. Art. 28.) The method of experimental illustration 

 by means of smoke-rings f is too well-known to need description 

 here. A beautiful variation of the experiment consists in forming 

 the rings in water, the substance of the vortices being coloured J. 



For further theoretical researches on the motion of vortex- 

 rings, including the question of stability, and the determination of 

 the small oscillations, we must refer to the papers cited below . 



The motion of a vortex-ring in a fluid limited (whether 

 internally or externally) by a fixed spherical surface, in the case 



* The corresponding case in two dimensions appears to have been worked out 

 very completely by Grobli; see Winkelmann, Handbuch der Physik, t. i., p. 447. 

 The same question has been discussed quite recently by Love, " On the Motion of 

 Paired Vortices with a Common Axis," Proc. Lond. Math. Soc.,t. xxv., p. 185 (1894). 



t Eeusch, "Ueber Ringbildung der Fliissigkeiten, " Pogg. Ann., t. ex. (1860); 

 see also Tait, Recent Advances in Physical Science, London, 1876, c. xii. 



J Reynolds, "On the Resistance encountered by Vortex Rings &c.", Brit. Ass. 

 Rep., 1876, Nature, t. xiv., p. 477. 



J. J. Thomson, 1. c. ante p. 239, and Phil. Trans., 1882. 



W. M. Hicks, "On the Steady Motion and the Small Vibrations of a Hollow 

 Vortex," Phil. Trans. 1884. 



Dyson, I. c. ante p. 166. 



The theory of ' Vortex- Atoms ' which gave the impulse to some of these investi- 

 gations was suggested by Sir W. Thomson, Phil. Mag., July, 1867. 



