RECENT PEOGlRESS IN HTDEODYNAMICS. 65 



The problem to determine this distribution, so that the motion shall be 

 steady and the ring remain of the same mean size and shape, is one of 

 extreme difficulty and has not yet been successfully attempted. If we 

 regard a ring whose axial section is small, compared with the radius 

 of the axis, ho shows that a single such ring moves forward, with 

 approximate constant aperture, with very great velocity in the direction 

 of the fluid motion through the ring itself. When there are two on the 

 same axis, and rotating in the same direction, they travel in the same 

 direction and thread each other alternately. If they have equal and 

 opposite rotations, they approach one another indefinitely, and widen 

 indefinitely as they do so. This is the case of a ring whose plane is 

 parallel to an infinite rigid plane in a fluid. But it is to be remarked 

 that, as stated thus, it is only partially true. This is clearly only the case 

 when the fluid motions through the rings are directed towards each 

 other. If they are directed from each other, they will move from each 

 other, contracting as they do so to a certain limiting radias, which is 

 determinate in terms of any simultaneous radius, strength, and distance.' 

 The permanent character of vortex motion in a perfect fluid has 

 suggested to Sir W. Thomson^ a theory of the constitution of matter in 

 which atoms consist of small vortex-rings, whose axes may be knotted 

 into any degree of multiple continuity, the bekuottedness (as Tait calls 

 it) ■^ being a permanent character of the atom. The lines of its spectrum 

 would thus depend on the vibrations of the atom, either of the section 

 or axis. In the paper in which this speculation was brought forward, 

 Thomson notes how the motion of the fluid may be set up by a surface 

 impulse over a diaphragm, across the opening of the ring. The ring will 

 carry forward with it a mass of fluid in irrotational motion, and the 

 whole impulse of the motion is equal to the resultant impulse on the 

 diaphragm. These ideas of the impulse of the motion, and the genera- 

 tion of cyclic motion, were worked out more fully in a paper presented 

 to the Royal Society of Edinburgh, of which only a fi'agmenf has been 

 published. In the latter part of this paper, tJae principles of vortex 

 motion are developed in a quite diflerent way from that followed by 

 Helmholtz, viz., from the fundamental proposition that the ' circulation ' 

 in a circuit in the fluid is the same for all circuits which can be con- 

 tinuously deformed into one another without leaving the irrotational 

 part of the fluid. From this all the known theorems are easily deduced. 

 Amongst fresh results may be mentioned the proof that the motion of a 

 liquid moving irrotationally within an n + 1 ply connected space is 

 determinate when the normal velocity at eveiy point of the boundary and 

 the values of the circulation in the n circuits are given ; and the theorem 

 that the circulation round any closed curve in the fluid is equal to twice 

 the surface-integral of the resolved part of the vortices perpendicular to 



' For very interesting practical illustrations, the reader is referred to the follow- 

 ing papers, by W. B. Rogers: — American Journal of Science and Art (2), xxvi. p. 246. 

 This was published in 1858, without knowledge of Helmholtz's mathematical 

 researches. Reusch, Pogg. Ann. ex. p. 30D (1860) ; Osborne Reynolds, Nature, xiv. p. 

 477 (1876), also Proc. Royal Institution, viii. p. 272 ; Oberbeck, Wied. Ann. ii. p. 1 

 (1877)-. This deals with jets, but contains interesting examples of the formation of 

 rings by incipient jets ; R. S. Ball, Transactions of the Royal Irish Academy, sxv. 

 p. 13o. 



■ ' Vortex Atoms,' Proc. Roy. Soc. Edin., vi. p. 94 (1867) ; Phil, Mag. (4) 34. 



' ♦ On Knots,' Trans. Roy. Soc. Edin., xxviii. p. 145. 



* ' Vortex Motion,' Trans. Roy. Soc. Edin. xxv. (1860). 



1881. F 



