NATURE 



193 



THURSDAY, DECEMBER 27, \l 



VORTEX RINGS 

 The Motion of Vortex Rings. By J. J. Thomson. 

 (London : Macniillan and Co., 1S83 ) 



BOTH as regards the interest of the subject and the 

 treatment it has received at the hands of the author 

 we do not doubt that the essay before us is destined to 

 take a foremost place amongst the essajs which have 

 been called forth, or at all events distinguished, by the 

 Adams Prize. 



The fact that these essays are upon set subjects pre- 

 cludes the possibility of the prize being awarded for a 

 distinctly original conception. It is almost a necessity 

 that the subjects chosen should involve the e.xtension of 

 some mathematical investigation which has already been 

 carried a certain length. 



The subject of the present essay is distinctly of this 

 class ; it involves an e.xtension of the investigation of the 

 theory of vortex motion in an ideal fluid, founded 

 by Helmholtz and continued chiefly by Sir William 

 Thomson. 



At the time Helmholtz conceived the fundamental 

 principle, ideal hydrodynamics had no other interest, 

 besides its mathematical interest, than it derived from 

 the somewhat casual explanations it affords of the pheno- 

 mena met with in the motion of actual fluids. Helm- 

 holtz's investigation had some relation to the observed 

 phenomena of actual vortices, particularly to the pheno- 

 mena of smoke rings, of which it afforded a general 

 explanation. But between the fundamental equations 

 which Helmholtz gave and their application to an actual 

 vortex ring certain integrations were necessary, and 

 these integrations presented mathematical difficulties. 

 If we consider the line of smoke which forms the ring as 

 indicating the portion of air in which vortex motion 

 exists, we may say that the difficulties of integration at 

 which Helmholtz stopped arise from the thickness of this 

 line of smoke, or, calling this the circular core of the ring, 

 from the finite area of the section of this core. Helm- 

 holtz contented himself with applying his theory to an 

 indefinitely thin core ; and the fact that the results of a 

 theory based on a frictionless fluid would only have an 

 imperfect relation to the motions of viscous fluids, 

 together with the fact that such rings, although they may 

 be produced by artificial apparatus, are short-lived, and 

 have no existence in the general motion of fluids, offered 

 but little inducement for farther prosecution of the 

 subject. The case however was altered when it was con- 

 ceived by Sir William Thomson that the atoms of matter 

 may be such rings moving in a perfect universal fluid. 

 Smoke rings, although their behaviour seems to have 

 suggested the idea, could not, owing to the viscosity of 

 the air, by any means be made to afford an experimental 

 verification of the capabilities of such an hypothesis. 

 The only way was to integrate Helmholtz's equations, and 

 thus arrive at the theoretical behaviour of such rings. 

 Unfortunately the mathematical difficulties are such that 

 there is little hope of obtaining a complete theory of 

 vortex rings having cores of any finite area. Sir William 

 Vol. XXIX.— No. 739 



Thomson, however, started an approximate theory as a 

 step towards this ; he succeeded in approximately inte- 

 grating the equations for rings the cores of which had 

 sections finite but small compared with the openings of 

 the rings, and with such rings it appears that his theory 

 can be tested as regards matter in the gaseous state. 



To do this, however, it is necessary to do more than 

 work out the theory of a single circular ring having a 

 core of circular section. The phenomena of gases depend 

 on the internal vibration of the atoms and on the influence 

 which they exert on each other by collisions or otherwise. 

 It was necessary therefore to obtain the theory of the 

 vibrations of these rings, also of the effect of what may 

 be called collisions. 



Sir William Thomson took many steps towards the 

 theory of vibrations. But the theory of collisions was 

 left for Mr. J. J. Thomson. 



Mr. Thomson has notj however, confined his attention 

 to the point set for the prize, but, starting from the founda- 

 tion laid by Helmholtz, has recast the theory to his own 

 method. 



Having deduced general expressions for the momentum, 

 moment of momentum, and energy in a mass of fluid in 

 which there is vortex motion, which expressions are better 

 adapted for his purpose than any previously obtained, he 

 proceeds to the theory of a solitary vortex ring subject to 

 the same limitation as that treated by Sir William 

 Thomson, i.e. the diameter of the core small compared 

 with the opening of the ring, but of more general shape, 

 in that it may have any small deviation from the circular 

 form. He obtains results which, where they correspond, 

 agree very approximately with those previously obtained 

 by Sir William Thomson. 



The author then proceeds to the immediate subject of 

 the essay — the action upon each other of two rings. 



In dealing with this subject he introduces another im- 

 portant limitation, i.e. that the rings shall not approach 

 each other by a distance which is large compared with 

 the openings of the rings. 



With this limitation, by means of a very powerful piece 

 of mathematical work, the theory of the mutual action of 

 such rings is deduced, both as regards mean motion and 

 vibration ; and he has thus carried the theory of vortex 

 atoms to such a stage that in certain general respects it 

 can be applied to the theory of gases. 



The essay, however, does not end here, for, although 

 outside the set subject, the author proceeds to consider 

 the theory of "linked rings." This term does not seem 

 well chosen, for it conveys the idea of rings liiiked as in a 

 chain, whereas what it is used to express is a ring of 

 which the core is compounded of several separate cores 

 wrapped in a spiral manner round each other like a ring 

 composed of twisted wire. 



In the treatment of this branch of his subject he has- 

 been no less successful than in the earlier parts. 



From the general scheme of his essay it is clear that 

 the author has had in his mind as a general object the 

 verification of the vortex atom theory ; and although he 

 avowedly refrains from going at length into such a vortex 

 atom theory of gases as might be built upon his work, 

 he adds a chapter at the end in which he discusses 

 certain results of his work, which may be applied without 

 further calculation to the vortex atom theory of gases. 



