420 



NA TURE 



[August 28, 1884 



solid which will have the property that the direction of vibration 

 in waves of rectilinear vibrations propagated through it shall 

 turn round the line of propagation of the waves, just as Faraday's 

 observation proves to be done by the line of vibration of light in 

 a dense medium between the poles of a powerful magnet. The 

 case of wave front perpendicular to the lines of resultant moment 

 of momentum (that is to say, the direction of propagation being 

 parallel to these lines) corresponds, in our mechanical model, to 

 the case of light travelling in the direction of the lines of force in 

 a magnetic field. 



In these illustrations and models we have different portions of 

 ideal rigid matter acting upon one another, by normal pressure 

 at mathematical points of contact — of course no forces of friction 

 are supposed. It is exceedingly interesting to see how thus, 

 with no other postulates than inertia, rigidity, and mutual im- 

 penetrability, we can thoroughly model not only an elastic solid, 

 and any combination of elastic solids, but so complex and recon- 

 dite a phenomenon as the passage of polarised light through a 

 magnetic field. But now, with the view of ultimately discarding 

 the postulate of rigidity from all our materials, let us suppose 

 some to be absolutely destitute of rigidity, and to possess merely 

 inertia and incompressibility, and mutual impenetrability with 

 reference to the still remaining rigid matter. With these postu- 

 lates we can produce a perfect model of mutual action at a dis- 

 tance between solid particles, fulfilling the condition, so keenly 

 desired by Newton and Faraday, of being explained by con- 

 tinuous action through an intervening medium. The law of the 

 mutual force in our model, however, is not the simple Newtonian 

 law, but the much more complex law of the mutual action be- 

 tween electro-magnets — with this difference, that in the hydro- 

 kinetic model in every case the force is opposite in direction to 

 the corresponding force in the electro-magnetic analogue. Ima- 

 gine a solid bored through with a hole and placed in our ideal 

 perfect liquid. For a moment let the hole be stopped by a 

 diaphragm, and let an impulsive pressure be applied for an 

 instant uniformly over the whole membrane, and then instantly 

 let the membrane be dissolved into liquid. This action origin- 

 ates a motion of the liquid relatively to the solid, of a kind to 

 which I have given the name of " irrotational circulation," which 

 remains absolutely constant however the solid be moved through 

 the liquid. Thus, at any time the actual motion of the liquid 

 at any point in the neighbourhood of the solid will be the 

 resultant of the motion it would have in virtue of the 

 circulation alone, were the solid at rest, and the motion 

 it would have in virtue of the motion of the solid itself. 

 had there been no circulation established through the aper- 

 ture. It is interesting and important to remark in passing 

 that the whole kinetic energy of the liquid is the sum of the 

 kinetic energies which it would have in the two cases separately. 

 Now, imagine the whole liquid to be inclosed in an infinitely 

 large, rigid, containing vessel, and in the liquid, at an infinite 

 distance from any part of the containing vessel, let two perforated 

 solids, with irrotational circulation through each, be placed at 

 rest near one another. The resultant fluid motion due to the 

 two circulations will give rise to fluid pressure on the two bodies, 

 which, if unbalanced, will cause them to move. The force 

 systems — force-and -torques, or pairs of forces — required to pre- 

 vent them from moving will be mutual and opposite, and will be 

 the same as, but opposite in direction to, the mutual force sys- 

 tems required to hold at rest two electro-magnets fulfilling the 

 following specification. The two electro-magnets are to be of 

 the same shape and size as the two bodies, and to be placed in 

 the same relative positions, and to consist of infinitely thin layers 

 of electric currents in the surfaces of solids possessing extreme 

 diamagnetic quality — in other words, infinitely small permea- 

 bility. The distribution of electric current on each body may 

 be any whatever which fulfils the condition that the total current 

 across any closed line drawn on the surface once through the 

 aperture is equal to 1/4* of the circulation l through the aperture 

 in the hydro-kinetic analogue. 



_ It might be imagined that the action at a distance thus pro- 

 vided for by fluid motion could serve as a foundation for n 

 theory of the equilibrium, and the vibrations, of elastic solids, 

 and the transmission of waves like those of light through an 

 extended quasi-elastic solid medium. But unfortunately for this 



The integral of tangential component velocity all round any closed 

 curve, passing once through the aperture, is defined as the " cyclic constant," 

 or the "circulation "("Vortex Motion," 5 60 (a), Trans. R. S. E. April 29, 

 1867). It has the same value for all closed curves passing just once through 

 the aperture, and it remains constant through all time, whether the solid 

 body be in motion or at rest. 



idea the equilibrium is essentially unstable, both in the case of 

 magnets, and, notwithstanding the fact that the forces are oppo- 

 sitely directed, in the hydro-kinetic analogue also, when the 

 several movable bodies (two or any greater number) are so 

 placed relatively as to be in equilibrium. If, however, we 

 connect the perforated bodies with circulation through them in 

 the hydro-kinetic system, by jointed rigid connecting links, we 

 may arrange for configurations of stable equilibrium. Thus 

 without fly-wheels, but with fluid circulations through apertures, 

 we may make a model spring balance, or a model luminiferous 

 ether, either without or with the rotational quality correspond; 

 ing to that of the true luminiferous ether in the magnetic fluid 

 — in short, do all by the perforated solids with circulations 

 through them that we saw we could do by means of linked 

 gyrostats. But something that we cannot do by linked gyrostats 

 we can do by the perforated bodies with fluid circulation : we 

 can make a model gas. The mutual action at a distance, re- 

 pulsive or attractive according to the mutual aspect of the two 

 bodies when passing within collisional distance 1 of one another, 

 suffices to produce the change of direction of motion in collision, 

 which essentially constitutes the foundation of the kinetic theory 

 of gases, and which, as we have seen before, may as well be due 

 to attraction as to repulsion, so far as we know from any investi- 

 gation hitherto made in this theory. 



There remains, however, as we have seen before, the difficulty 

 of providing for the case of actual impacts between the solids, 

 which must be done by giving them massless spring buffers, or, 

 which amounts to the same thing, attributing to them repulsive 

 forces sufficiently powerful at very short distances to absolutely 

 prevent impacts between solid and solid ; unless we adopt the 

 equally repugnant idea of infinitely small perforated solids, with 

 infinitely great fluid circulations through them. Were it not for 

 this fundamental difficulty, the hydro-kinetic model gas would be 

 exceedingly interesting ; and, though we could scarcely adopt it 

 as conceivably a true representation of what gases really are, it 

 might still have some importance as a model configuration of 

 solid and liquid matter, by which without elasticity the elasticity 

 of a tme gas might be represented. 



But lastly, since the hydro-kinetic model gas with perforated 

 solids and fluid circulations through them fails because of the 

 impacts between the solids, let us annul the solids and leave the 

 liquid performing irrotational circulation round vacancy, 2 in the 

 place of the solid cores which we have hitherto supposed ; or let 

 us annul the rigidity of the solid cores of the rings and give them 

 molecular rotation according to Helmholtz's theory of vortex 

 motion. For stability the molecular rotation must be such as to 

 give the same velocity at the boundary of the rotational fluid core 

 as that of the irrotationally circulating liquid in contact with it, 

 because, as I have proved, frictional slip between two portions ot 

 liquid in contact is inconsistent with stability. There is a 

 further condition, upon which I cannot enter into detail just now, 

 but which may be understood in a general way when I say that 

 it is a condition of either uniform or of increasing molecular 

 rotation from the surface inwards, analogous to the condition 

 that the density of a liquid, resting for example under the in- 

 fluence of gravity, must either be uniform or must be greater 

 below than above for stability of equilibrium. All that I have 

 said in favour of the model vortex gas composed of perforated 

 solids with fluid circulations through them holds without modifi- 

 cation for the purely hydro-kinetic model, composed of either 

 Helmholtz cored vortex-rings or of coreless vortices, and we are 

 now troubled with no such difficulty as that of the impacts 

 between solids. Whether, however, when the vortex theory of 

 gases is thoroughly worked out. it will or will not be found to 

 fail in a manner analogous to the failure which I have already 

 pointed out in connection with the kinetic theory of gases com- 

 posed of little elastic solid molecules, I cannot at present under- 

 take to speak with certainty. It seems to me most probable 

 that the vortex theory cannot fail in any such way, because all I 

 have been able to find out hitherto regarding the vibration of 



1 According to this view there is no precise distance, or definite condition 

 respecting the distance, between two molecules, at which apparently they 

 come to be in collision, or when receding from one another they cease to be in 

 collision. It is convenient, however, in the kinetic theory of gases, to adopt 

 arbitrarily a precise definition of collision, according to which two bodies or 

 particles mutually acting at a distance may be said to be in collision when 

 their mutual action exceeds some definite arbitrarily assigned limit, as, for 

 example, when the radius of curvature of the path of either body is less than 

 a stated fraction (1/100, for instance) of the distance between them. 



2 Investigations respecting coreless vortices will be found in a paper by the 

 author, " Vibrations of a Columnar Vortex," Proc. R. S. E., March 1 r88o : 

 and a paper by Hicks, recently read before the Royal Society. 



