454 



NA TURE 



[September 6, 1900 



or dynamically. Thus, to take the simplest case, the dynamics 

 of a solid body can be confined to a discussion of its three com- 

 ponents of translation and its three components of rotation, in- 

 stead of the motion of each element of its mass. With the number 

 of independent co-ordinates thus diminished, when the initial 

 state of the motion is specified the subsequent course of the 

 complete system can be traced ; but the course of the changes in 

 any part of it can only be treated in relation to the motion of 

 the system as a whole. It is just this mode of treatment of a 

 system as a whole that is the main characteristic of modern 

 physical analysis. The way in which Maxwell analysed the 

 interactions of a system of linear electric currents, previously 

 treated as if each were made up of small independent pieces or 

 elements, and accumulated the evidence that they formed a 

 single dynamical system, is a trenchant example. The inter- 

 actions of vortices in fluid form a very similar problem, which is 

 of special note in that the constitution of the system is there 

 completely known in advance, so that the two modes of 

 dynamical exposition can be compared. In this case the older 

 method forms independent equations for the motion cf each 

 material element of the fluid, and so requires the introduction 

 of the stress — here the fluid pressure — by which dynamical eff"ect 

 is passed on to it from the surrounding elements : it corresponds 

 to a method of contact action. But Helmholtz opened up new 

 ground in the abstract dynamics of continuous media when he 

 recognised (after Stokes) that, if the distribution of the velocity 

 of spin at those places in the fluid where the motion is vortical 

 be assigned, the motion in every part of the fluid is therein 

 kineniatically involved. This, combined with the theorem of 

 Lagrange and Cauchy, that the spin is always confined to the 

 same portions of the fluid, formed a starting-point for his theory 

 of vortices, which showed how the subsequent course of the 

 motion can be ascertained without consideration of pressure or 

 other stress. 



The recognition of the permanent state of motion constituting 

 a vortex ring as a determining agent as regards the future 

 course of the system was in fact justly considered by Helmholtz 

 as one of his greatest achievements. The principle had entirely 

 eluded the attention of Lagrange and Cauchy and Stokes, who 

 were the pioneers in this fundamental branch of dynamics, and 

 had virtually prepared all the necessary analytical material for 

 Helmholtz's use. The main import of this advance lay, not in 

 the assistance which it afforded to the development of the 

 complete solution of special problems in fluid motion, but in 

 the fact that it constituted the discovery of the types of per- 

 manent motion of the system, which could combine and inter- 

 act with each other without losing their individuality,^ though 

 each of them pervaded the whole field. This rendered possible 

 an entirely new mode of treatment ; and mathematicians who 

 were accustomed, as in astronomy, to aim directly at the 

 determination of all the details of the special case of motion, 

 were occasionally slow to apprehend the advantages of a pro- 

 cedure which stopped at formulating a description of the nature 

 of the interaction between various typical groups of motions 

 into which the whole disturbance could be resolved. 



The new train of ideas introduced into physics by Faraday 

 was thus consoHdated and emphasised by Helmholtz's investig- 

 ations of 1858 in the special domain of hydrodynamics. In 

 illustration let us consider the fluid medium to be pervaded by 

 permanent vortices circulating round solid rings as cores : the 

 older method of analysis would form equations of motion for 

 each element of the fluid, involving the fluid pressure, and by 

 their integration would determine the distribution of pressure on 

 each solid ring, and thence the way it moves. This method is 

 hardly feasible even in the simplest cases. The natural plan is 

 to make use of existing simplifications by regarding each vortex 

 as a permanent reality, and directly attacking the problem of 

 its interactions with the other vortices. The energy of the 

 fluid arising from the vortex motion can be expressed in terms 

 of the positions and strengths of the vortices alone ; and then 

 the principle of Action, in the generalised form which includes 

 steady motional configurations as well as constant material con- 

 figurations, affords a method of deducing the motions of the 

 cores and the interactions between them. If the cores are thin 

 they in fact interact mechanically, as Lord Kelvin and Kirchhoff" 

 proved, in the same manner as linear electric currents would 

 do ; though the impulse thence derived towards a direct hydro- 

 kinetic explanation of electro-magnetics was damped by the fact 



1 We may compare G. W. Hill's more recent introduction of the idea of 

 permanent orbils into physical astronomy. 



NO. 1610, VOL 62] 



that repulsion and attraction have to be interchanged in the 

 analogy. The conception of vortices, once it has been arrived 

 at, forms the natural physical basis of investigation, although 

 the older method of determining a distribution of pressure-stress 

 throughout the fluid and examining how it affects the cores is 

 still possible ; that stress, however, is not simply transmitted, 

 as it has to maintain the changes of velocity of the various 

 portions of the fluid. But if the vortices have no solid cores we 

 are at a loss to know where even this pressure can be considered 

 as applied to them ; if we follow up the stress, we lose the 

 vortex ; yet a fluid vortex can nevertheless illustrate an atom of 

 matter, and we can consider such atoms as exerting mutual 

 forces, only these forces cannot be considered as transmitted 

 through the agency of fluid pressure. The reason is that the 

 vortex cannot now be identified with a mere core bounded by a 

 definite surface, but is essentially a configuration of motion 

 extending throughout the medium. 



Thus we are again in face of the fundamental question 

 whether all attempts to represent the mechanical interactions of 

 electro-dynamic systems, as transmitted from point to point by 

 means of simple stress, are not doomed to failure ; whether 

 they do not, in fact, introduce unnecessary and insurmountable 

 difficulty into the theory. The idea of identifying an atom with 

 a state of strain or motion, pervading the region of the zether 

 around its nucleus, appears to demand wider views as to what 

 constitutes dynamical transmission. The idea that any small 

 portion of the primordial medium can be isolated, by merely 

 introducing tractions acting over its surface and transmitted 

 from the surrounding parts, is no longer appropriate or con- 

 sistent : a part of the dynamical disturbance in that element of 

 the medium is on this hypothesis already classified as belonging 

 to, and carried along with, atoms that are outside it but in its 

 neighbourhood — and this part must not be counted twice over. 

 The law of Poynting relating to the paths of the transmission of 

 energy is known to hold in its simple form only when the 

 electric charges or currents are in a steady state ; when they are 

 changing their positions or configurations their own fields of 

 intrinsic energy are carried along with them. 



It is not surprising, considering the previous British familiarity 

 with this order of ideas, that the significance for general physics 

 of Helmholtz's doctrine of vortices was eagerly developed in 

 this country, in the form in which it became embodied through 

 Lord Kelvin's famous illustration of the constitution of the 

 matter, as consisting of atoms with separate existence and 

 mutual interactions. This vortex-atom theory has been a main 

 source of physical suggestion because it presents, on a simple 

 basis, a dynamical picture of an ideal material system, atomic- 

 ally constituted, which could go on automatically without 

 extraneous support. The value of such a picture may be held 

 to lie, not in any supposition that this is the mechanism of the 

 actual world laid bare, but in the vivid illustration it affords of 

 the fundamental postulate of physical science, that mechanical 

 phenomena are not parts of a scheme too involved for us to 

 explore, but rather present themselves in definite and consistent 

 correlations, which we are able' to disentangle and apprehend 

 with continually increasing precision. 



It would be an interesting question to trace the origin of our 

 preference for a theory of transmission of physical action over 

 one of direct action at a distance. It may be held that it rests 

 on the same order of ideas as supplies our conception of force ; 

 that the notion of effort which we associate with change of the 

 motion of a body involves the idea of a mechanical connection 

 through which that effort is applied. The mere idea of a trans- 

 mitting medium would then be no more an ultimate foundation 

 for physical explanation than that of force itself. Our choice 

 between direct distance action and mediate transmission would 

 thus be dictated by the relative simplicity and coherence of the 

 accounts they give of the phenomena : this is, in fact, the basis 

 on which Maxwell's theory had to be judged until Hertz 

 detected the actual working of the medium. Instantaneous 

 transmission is to all intents action at a distance, except in so 

 far as the law of action may be more easily formulated in terms 

 of the medium than in a direct geometrical statement. 



In connection with these questions it may be permitted to 

 refer to the eloquent and weighty address recently delivered by 

 M. Poincare to the International Congress of Physics. M. 

 Poincare accepts the principle of Least Action as a trustworthy 

 basis for the formulation of physical theory, but he imposes the 

 condition that the results must satisfy the Newtonian law of 

 equality of action and reaction between each pair of bodies 



