624 REPORT— 1900. 



dynamical effect is passed on to it from the surrounding elements : it correspdnds 

 to a method of contact action. But Helmholtz opened up new pfround in the abstract 

 dynamics of continuous media when he recognised (after Stokes) that, if the dis- 

 tribution 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 liinemati- 

 cally 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 aflbrded to the development of the 

 complete solution of special problems in fluid motion, but in the fact that it con- 

 stituted the discovery of the types of permanent motion of the system, which could 

 combine and interact 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 procedure 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 consolidated 

 and emphasised by Helmholtz's investigations of 1858 in the special domain of hydro- 

 dynamics. In illustration let us consider the fluid medium to be pervaded by per- 

 manent 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 vor- 

 tices alone ; and then the principle of Action, in the generalised form which 

 includes steady motional configurations as well as constant material configura- 

 tions, 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 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 diitribution of pressure-stress throughout the fluid and 

 examining how it aff'ects the cores is still possible ; that stress, however, is not simply 

 transmitted, as it has to maintain thechangesofvelocityof 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 consi- 

 dered 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 



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

 orbits into physical astronomy. 



