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SCIENCE. 



[N. S. Vol. XII. No. 299. 



method is still fully applicable to those 

 problems of gravitational astronomy in 

 which dynamical explanation was first suc- 

 cessful on a grand scale, the planets being 

 treated as point-masses, each subject to the 

 gravitational attraction of the other bodies. 

 But the more recent development of the 

 dynamics of complex systems depends on 

 the fact that analysis has been able to re- 

 duce within manageable limits the number 

 of varying quantities whose course is to be 

 explicitly traced, through taking advantage 

 of those internal relations of the parts of 

 the system that are invariable, either 

 geometrically or dynamically. Thus, to 

 take the simplest case, the dynamics of a 

 solid body can be confined to a discussion 

 of its three components of translation and 

 its three components of rotation, instead of 

 the motion of each element of its mass. 

 With the number of independent co-ordi- 

 nates thus diminished when the initial 

 state of the motion is specified the subse- 

 quent 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 treatmant of a sys- 

 tem as a whole that is the main character- 

 istic of modern physical analysis. The way 

 in which Maxwell analyzed 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 interactions of 

 vortices in fluid form a very similar prob- 

 lem, which is of special note in that the 

 constitution of the system is there com- 

 pletely known in advance, so that the two 

 modes of dynamical exposition can be com- 

 pared. In this case the older method 

 forms independent equations for the mo- 

 tion of each material element of the fluid, 

 and so requires the introduction of the 



stress — here the fluid pressure — by which 

 dynamical efiect 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 dy- 

 namics of continuous media when he recog- 

 nized (after Stokes) that, if the distribu- 

 tion 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 kinematically involved. 

 This, combined with the theorem of La- 

 grange 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 de- 

 termining agent as regards the future course 

 of the system was in fact justly considered 

 by Helmholtz as one of his greatest achieve- 

 ments. The principle had entirely eluded 

 the attention of Lagrange and Cauchy and 

 Stokes, who were the pioneers in this fun- 

 damental branch of dynamics, and had 

 virtually prepared all the' necessary ana- 

 lytical material for Helmholtz 's use. The 

 main import of this advance lay, not in the 

 assistance which is aiibrded to the develop- 

 ment of the complete solution of special 

 problems in fluid motion, but in the fact 

 that it constituted 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 mathema- 

 ticians who were accustomed, as in as- 

 tronomy, to aim directly at the determina- 



* We may compare G. W. Hill's more recent in- 

 troduction ot the idea of permanent orbits into phys- 

 ical astronomy. 



