Sbptembee21, 1900.] 



SCIENCE. 



427 



the limitations which restrict the connec- 

 tions of the system, at the expense, how- 

 ever, of introducing new variables ; if, in- 

 deed, it does not introduce gratuitous com- 

 plexity for purposes of physics to attempt 

 to do this. However these questions may 

 stand, this principle of straightness or di- 

 rectness of path forms, whenever it applies, 

 the most general and comprehensive for- 

 mulation of purely dynamical action : it 

 involves in itself the complete course of 

 events. In so far as we are given the alge- 

 braic formula for the time-integral which 

 constitutes the Action, expressed in terms 

 of any suitable coordinates, we know im- 

 plicitly the whole dynamical constitution 

 and history of the system to which it ap- 

 plies. Two systems in which the Action is 

 expressed by the same formula are mathe- 

 matically identical, are physically precisely 

 correlated, so that they have all dynamical 

 properties in common. When the struc- 

 ture of a dynamical system is largely con- 

 cealed from view, the safest and most direct 

 way towards an exploration of its essential 

 relations and connections, and in fact to- 

 wards answering the prior question as to 

 whether it is a purely dynamical system at 

 all, is through this order of ideas. The 

 ultimate test that a system is a dynamical 

 one is not that we shall be able to trace 

 mechanical stresses throughout it, but that 

 its relations can be in some way or other 

 consolidated into accordance with this prin- 

 ciple of minimum Action. This definition 

 of a dynamical system in terms of the 

 simple principle of directness of path may 

 conceivably be subject to objection as too 

 wide ; it is certainly not too narrow ; and 

 it is the conception which has naturally 

 been evolved from two centuries of study 

 of the dynamics of material bodies. Its 

 very great generality may lead to the ob- 

 jection that we might completely formulate 

 the future course of a system in its terms, 

 without having obtained a working famili- 



arity with its details of the kind to which 

 we have become accustomed in the analysis 

 of simple material systems ; but our choice 

 is at present between this kind of formu- 

 lation, which is a real and essential one, 

 and an empirical description of the course 

 of phenomena combined with explanations 

 relating to more or less isolated groups. 

 The list of great names, including Kelvin, 

 Maxwell, Helmholtz, that have been asso- 

 ciated with the employment of the prin- 

 ciple for the elucidation of the relations of 

 deep-seated dynamical phenomena, is a 

 strong guarantee that we shall do well by 

 making the most of this clue. 



Are we then justified in treating the ma- 

 terial molecule, so far as revealed by the 

 spectroscope, as a dynamical system com- 

 ing under this specification ? Its intrinsic 

 energy is certainly permanent and not sub- 

 ject to dissipation; otherwise the molecule 

 would gradually fade out of existence. The 

 extreme precision and regularity of detail 

 in the spectrum shows that the vibrations 

 which produce it are exactly synchronous 

 whatever be their amplitude, and in so far 

 resemble the vibrations of small amplitude 

 in material systems. As all indications 

 point to the molecule being a system in a 

 state of intrinsic motion, like a vortex ring, 

 or a stellar system in astronomy, we must 

 consider these radiating vibrations to take 

 place around a steady state of motion 

 which does not itself radiate, not around a 

 state of rest. Now not the least of the 

 advantages possessed by the Action prin- 

 ciple, as a foundation for theoretical phys- 

 ics, is the fact that its statement can be 

 adapted to systems involving in their con- 

 stitution permanent steady motions of this 

 kind, in such a way that only the variable 

 motions superposed on them come into con- 

 sideration. The possibilities as regards 

 physical correlation of thus introducing 

 permanent motional states as well as per- 

 manent structure into the constitution of 



