TfiANSACTIONS OF SECTION A, 62l 



not only for mathematical analysis, but as. a guide to physical exploration, remains 

 fundamental. When the principles of the dynamics of material systems are refined 

 down to their ultimate common basis, this principle of minimum is what remains. 

 Hertz preferred to express its contents in the form of a principle of straifrhtness of 

 course or path. It will be recognised, on the lines already indicated, that this is 

 another mode of statement of the same fundamental idea : and the general equiva- 

 lence is worked out by Hertz on the basis of Hamilton's development of the prin- 

 ciples of dynamics. The latter mode of statement may be adaptable so as to avoid 

 the limitations which restrict the connections of the system, at the expense, however, 

 of introducing new variables ; if, indeed, it does not introduce gratuitous complexity 

 for purposes of physics to attempt to do this. However these questions may stand, 

 this principle of straightness or directness of path forms, wherever it applies, the 

 most general and comprehensive formulation of purely dynamical action : it involves 

 in itself the complete course of events. In so far as we are given the algebraic 

 formula for the time-integral which constitutes the Action, expressed in terms of 

 any suitable coordinates, we know implicitly the whole dynamical constitution and 

 history of the system to which it applies. Two systems in which the Action is 

 expressed by the same formula are mathematically identical, are physically pre- 

 cisely correlated, so that they have all dynamical properties in common. When 

 the structure of a dynamical system is largely concealed from view, the safest and 

 most direct way towards an exploration of its essential relations and connections, 

 and in fact towards 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 principle 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 objection 

 that we might completely formulate the future course of a system in its terms, with- 

 out having obtained a working familiarity with its details, of the kind to which we 

 have become accustomed in the analysis of simple material systems ; but our choice 

 18 at present between this kind of formulation, which is a real and essential one, 

 and an empirical description of the course of phenomena combined with expla- 

 nations relating to more or less isolated groups. The list of great names, including 

 Kelvin, Maxwell, Helmholtz, that have been associated with the employment of 

 the principle for the elucidation of the relations of deep-seated dynamical pheno- 

 mena, is a strong guarantee that we shall do well by making the most of this clue. 

 Are we then justified in treating the material molecule, so far as revealed by 

 the spectroscope, as a dynamical system coming under this specification ? Its 

 intrinsic energy is certainly permanent and not subject to dissipation ; otherwise 

 the molecule would gradually fade out of existence. The extreme precision and 

 regularity of detail iu 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 physics, is the fact that its statement can be 

 adapted to systems involving in their constitution permanent steady motions of 

 this kind, in such a way that only the variable motions superposed on them come 

 into consideration. The possibilities as regards physical correlation of thus intro- 

 ducing permanent motional states as well as permanent structure into the con- 

 stitution of our dynamical systems have long been emphasised by Lord Kelvin ; ^ 



' For a classical exposition see his Brit. Assoc. Address of 1881 on ' Steps towards 

 Kinetic Theory of Matter,' reprinted in ' Popular Lectures and Addresses,' vol. i. 



