214 ME. G. H. LIVENS ON THE 



both volume integrals being taken throughout the entire field : the derivation of the 

 dynamical and field equations is then accomplished by an application of one or the 

 other of the well-known processes of analytical dynamics. The interpretation of the 

 same results for the case when the kinetic energy is given by the usual expression 



87rJ 



E 2 dv 



was given by the present author.* 



Whilst the theoretical simplicity of these discussions, which results from their 

 interpretation in terms of the simple electronic hypothesis, is a great point in their 

 favour, it seemed, nevertheless, of theoretical interest at least to attempt to formulate 

 the problem under less restricted conditions, especially in view of the pronounced 

 tendency exhibited in some quarters to deny the adequacy of the Maxwellian theory 

 as a complete microscopic theory. Besides the more general discussion in the form in 

 which it is here presented emphasises certain difficulties inherent in the usual formula- 

 tions which have not hitherto received adequate attention. 



5. The most general dynamical principle which determines the motion of every 

 material system is the Law of Least Action expressible in the usual form 



wherein T denotes the kinetic energy and W the potential energy of the system in 

 any configuration and formulated in terms of any co-ordinates that are sufficient to 

 specify the configuration in accordance with its known properties and connexions, and 

 where the variation refers to a fixed time of passage of the system from the initial 

 to the final configuration. This is the ordinary form of HAMILTON'S principle, but it 

 involves in any case a complete knowledge of the constitution of the systems, because, 

 before it can be applied it is necessary to know the exact values of the kinetic and 

 potential energies expressed properly in terms of the co-ordinates and velocities. As 

 however we have frequently to deal with systems whose ultimate constitution is 

 either wholly or partly unknown it is necessary to establish a modified form of the 

 principle allowing for a possible ignorance of the constitution of the systems with 

 which we may have to deal. The modification is fully discussed in most works on 

 analytical dynamics,! and we may here content ourselves by merely presenting the 

 results, interpreting them however in a manner somewhat different from that usually 

 given, in order to throw some light on certain questions which arise in the subsequent 

 application in our present theory. Suppose then that it has been found impracticable 

 to express the Lagrangian function L in terms of the chosen co-ordinates of the 

 system, the typical one of which we may denote by q ; but that it is expressed in 



* 'Phil. Mag.,' vol. 32 (1916), p. 195. 



t E.g., ' Treatise on Analytical Dynamics' (2nd ed., Cambridge, 1918), by E. T. WHITTAKER. 



