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 tlie case when the kinetic energy is given by the usual expression 
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 then- 
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. 
