Mr Pocklington, On the Kinetic Theory of Matter. 283 



On the Kinetic Theory of Matter. By H. C. Pocklington, 

 M.A., St John's College. 



[Read 26 October 1903.] 



1. In formulating a kinetic theory of matter we, at least 

 tacitly, make the assumption (A) that matter forms, or, if the 

 ether influences the phenomena, that matter and ether together 

 form a mechanical system, i.e. a system the motion of which is 

 given by Lagrange's equations. These equations of motion will 

 contain undetermined multipliers if the number of coordinates 

 chosen to specify the configuration of the system exceeds the 

 number of independent velocities, and there must then be relations 

 between the velocities of the coordinates chosen. Assumption A 

 is equivalent to the assumption that the motion is deducible from 

 the principle of Least Action, and also to the assumption that the 

 ultimate particles of the system obey Newton's Laws of Motion, 

 and that there are no forces of the nature of slipping friction. 



It is usual to make the further assumption (B) that the 

 number of independent coordinates required to specify the con- 

 figuration of the system is equal to the number of independent 

 coordinates required to specify its velocity. In §§ 2 — 5 this 

 assumption is made, and by attacking the problem from a new 

 standpoint the difficulties hitherto met with are avoided, with 

 the result that we prove that we cannot explain the phenomena 

 exhibited by an ordinary gas if Assumption B is accepted. 



In §§ 6 — 11 Assumption B is accordingly rejected, and we 

 suppose that the number of independent velocities is reduced by 

 the existence of one or more equations of condition in the form 

 of non-integrable linear differential equations of the first order. 

 The class of system considered thus includes the simpler cases of 

 bodies rolling on one another, some of which are chosen as illus- 

 trative examples. 



In particular cases the Jacobian of the independent velo- 

 cities (or of functions linear in them) and the coordinates with 

 respect to their original values is unity, but this does not hold 

 generally, and the theorems proved about the sharing of energy, 

 etc., for the case where Assumption B is made require further 

 examination. 



A general treatment is impossible, and hence two particular 

 cases are discussed. It is shown that the whole of the energy 

 of the system can pass to a few (or even one) of its degrees of 



