284 Mr Pochlington, On the Kinetic Theory of Matter. 



freedom, and that a number of systems started from all possible 

 configurations with all possible velocities can divide into discrete 

 sets, in each of which some function of the coordinates and 

 velocities has a definite value, different for the different sets. 

 The relation of these facts to the known phenomena of matter 

 is discussed in § 10, and the conclusion is drawn that if Assump- 

 tion B is rejected, the kinetic theory of matter may probably be 

 found capable of explaining them. No equations are suggested 

 as forming the mathematical basis of the theory, but the kind of 

 coordinates entering into the equation of condition is discussed, 

 without however arriving at any definite result. 



2. The system under investigation is taken to be a certain 

 quantity of gas, with ether, contained in a bottle (the inner surface 

 of which is not a surface of revolution) which is perfectly rigid 

 and smooth, and perfectly reflecting to all disturbances of the 

 ether. We first assume that the configuration of the system as 

 to position and velocity {i.e. its phase) is defined by a certain 

 number of independent coordinates q 1 ... and an equal number of 

 independent velocities. We define the momenta p 1 ... as usual 

 as the partial differential coefficients of the kinetic energy with 

 respect to the velocities, and then prove that J, the Jacobian of 

 the coordinates and momenta with respect to their initial values, 

 is unity. This may be done either via Hamilton's Characteristic 

 Function, or by evaluating (dJ/dt)/J = Xdqjdq + Xdp/dp by means 

 of Hamilton's equations of motion. 



We now suppose an infinite number of copies of the original 

 system to be made, and that these are initially in all possible 

 phases consistent with the total energy lying between certain 

 limits, and with the condition that each phase can be reached from 

 a given one without passing through a configuration in which the 

 potential energy exceeds the total energy of the phase to be 

 reached. Let the number of systems that lie between q lf q 1 + dq lt 

 ... p 1} p x + dp 1} ... be equal to kdq ± dq 2 ... dp^p 2 ... initially, where 

 k is a constant. Then by the Jacobian equation, this state is 

 permanent. 



3. Let new functions of the coordinates and momenta, linear 

 in the latter, be found such that the kinetic energy is the sum of 

 their squares. These functions have been called momentoids ; an 

 infinite number of sets of momentoids can be found. It can now 

 be proved that at any moment the mean taken over the different 

 systems (or say for the sake of brevity, the system-mean) of the 

 square of any one of the momentoids is equal to 1/n of the system- 

 mean of the kinetic energy, where n is the number of degrees of 

 freedom. 



We now select a particular set of momentoids. Since they are 

 linear functions of the momenta, they are also linear functions 



