Mr Pocklington, On the Kinetic Theory of Matter. 285 



of the velocities, and we may find momentoids by resolving the 

 kinetic energy into sums of squares of linear functions of the velo- 

 cities. One such resolution is of the form 



T = (aq 1 + {3q 2 +r / q s +...) 2 



+ (7' , <Za + ...) 2 

 -fete. 



where the first bracket contains all the velocities, and each suc- 

 ceeding one, one less. Hence the quantities in brackets are 

 momentoids; call them w x , u 2 , etc. 



The diminution of kinetic energy produced by putting q^ and 

 q 2 equal to zero without altering the other velocities is 



u*+u.*-(yq. i +...y-(ry'q 3 +...y 



and is not greater than the kinetic energy corresponding to the 

 momentoids u x ,u. 2 . In general, the diminution of the kinetic energy 

 produced by putting the first r velocities equal to zero is not 

 greater than that corresponding to the r momentoids, and hence 

 its system-mean is not greater than r/n of the system-mean of the 

 original kinetic energy. 



4. This theorem is true at any moment, and in particular 

 after the lapse of a long time. Let us consider what the state 

 of the systems then is. If an atom cannot be made, destroyed 

 or altered in its fundamental properties without breaking the 

 geometrical constraints of the system, and the gas chosen is 

 monatomic, each system contains the same number of atoms of 

 the same kind, and we conclude from the known physical pro- 

 perties of such gases that the systems are finally indistinguishable 

 from each other by any physical test, excepting of course that 

 their energies vary between the original limits. 



Let us now take the r velocities mentioned at the end of § 3 

 to be the component velocities of translation of the atoms. Then 

 r is three times the number of atoms. But n is the total number 

 of degrees of freedom of ether and gas, and if not infinite is at 

 least many times larger than r. Hence the system-mean of the 

 change in kinetic energy produced by stopping the motion of 

 translation of the atoms without altering any other velocity is 

 at most a very small fraction of the system-mean of the kinetic 

 energy. Since the systems are ultimately indistinguishable, the 

 same is true of each system. This is entirely opposed to what is 

 known of gases ; in fact, on stopping the motion of translation of 

 the atoms of a monatomic gas in its steady state nearly the whole 

 of its energy disappears. 



5. We must now critically examine the statement that the 

 final states of all the systems are the same. This was deduced 



VOL. XII. PT. IV. 19 



