Probabilities to the Movement of Gas-Molecules. 257 



simplified case of two dimensions, if U and xi be the velocities 

 prior to encounter let U^, u x designate the velocities just 

 after they have begun to influence each other sensibly ; let 

 U 2 , u 2 ; U3, u z ... be the pairs at later stages ; stages being- 

 measured by equal intervals of distance or, better, of time. 

 Given the law of repulsion, we can (theoretically) calculate 

 ~Ut and \i t the velocities at any time (say before that of least 

 distance between the mass-centres) in terms of UT, u, and t ; 

 and thence U, u in terms of U*, u t} and t. To obtain the 

 correlation between U* and u t from the frequency-function 

 of U and u, viz. Const, exp — \(MU 2 + ?nw 2 ), it is proper to 

 substitute for U and u their values in terms of U* and u t 

 (we need not trouble about the differential factors, since, by 

 Liouville's theorem dU t dii t = d\Jt du t '). There would result 

 a complicated expression showing interdependence between 

 the velocities at any assigned stage of the encounter, but 

 not normal correlation. But if the averages of the velocities 

 over the period of approach (or regression) are respectively 

 jU and x u, then between those variables there will be normal 

 correlation (of the form (2)). (Cp. above (14) and refer- 

 ence). Correlation in this sense may be expected throughout 

 that small part of the field which is occupied by molecules in 

 encounter. 



4. Relations between the arguments. — The third argument 

 proves in one respect more, in another less, than the first. 

 The third argument proves that perfect stability cannot exist 

 unless the velocities of the mass-centres are distributed 

 according to the normal law of frequency in its simplest 

 form without terms implying correlation between those 

 velocities (of the kind shown in (2)). The first argument 

 cannot prove this ; but it proves that normal distribution will 

 be set up approximately at least in the medley. Whereas 

 the conditional conclusion of the third argument does not by 

 itself confer the power of prediction. Could data such as 

 those which form the premisses of the third argument be 

 supposed, and yet stability not result ? Certainly a non- 

 conservative system may be imagined, such that, in addition 

 to the conservation of Momentum MU + mw (in the case of 

 one dimension), there should be secured the conservation — 

 not of energy, but — of the quantity MU 4 + mw 4 . It might 

 prima facie be argued as above that the system could not be 

 perfectly stable unless the velocities are distributed according 

 to the law of frequency Const, exp — (MU 4 + jme 4 ). But 

 there is not fulfilled the Liouville condition required by 

 the third as well as the second argument (II. o). The 

 second method has an advantage over the others in the 



Phil. Mag. S. 6. Vol. 43. No. 254. Feb. 1922. S 



