90 REPORT— 1891. 



effect of both refractions on the ultimate motion is the same as that of a 

 collision in which the line of centres is perpendicular to the direction of 

 relative motion between the two refractions. And by making the relative 

 velocity between the refractions very great, the duration of the encounter 

 may be made very small. Boltzmann points out that an encounter involving 

 three or more particles will sometimes have the effect of leaving two (or 

 more) particles permanently entangled together, and the number of such 

 double molecules Avill increase as the temperature and volume are decreased, 

 thus suggesting an explanation of the phenomena of dissociation and lique- 

 faction. 



In the case of polyatomic molecules specified by generalised co-ordinates, 

 the collision foi-mula of § 33 cannot be directly applied to an encounter of 

 this kind, owing to the changes of position of the molecules between the 

 two refractions. But there would be no difficulty in here proving the 

 Boltzmann-lNIaxwell Law by taking separate account of the two refractions 

 and the free motion between them as explained in § 32. 



50. The effect of double, treble, and multiple encounters in relation to 

 the Boltzmann-Maxwell Law has been investigated by Natanson.' He 

 arrives at conclusions entirely in accordance with what has been said 

 above in § 30, and shows that the translational velocities of the c.m.'s of 

 two or more molecules during an encounter follow the Boltzmann-Maxwell 

 distribution. This may be readily verified as follows. 



If the frequencies of distribution among two sets of molecules of 

 masses 7n^, m^ are proportional to 



e-MT,+x,) and 6-'"^^+''-' 



then, as in § 30, the fi'equency of distribution of pairs of such molecules in 

 the course of an encounter is proportional to 



g-;.(T,+Tj+x,+x,+x,a) ..... (40) 



where x,, is the mutual potential energy due to the encountei", so that Xi2 

 vanishes when the molecules are beyond the range of their mutual influence. 

 Now let 7t,,v,,Tf,, U2,Vo,w.2 be the translational velocities of the mole- 

 cules, u, V, w those of their cm., ■?6,., i',., tv,. the components of relative velocity. 

 Then from 



we have total kinetic energy of translation of two molecules parallel to x 



* ' - ' -m,+TO2 



•whence it readily follows from (40) that the mean energy of translation of 

 the whole mass m, -Fmg collected at the cm. is equal to 3/'2h, and is equal 

 to the mean translational energy of either of the separate molecules. And 

 by combining the translational energy of the pair with that of a third 

 molecule the result can be extended to any number of molecules. 



In my first Report, § 43, I alluded to some difiiculties i-aised by Tait 

 regarding the question of temperature, but the present conclusions show 

 that no such difficulties arise in connection with the Boltzmann-Maxwell 



' ' Ueber die kinetische Theorie unvoUkommener Gase,' Annalen der Phi/sik und. 

 Chemie, xxxiii. (1885), p. 685. 



