70 PROFESSOR TAIT ON THE 



different position on its spherical locus. Similarly, impact of equal particles 

 does not alter the distribution of velocity along the line of centres, nor along 

 any line perpendicular to it. But it does, in general, produce alterations in 

 the distribution parallel to any line other than these. 



Hence impacts, in all of which the line of centres is parallel to one common 

 line, produce no change in the arrangement of velocity-components along that 

 line, nor along any line at right angles to it. But there will be, in general, 

 changes along every other line. It is these which lead gradually (though very 

 rapidly) to the final result, in which the distribution of velocity-components is 

 the same for all directions. 



When this is arrived at, collisions will not, in the long run, tend to alter it. 

 For then the uniformity of distribution of the spheres in space, and the 

 symmetry of distribution of velocity among them, enable us (by the principle 

 of averages) to dispense with the only limitation above imposed ; viz., the 

 parallelism of the lines of centres in the collisions considered. 



5. In what precedes nothing whatever has been said as to the ratio of the 

 diameter of one sphere to the average distance between two proximate spheres, 

 except what is implied in the preliminary assumption that the sum of the 

 volumes of the spheres is only a very small fraction of the space in which they 

 are free to move. It is probable, though not (so far as I know) thoroughly 

 proved, that if this fraction be exceedingly small the same results will ulti- 

 mately obtain, but only after the lapse of a proportionately long time ; while, if 

 it be infinitely small, there will be no law, as there will be practically no colli- 

 sions. On the other hand, if the fraction be a large one {i.e., as in the case of 

 a highly compressed gas), it seems possible that these results may be true, at 

 first, only as a very brief time-average of the condition of the spheres in any 

 region large enough to contain a great number : — that, in fact, the distribution 

 of particles and speeds in such a region will be for some time subject to con- 

 siderable but extremely rapid fluctuations. Reasons for these opinions will 

 be seen in the next section of the paper. But it must also be noticed that 

 when the particles fill the greater part of the space in which they move, 

 simultaneous impacts of three or more will no longer be of rare occurrence ; 

 and thus a novel and difficult feature forces itself into the question. 



Of course with infinitely hard spheres the probability of such multiple 

 collisions would be infinitely small. It must be remembered, hoAvever, that the 

 investigation is meant to apply to physical particles, and not to mere mathe- 

 matical fictions ; so that we must, in the case of a highly compressed gas, take 

 account of the possibility of complex impacts, because the duration of an 

 impact, though excessively short, is essentially finite. 



