FOUNDATIONS OF THE KINETIC THEORY OF GASES. 67 



regarded as hard smooth spheres whose coefficient of restitution is unity. The 

 influence of external forces, such as gravity, is neglected; and so is that of 

 internal (molecular) forces. The number of spheres is regarded as extremely 

 great (say of the order 10 20 per cubic inch) : but the sum of their volumes is 

 regarded as very small in comparison with the space through which they are 

 free to move; as, for instance, of the order 10" 3 or 10~ 4 . It will be seen that 

 several of the fundamental assumptions, on which the whole investigation rests, 

 are justified only by reference to numbers of such enormous magnitude, or such 

 extreme minuteness, as the case may be. The walls of the containing vessel 

 are supposed simply to reverse the normal velocity of every sphere impinging 

 on them. 



I. One set of Equal Spheres. 



1. Very slight consideration is required to convince us that, unless we 

 suppose the spheres to collide with one another, it would be impossible to apply . 

 any species of finite reasoning to the ascertaining of their distribution at each 

 instant, or the distribution of velocity among those of them which are for the 

 time in any particular region of the containing vessel. But, when the idea of 

 mutual collisions is introduced, we have at once, in place of the hopelessly com- 

 plex question of the behaviour of innumerable absolutely isolated individuals, 

 the comparatively simple statistical question of the average behaviour of the 

 various groups of a community. This distinction is forcibly impressed even 

 on the non-mathematical, by the extraordinary steadiness with which the 

 numbers of such totally unpredictable, though not uncommon, phenomena as 

 suicides, twin or triple births, dead letters, &c, in any populous country, are 

 maintained year after year. 



On those who are acquainted with the higher developments of the mathe- 

 matical Theory of Probabilities the impression is still more forcible. Every one, 

 therefore, who considers the subject from either of these points of view, must 

 come to the conclusion that continued collisions among our set of elastic spheres 

 will, provided they are all equal, produce a state of things in which the per- 

 centage of the whole which have, at each moment, any distinctive property 

 must (after many collisions) tend towards a definite numerical value ; from 

 which it will never afterwards markedly depart. 



This principle is of the utmost value, when legitimately applied; but the 

 present investigation was undertaken in the belief that, occasionally at least, its 

 powers have been to some extent abused. This appears to me to have arisen 

 from the difficulty of deciding, in any one case, what amount of completeness or 

 generality is secured when the process of averaging is applied in successive 

 steps from the commencement to the end of an investigation, instead of being 

 reserved (as it ought to be) for a single comprehensive step at the very end. 



