124 
Proceedings of Royal Society of Edinburgh, 
SESS. 
we suppose the spheres to collide with one another, it would be 
impossible to apply any species of finite reasoning to the ascertain- 
ing 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 complex question of the behaviour of innumerable abso- 
lutely isolated individuals , the comparatively simple statistical 
question of the average behaviour of the various groups of a com- 
munity. 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 mathematical 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 
percentage of the whole which have, at each moment, any dis- 
tinctive property must (after many collisions) tend towards a 
definite numerical value ; from which it will never afterwards 
markedly depart.” 
“When [the final result, in which the distribution of velocity- 
components is the same for all directions] is arrived at, collisions 
will not, in the long run, tend to alter it. Eor then the uniformity 
of distribution of the spheres in space, and the symmetry of dis- 
tribution 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.” 
2. ISTow, considering the 3.10 20 absolutely equal particles in each 
cubic inch of a gas, where could we hope to find a more perfect 
example of such a community ? Where a more apt subject for the 
application of the higher parts of the Theory of Probabilities^ If 
we are ever to find an approach to statistical regularity, it is surely 
here, where all the most exacting demands of the mathematician 
are fully conceded. 
