1886.] Professor Tait on the Kinetic Theory of Gases. 
21 
The following Communications were read : — 
2. Astronomical Tables for facilitating the computation of 
Differential Refraction, for Latitudes 56° and 57° ‘30. 
By the Hon. Lord M‘Laren, 
3. On the Foundations of the Kinetic Theory of Gases. 
By Professor Tait. 
In a former paper, printed in Trans. Roy. Soc. Edin ., 1886, I 
showed that the recovery of the “ special ” state by a gas supposed 
to consist of equal hard spheres takes place, at ordinary pressures 
and temperatures, in a period of the order of 10" 9 seconds, at 
highest. 
This forms the indispensable preliminary to the present investiga- 
tion. For it warrants us in assuming that, except in extreme cases 
in which the causes tending to disturb the “ special ” state are at 
least nearly as rapid and persistent in their action as is the tendency 
to recovery, a local “ special ” state is maintained in every region of 
the space occupied by a gas or gaseous mixture. This may be, and 
in the cases now to be treated is, accompanied by a common 
translatory motion of the particles (or, of each separate class of 
particles) in the region — a motion which at each instant may vary 
continuously from region to region, and may in any region vary 
continuously with time. 
A troublesome part of the investigation is the dealing -with a 
number of complicated integrals which occur in it, and which (so far 
as I know) can be treated only by quadratures. All are of the form 
/' c ° vyV 
J 0 T ’ 
where v is that fraction of the whole number of particles of one 
kind per cubic unit whose speeds (relatively to those of the same 
kind, in the same region, as a whole ) lie between v and v + dv, 
and 1/e is the mean free path of a particle whose speed is v. 
Throughout the paper regard has been had to the fact that e must 
