MR. J. H. JEANS ON THE DISTRIBUTION OF MOLECULAR ENERGY. 
401 
§ 5. In calculating the nnmher ot collisions of tliis kind wliich are to be expected 
in the interval of time dt, a consideration enters, which does not enter in the simpler 
case in which the sjiheres are symmetrically loaded. 
From the co-ordinates of the two molecnles just before collision, we can trace back, 
as far as the previous collisions, the paths by which the molecnles arrived at tliis 
position. If these j^aths are such that the spaces occupied by the two molecules, at 
any two corresjDonding points of these paths, are found to overlap:), then it is clear that 
a collision of the kind we are investigating can only occur, either when the same two 
molecules have previously collided, or when one of them has collided with a third 
molecule within a certain small interval previous to the collision in question. In 
either case it would be wrong to calculate the probability of such a collision upon the 
assumption that the molecules of the gas are, in Boltzmann’s sense, nngeordnet. 
When, however, terms of degree higher than v~ are neglected, it will be legitimate 
to ignore this consideration altogether. For the number of collisions to which it 
applies will vanish with r, so that if equation (i.) be summed over all collisions, the 
terms on the right-hand side which are influenced by this consideration will be of 
a higher order in r than ?•“, and may accordingly he inaccurately calculated, without 
invalidating the result as far as terms in r^. 
§ 6. When we agree to ignore this consideration, we may at once average equation 
(i.) over all values of the six variables of orientation. The probability of these 
variables having specified values at a collision is not independent of the velocities 
of the collision, but will be the same for all collisions such as we are now considering, 
in wlilch these velocities have specified values. In this way we find that tlie mean 
increase in c~ -b c'-^ at a collision at which tlie velocities are u, i\ w, ct, u, v\ w', ct', is 
of the form 
r'[a 2 Y“ /lo (ct" "k -f- terms of a liigher order in . . . (ii.), 
in which a,,, are constants. 
Now if we suppose that the gas has reached its present state through a series 
of natural processes, the law of distribution of velocities will depend only on 
c" and crh In the case in which r = 0, this law is known to be 
e~'''""y(77T) dll dv div drs . (ii'-)* 
Hence in the case in which r is small, it may be taken to be 
Y (c,-ni) da dv div drs . 
where F is a function of which the coefficients involve r, Init is such that (Iv.) reduces 
to (iii.) when r ~ 0, 
* Direct calculation shows that the values of ao, /Jo are ao = - /Jo == where k is the radius of 
OK“ .O 
gyration of a molecule about a line perpendicular to the axis of symmetry. 
VOL. CXCVI.—A. d F 
