418 
MR. J. H. JEANS ON THE DISTRIBUTION OF MOLECULAR ENERGY. 
Here, as throughout the paper, a single tall integral sign denotes integration over 
all values of the variables, of which the differentials occur after the sign of integration. 
For those values of x'\ y", z", for which X is different from zero, we have seen that 
is, for every co-ordinate ^ of the same order of small quantities as d^, and hence 
it follows tliat 
\ A [Xe-” ¥¥']//' dp du dp' dP' dll' dx" dy" dz" 
t. ' 
is of the same order of small quantities. 
In the limit, when the differentials such as r/f are supposed to vanish, this integral 
will vanish also, so that we may put X = 0 in expre.ssion (vii.). 
The total gain to Nq from all encounters is therefore F/P, where 
I = ^Afe-” FF'Ij^^' dp du dp dP’ da dx" dy" dz" .... (viii.). 
§ 21. Now we have seen (§ 19) that the change in Nq in time dt will arise from 
three causes— 
(a) the change arising from collisions, 
(/3) the change arising from encounters otlier tlian collisions, 
(y) the change arising from the change in the co-ordinates of the various mole¬ 
cules during the time dt, which would occur if there was no interaction 
lietween different molecules. 
Now any molecule. A, will collide with a second molecide, B, in the interval of 
time dt, provided that at the beginning of this time the centre of A lies within a 
certain region of space ; provided, tliat is, that x" y" z" lie within certain limits. It 
is easily seen that these limits are arrived at by writing dx" = u" dt, and allowing 
for y" z" a range of values corresponding to points inside a certain circle of diameter 
equal that of a molecular sphere. Hence the increase in Nq arising from cause (a) 
will be Jc/P dt where 
J = \ Ale-" PP']ff' dp du dp' dP' du' u" dy" dz" .... (ix.), 
and A denotes the Increase due to a collision. 
The Increase arisino- from encounters other than collisions has alreadv been found 
to be h/P (see viii.), where A must denote the increase due to an encounter of which 
the duration is dt. Since dt is to be very small, we may replace A by dt H/D/, 
where D Di denotes the rate of increase under tlie influence of intermolecular forces 
only. The increase to Nq arising from cause (^) will therefore be Kc/Pt/h where 
K 
[e ’’FF'] ff dpdu dp' t/P 'du' dx" dy" dz 
(x.). 
