.AIR. J. H. JEANS ON THE DISTRIBUTION OF MOLECUEAR ENERGY. 
414 
no force excej:)! its own internal forces for a time dt, and let all the collisions 
^\•llicll would occur in this time he supposed to occur. After this imagine the 
molecules divided into pairs in every possible way and sujDpose an encounter to 
occur bet’s^'een the two molecules of every such pair. The duration of the encounter 
is to be dt ; during it the co-ordinates of each molecule are to change only on account 
of the forces of the encounter; that is, on account of the intermolecular forces 
existing between the two molecules under consideration. These encounters are to 
take place consecutively, not simultaneously. 
It is easily seen that each molecule has now been acted upon by exactly the same 
forces by which in the actual course of events it would have been acted upon in the 
interval dt. Hence, since there is no limit to the smallness of dt, the final state of 
the gas is independent of the order in which this series of events takes place, and is 
identical with the state in which the ^as would have been found if the forces had 
acted simultaneously. 
Two points deserve attention in connection with this argument. Firstly, it might 
be objected that the changes in the co-ordinates of molecules which exjoerience an 
actual collision are not additive, inasmuch as one of these changes is not infinitesimal. 
It is, however, clear that there is no necessity to take the infinitesimal changes into 
account at all in the case of these molecules, for the number of these molecules 
vanishes in comparison with the total number when dt is made to vanish. Secondly, 
it is true that the number of molecules within any specified limits will not always 
consist of the same individual molecules. But it is a fundamental assumption of the 
kinetic theory that any N molecules which have nothing in common except that 
certain co-ordinates have specified values, will behave exactly like N other similarly 
conditioned molecules. 
We can therefore reduce the continuous chano-es of the co-ordinates of molecules 
O 
whicli arise from the action of intermolecular forces, to a series of encounters of the 
kind described in § 18. 
§ 20. To completely specify an encounter, we require the values of all the co-ordi¬ 
nates enumerated in § 17, of both molecules. It will, however, be convenient to 
write 
t 
X =. X — X, 
with a similar notation for y, z, n, v, w, and to specify a collision by the values of 
x", . . . instead of the values of a;' . . . 
As regards the law of distribution of the various co-ordinates, we notice that 
whatever the values of the y, r, s, and it co-ordinates may be, the probability that the 
■p co-ordinates lie within the limits dp may, in the absence of external forces, be 
taken to be fi{p)dp where 
f{p) dp = 1. 
Since the internal energy is only slightly 
changed by encounters, we shall again suppose that the distribution of internal 
