ON OUR KNOWLEDGE OF THERMODYNAMICS. 77 



in addition to the equation of energy, any number of integrals 7ti=const., 

 7<,^=const., itc, and if at any given instant the systems are so distributed 

 that the number of them whose co-ordinates and momenta lie within given 

 small limits is proportional to any function whatever of E, h^, h^ . . ., 

 then the distribution will be permanent, the systems being similarly distri- 

 buted at every subsequent instant of time. 



(ii) If the kinetic energy be expressed as the sum of squares, and if 

 the frequencj'-function involves no integrals of the equations of motion 

 containing velocities or momenta with the exception of the energy, then 

 the mean values of the different squares are all equal to one another. 

 [3faxiceU's Law of Partition of Kinetic Energy.^ 



I 



SECTioJf II. — Systems of Colliding Molecules. 

 Applicability of the Preceding Investigations to Gases, 



3.0. Before considering in detail those investigations in which col- 

 lisions and encounters between the molecules of a gas are taken specially 

 into account, it may be well to examine briefly how far the general results 

 established in Section I. can be applied to the problem of the Kinetic 

 Theory of Gases (see also Boltzmann's Appendix, infra). 



The ' independent systems ' considered above may be chosen in several 

 different ways. 



(i) We may take each ' system ' to represent a single molecule of gas 

 moving about freely or in a field of external force. The above investi- 

 gations will apply so long as the molecules considered do not encounter 

 or collide with other molecules, and we conclude that in the absence of 

 such encounters any distribution determined by the expression (14) of 

 § 15 will be permanent if the 2n quantities 7? , , . . . q,^ represent the 

 n momenta and n co-oi'dinates of a single molecule. 



This is most important. It is not sufficient in the Kinetic Theory 

 to investigate a law of distribution which is unaffected by collisions or 

 encounters any more than it is sufficient to investigate a law which is. 

 permanent in their absence. It is necessary to satisfy conditions of 

 permanence in both cases. 



(ii) We may take each ' system ' to represent a pair of molecules or a. 

 group of several molecules in the course of a binary or multiple encounter, 

 it being assumed that the intermolecular forces remain finite during 

 encounter, and that at each instant there are sufficient encounters of the 

 same kind to give rise to a law of permanent distribution among the 

 encountering sets of molecules. Here the quantities g,, . . . q,^ will 

 have to include all the co-ordinates of all the molecules in the group, 

 considered. Then any distribution determined by (8) will be permanent 

 so long as the molecules of any one group do not encounter any molecules 

 not in that group. 



But in a gas each molecule will encounter various molecules in suc- 

 cession, so that the same set of molecules cannot be considered perma- 

 nently as a system apart from the rest. From this we find at once that 

 if the frequency of distribution i is a function of the energy alone ^ it 



' This is not the case if the mass of gas has a perceptible motion of translation 

 or rotation (see § 45 and Appendix B). 



