OF ENERGY IN A SYSTEM OP MATERIAL POINTS. 729 



expression is no longer approximate when V is nearly as great as E, and it 

 does not vanish, as it ought to do, when V=E, 



Hence when the potential energy of the system in the given configuration 

 is very small compared with its kinetic energy, we may use the approximately 

 correct statement, that the number of systems in a given configuration is 

 inversely proportional to the exponential function, the index of which is half 

 the potential energy of the system in the given configuration divided by the 

 average kinetic energy corresponding to each variable of the system. 



If we divide the system into any two parts, A and B, we may consider 

 V, the potential energy of the whole system, as made up of three parts, 

 V A and V B , the potential energy of A and B, each on itself, and W, that of 

 B with respect to A. 



When, as in the case of a gas, the parts of a system are in a great 

 degree independent of each other, the average values of V A and V s may be 

 treated as constants, and the variations of V will be the same as those of W, 

 so that the variable part of the exponential function will be reduced to 



w 

 e (57). 



If we suppose that A denotes a single molecule of a particular kind of 

 gas, and that B denotes all the other molecules, of whatever kind, in the 

 system, then, since there are many molecules similar to A, we may pass, from 

 the number of systems in which A is within a given element of volume, to 

 the average number of molecules similar to A which are within that element, 

 or, in other words, the average density of the gas A within that element. 



We may therefore interpret the expression (57) as asserting that the density 

 of a particular kind of gas at a given point is inversely proportional to the 

 exponential function whose index is half the potential energy of a single 

 molecule of the gas at that point, divided by the average kinetic energy 

 corresponding to a variable of the system. 



We must remember that since the centre of mass of a molecule is 



determined by three variables, the mean kinetic energy of agitation of the 



centre of mass of a molecule is three times the quantity K which denotes 

 the mean kinetic energy of a single variable. 



VOL. H. 92 



