RECENT STATISTICAL THEORIES 699 



try out the new statistics upon a material gas, would there be any 

 way of avoiding the inadmissible conclusion that the number of atoms 

 in such a gas is also absolutely fixed by temperature and volume? 



There is such a way. One might replace the distribution proposed 

 in equation (46) by a more general one involving a disposable constant 

 B, as follows: 



Zu = a,e-'^-'^'"^^. . (54) 



On substituting this into the expression for {k log W) we get instead 

 of (47) the equation: 



b{k log W*) = - -feLCl + log a.)E5Z,. + kBZZiSZ^s 



s i si 



-\--^ll^sY.i^Zis. (55) 



■>- s i 



The right-hand member must as before reduce to bEjT if the distri- 

 bution (54) is acceptable; and this it will do, provided that not only 

 X!i5Zjs but also ^sUd^Zis is zero. Now the second of these quanti- 

 ties is zero for all variations in which the total number of particles 

 remains the same. The distribution (54) enjoys a greater entropy 

 than any other which is compatible with the same total energy and 

 the same total number of particles. The distribution (46) was still 

 more exalted; it enjoyed a greater entropy than any other compatible 

 with the same total energy, even including those for which the total 

 number of particles was somewhat different. But the distribution 

 (54) is sufficiently distinguished to be qualified as the most probable 

 distribution for a material gas. It seems rather singular that the 

 distribution (46) is required for radiation-gas. Here is evidently one 

 of the deep differences between matter and radiation. 



Following the same routine as before, we arrive at the following 

 expression for the number of particles in the shell s\ 



^I^ = Q^ ,b^Jt _ 1 ' (56) 



and in dealing with radiation-gas we have put B = Q and have taken 

 the value of Qs from equation (37). If in dealing with a material 

 gas we take the value of Qs from equation (25) instead and put // 

 = Jfj V we obtain : 



Ms = j^, {2.my^ J::!,lrt\ - F(^^)de„ (57) 



and now it is obvious that we must evaluate B in terms of the total 



