﻿476 Messrs. G. Gr. Darwin and R. H. Fowler on 



definition of an integral, in the limit A($)-»H($), where 



H(*)= ^e-^W^d* ... d n . (12-3) 



The integration is over the volume V and over all values of 

 the p's from -co to + co . This gives at once 



(27nn)3/*V 



In the formulae the functions dh/dS and d 2 h/d§ 2 also occur, 

 and it is easily shown directly that their limits are dH/d$ 

 and d 2 K/d$ 2 . We may therefore use (12*4) throughout for 

 the real assembly; and at once obtain the following ex- 

 pressions : 



ET c= g|e/5 =* P * T ' • (12 ' 5) 



_ _ p ^niogjiy * e _ i{iog 1/S) „ ( „ 2+o2+K2) dw dy dz du dv dw ^ 



(12-51) 



= P(^JV^<" 2+ ^>&....^. . . . (12-52) 

 The temperature $ is determined by 



E = M4log/(3)+fP— L^, . (12-53) 



and the fluctuation of energy of the molecules in a bath of 

 temperature S is given by 



(E -E c )?=a^E c =fPOT . . (12-54) 



If the fluctuation of c^ is evaluated, it takes the simple value 



(e t -Z t y = c ( , (12-55) 



whether it is in a bath or not ; for the second factor, 

 analogous to that in (8*4), can be omitted when the cell is 

 taken to be of small size. Thus in all cases the possession 

 of c t is a normal property of the assembly. These results 

 can be readily extended to cases where there is an external 

 field of force acting on the molecules. 



By means of this assembly we can establish the meaning 

 of 3 in terms of T, by observing that the gas itself constitutes 

 a constant volume gas thermometer. It is easy to show that 

 the pressure of a gas must be f of the mean kinetic energy 



