﻿836 Messrs. C. Gr. Darwin and R. H. Fowler on 



composed of groups of systems ; let there be M r systems 

 which have a partition function f r . This means that i£ the 

 possible conditions of one of these systems are that it should 

 be able to have energies € r> i, e r ,2, .. , and associated with 

 each of these states there is a weight factor p r ,i)Pr,2, . •• > 

 then 



or the limit of this expression if, as for mechanical as 

 opposed to quantized systems, all the dimensions of the 

 cells must tend to zero. In order to allow for changes 

 of condition other than those of temperature, we must 

 suppose that each e rtt is a function of the parameters 

 %i, #2, . . . ; for example, in the case of free molecules in 

 a vessel the wall may be represented by a local field of 

 strong repulsive force, and then the potential of this 

 repulsive force must be contained in € r>t . With these data 

 we find at once by (2*3) that 



E = 2M^~\og/ r , .... (7-11) 



and the average number of the rth group of systems in their 

 tth cell is by (2'5) 



a r , t = M rPrtt ^,tjf r ^ } a. u x%> ...). . . (7-2) 



We also require to evaluate the external work done by the 

 assembly in any small displacement represented by small 

 changes in the parameters. Now the potential energy due 

 to the external bodies is contained in e r ,t, and it will give 

 rise to reactions on the external bodies. If the positions of 

 the bodies are defined by the parameters a? 1? x^ ... , the 

 reactions will be a set of generalized forces of amounts 



A A 



for each single system of group r in the tth cell. The total 

 generalized force tending to alter the parameter x± will 

 thus be , 







Zr, t^r,t\ — S — e r, t h 

 \ 0#i / 











d 



its mean value will be 













Xl 



= ^.*v(~^*v'} 















\ OXi 



i)/m 



*i, 



a? 2 ; 



•■•), 







= lo*V^ M ' logm 



#i, x 2 , 



••)• 



• 



(7-3) 



