176 THERMODYNAMIC ANALOGIES. 



closely together that the most probable division may fairly 

 represent the whole. This is in general the case, to a very 

 close approximation, when n is enormously great ; it entirely 

 fails when n is small. 



If we regard dcfr/de as corresponding to the reciprocal of 

 temperature, or, in other words, de/d(f> as corresponding to 

 temperature, < will correspond to entropy. It has been denned 

 as log (d V/de). In the considerations on which its definition 

 is founded, it is therefore very similar to log F". We have 

 seen that d(j>/dlogV approaches the value unity when n is 

 very great.* . 



To form a differential equation on the model of the thermo- 

 dynamic equation (482), in which de/dcf) shall take the place 

 of temperature, and < of entropy, we may write 



da * + etc -> ( 489 > 



or <Z*= de + da 1 + da 2 + ete. (490) 



de da-L da 2 



With respect to the differential coefficients in the last equa- 

 tion, which corresponds exactly to (482) solved with respect 

 to drj 9 we have seen that their average values in a canonical 

 ensemble are equal to I/, and the averages of A l /, A 2 /, 

 etc.f We have also seen that de/dcfr (or d(f>/de) has relations 

 to the most probable values of energy in parts of a microca- 

 nonical ensemble. That (del da^)^, etc., have properties 

 somewhat analogous, may be shown as follows. 



In a physical experiment, we measure a force by balancing it 

 against another. If we should ask what force applied to in- 

 crease or diminish & x would balance the action of the systems, 

 it would be one which varies with the different systems. But 

 we may ask what single force will make a given value of a^ 

 the most probable, and we shall find that under certain condi- 

 tions (de/da^Q, a represents that force. 



* See Chapter X, pages 120, 121. 



t See Chapter IX, equations (321), (327). 



