70 AVERAGE VALUES IN A CANONICAL 



are also identical with those given by Clausius for the corre- 

 sponding quantities. 



Equations (112) and (181) show that if ty or ^r q is known 

 as function of S and x , a 2 , etc., we can obtain by differentia- 

 tion e or e q , and A ly A Zy etc. as functions of the same varia- 

 bles. We have in fact 



* = * f -i=:* f -e. (192) 



The corresponding equation relating to kinetic energy, 



which may be obtained in the same way, may be verified by 

 the known relations (186), (187), and (188) between the 

 variables. We have also 



etc., so that the average values of the external forces may be 

 derived alike from ty or from ty q . 



The average values of the squares or higher powers of the 

 energies (total, potential, or kinetic) may easily be obtained by 

 repeated differentiations of -\|r, ^, ^ p1 or e, e g , e^, with 

 respect to <t). By equation (108) we have 



c = J . . .J e <fe . . . dfc, (195) 



phases 



and differentiating with respect to , 



phases 



whence, again by (108), 



de _ ? \fe 

 d~~ 2 



