and the Partition of Energy in Continuous Media. 243 



where A is a constant. It follows at once that f Y 2 , f 2 2 , . . . 

 each have the same mean value, given by 



£=£=-£= 5 s - • • • < 21 > 



The next step is to find the potential energy of the trains 

 of waves f„ £ 2 , .., f„. If p is the equilibrium pressure, and 

 <j the condensation at any point, the potential energy V is 

 given by 



V = \j> \\^ a 2 dx dy dz, 



where Y is measured from the equilibrium configuration, and 

 all heat-energy is treated as kinetic. The condensation in 

 in the sth cell, sav <r s , is, by equation (19), 



so 



that 



a<. — a S. s co q ^ n i , . 0S7T 

 a s = = — = — 2 ]; q sin — 



a a ci oq -i n 



or, since p = Rj' T, 



^(6 , +B*+. ••&*), 



V=iRT-(f 1 3 +fe 3 +...|„ 2 ). 



Hence, by equation (21), each term in V has average 

 energy ^RT, 



19. We next consider the partition of kinetic energy. 



Suppose that of the N molecules, a number N' have 

 ^'-components of velocity which are intermediate between 

 uandu + ^u. In the " normal state " there is no corrella- 

 tion between velocity and positional coordinates, so that the 

 N' molecules will be distributed between the n ceils accord- 

 ing to the same laws as the N molecules in the analysis just 

 completed (§ 18). 



Of these N' molecules, let the numbers in the different 

 cells be b u b%, ... b n , and let rj^ rj 2 , ...rj n be given (cf. equa- 

 tions (.17)), by 



-=— + 2 ^ sin - y - (5=1,2,. ..n), . (22) 



co nco 7=i n v 



then the 77's will be distributed according to the law 

 {cf. equation (20)), 



A' e -*W+i-f+ - ^ d m di h ... d Vn} . . (23) 



where fc'=Q?/4:W. 



