and the Partition of Energy in Continuous Media. 239 



each particle has, on the average, kinetic energy -JUT, and 

 each element of stretched string has, on the average, potential 

 energy JRT. In the normal state, these kinetic and potential 

 energies are distributed at random, and without correlation, 

 about the common mean- value ^RT. 



In the present investigation, however, we are concerned 

 with expressing the energy of the normal state in terms of 

 the energy of trains of waves. 



It is readily found that the free vibrations, subject to 

 # =0, # n+ i=0, are given by 



*»=^^(?=l,2,...n), • • • (12) 

 and the frequency of this vibration (j:>) is given by 



p=2s/t sin ^ (13) 



V m 2(n + l) v 7 



If £i> f 2? • • • f » are the various principal coordinates, we 

 may, from equation (12), write 



a.Ji&sin-f^ (14) 



Expressed in terms of the principal coordinates, equations 

 (10) and (11) become 



2T = im(n + l)& + &+... + &), 

 2V^2^ + l)(^sin 2 ^^ + ^sin 2 ^^ + ... + ^sin^, 



20 + 1) bJj 2(> + l) '•■■'*■ 2 n + 1) 



We can now express the energy of the " normal state " in 

 terms of the energies of free vibrations, or of trains of waves. 

 Since each term in T and V has an average amount of energy 

 iRT, it follows that each free vibration has average energy RT. 

 From equation (12) it follows that the wave-length X of the 



2 

 qt\\ free vibration is — X (length of system). Hence the 



number of free vibrations for which X lies between X and 

 X + dX is 2X~ 2 dX per unit length of the system. The energy 

 per unit length, of wave-length intermediate between X and 

 X + dX, is accordingly 



2RTX- 2 dX (15) 



This is the one-dimensional analogue of formula (9). It 

 represents the energy of the random distribution of kinetic 



