and the Partition of Energy in Continuous Media. 237 



of wave-lengths intermediate between X and \ + d\ (where X 

 is large compared with the scale of structure of the medium, 

 if the medium is coarse-grained) must be 



CX~ 4 d\ (8) 



per unit volume, where C is a numerical quantity which 

 depends on the structure of the medium. It follows that 

 when the medium is in equilibrium with matter at tempe- 

 rature T, its vibrational energy of wave-lengths intermediate 

 between \ and \ + d\ must be 



CRT\-m (9) 



per unit volume. 



If the medium is aether, it is easily found (§ 5) that the 

 value of C must be Sir, and formula (3) follows at once. It' 

 the medium is gaseous (so that all vibrations are longitudinal) 

 the value of C is 4-77% while for an elastic solid medium 

 C = 12tt. 



14. If the medium is structureless, then formula (9) holds 

 down to the very shortest wave-lengths. The energy corre- 

 sponding to any finite value of T is infinite. Whatever the 

 value of T, the whole energy (except for an infinitesimal 

 fraction) is confined to vibrations of infinitesimal wave-length. 

 In this case the value of the Law of Equipartition is not so 

 much that it gives the final state of the medium (a state 

 reached only after infinite time (cf. § 24, below)), as that in 

 shows the tendency for the energy to run into vibrations of 

 infinitesimal wave-length. Or, what is the same thing, it 

 shows the tendency for regular trains of waves to become 

 dissipated into irregular disturbances (subject to a certain 

 limitation, cf. below). 



15. If the medium is coarse-grained, then formula (9) is 

 not applicable, when dealing with waves of length com- 

 parable with the scale of structure of the medium. Let 

 X denote (loosely speaking) the smallest w r ave-length pos- 

 sible, so that A is a length comparable with the scale of 

 coarse-grainedness of the medium. From formula (9), the 

 total energy per unit volume of the medium is 



!A=A o\q 



If J is the mechanical equivalent of heat, the " specific 

 heat " of the medium must be 



CR 



3J\ * 



per unit volume. Denoting this specific heat by <t, and 



J 



