CAPACITY FOR HEAT OF METALS AT LOW TEMPERATURES. 
345 
identical with elastic vibrations. By applying the classical theory of elasticity to the 
vibrations of a homogeneous isotropic sphere in the case where the pressures on the 
surface vanish,* Debye determined the number of characteristic types of vibrations 
up to or below a given frequency v, and obtained the result that 
z — v 3 vY, .(ll) 
where 
z is the number of sets of “ standing waves ” as in the analogous problem in 
“ radiation,” 
v, the volume of the sphere, 
F, a quantity calculated from the elastic constants and given by 
where 
F 
4:7T 
~3 
cq is the velocity of dilatational waves, 
a 2 ,, ,, torsional waves. 
( 12 ) 
Lord Rayleigh! has shown that a coupled system of N mass points with 3N 
degrees of freedom may be regarded as giving rise to 3N types of vibrations. 
Taking the N atoms in unit mass of a solid as such a system, Debye ascribes to 
each of these types of vibration the mean energy — 
1 
of a linear oscillator. 
The next assumption is the characteristic one of the entire theory. He applies 
equation (ll) deduced for a continuous medium in which an infinite number of 
vibrations is possible to the system of N atoms in which only 3N vibrations are 
possible. 
Since the form of the body considered has no influence on the result expressed in 
equation (ll) for unit volume, we have 
= 3N = .(13) 
where v m is the limit frequency. 
And the number of “ standing waves ” in the interval dv denoted by dz is given by 
dz = 3vF dv. 
(14) 
Hence the total energy of the IN atoms is 
E 
9N 
hv 
N 
hv 
-1 
. v 2 dv, 
(15) 
* A condition which is regarded as corresponding to the case of a sphere in thermal equilibrium with 
the outer surroundings, since no energy crosses the surface, 
t “ Theory of Sound,” vol. 1, p. 129, 1877. 
2 Y 
VOL. CCXIY.-A. 
