744 
a gas can be divided into elements of volume (the dimensions of 
which are small compared with / and large compared with s) which 
have a certain individuality, each element dilating or contracting 
and exerting a pressure on the neighbouring ones, as is admitted 
in ordinary aerodynamics. Things begin to change already when 4 
is no longer very large compared with s. We must then take account 
of the phenomena that are caused by the intermixing of adjacent 
elements of volume. The viscosity and the conduction of heat, the 
effects of this intermixing, lead to a departure from the simple 
laws which hold for large wave-lengths. Stationary waves of 
a length even smaller than s, and yet following more or less the 
ordinary rules, are entirely out of the question. Indeed under these 
circumstances the greater part of the molecules that enter a layer 
of thickness 42 or $4 would traverse it without a collision. We 
cannot. say any longer that one layer exerts a pressure on an other; 
on the contrary, the molecular motion will cause a rapid mixing 
up of the layers. 
Now. the smallest wave-length 2, introduced by Lenz is not much 
greater than the distance d of the molecules, while the mean free 
path s can be a considerable multiple of 6. We therefore come to 
the conclusion that, of the modes of vibration which he considers 
in his theory, those with a wave-length near the lower limit 4, 
cannot really exist. 
SOMMERFELD *°) has tried to meet this objection by observing that 
neither at somewhat high temperatures, nor at very low ones we 
need fear considerable errors in Lenz's formulae. For high tempera- 
tures they agree with those which may be derived from the ordinary 
theory of gases and Lxnz’s equations show that at low temperatures 
the energy becomes more and more concentrated in the modes of 
vibration of large wave-length to which our objection does not 
apply. This is so indeed, but a simple calculation shows that it is 
not until the temperature is extremely low, that the greater part 
of the energy will have shifted to waves considerably longer than 4,. 
According to Lenz's theory the energy belonging to the modes of 
vibration with wave-lengths between 4 and / + d2 is given by 
1 dh 
Anhev ee 
2he Fie 
ekhT—] 
We shall use this expression to seek a certain: mean wave-length 
2 which we define by the condition that the energy corresponding 
1) Lc, p. 141, 142. 
