1332 
chosen limit are to each other in ratio of (4+ 2u)~‘* to 2u". As 
this is independent of » it is also the ratio between the fractions g 
and 4. Performing the calculations indicated we find for the sum (16) 
| d log (AH 2) let 20e 
=| 4 og (A42) "He +a |e. 
a d log v 
Òlogv 
To derive from this the sum 
Ou, Ou, 4 _Òus 
Òlogv _ Òlogv "05 Òlog v 
which oceurs in (11) we have still to extend the summation to all 
the modes of motion of different frequencies. As now = Ne, is the 
total energy E of the heat motion, (11) becomes 
d log (A+ 2u)—2+ 2u—*/s 
paola et g AH 2) + 2u ij ln 
d log v 
(17) 
In this formula we must give as well to x as to the elastic con- 
stants 2 and u the values they would have if there were no heat 
motion, the volume being v, and strictly speaking it ought to be 
taken into account that these quantities and therefore the coefficient 
by which Z is’ multiplied are more or less dependent on that volume; 
by this the equation becomes rather complicated. The simplest results 
will be obtained for very low temperatures. For these £ is propor- 
tional to 7. Hence, if we assume that the coefficient of may be 
represented by a series 
CAL No aha oe 
we may conclude that quite near the absolute zero v—v, is propor- 
1 dv 
tional to 7'* and the coefficient of dilatation AP io “a 2 
0 
§ 9. The equation obtained for the dilatation can be still further 
simplified if one makes the assumption, rather arbitrary of course, 
that by an isotropic dilatation the coefficients A and u are made to 
change proportionally to each other. The coefficient of compressibility 
(for an infinitesimal change of volume) which has the value 
3 
32 + 2e 
and with which, in a rough approximation, we may identify the 
coefficient x occurring in our formula, will then change in the 
inverse ratio to u. We may also say that the quantity of which 
the logarithm appears in the numerator of the first fraction in (17) 
changes proportionally to x‘. Hence, denoting the pressure by p 
and using the relation 
