1330 
may confine ourselves to terms of the second order with respect 
to the inner coordinates, and even if this were allowed, the difficulty 
would remain that we do not know enough about the forces acting 
between the particles to calculate the differential coefficients ef 
Vv 
For the modes of vibration in which the wave-length is many 
times greater than the distances between neighbouring particles, 
these forces, so far as they have to be considered here, are determined 
by the ordinary elastic constants. If, however, the wave-length 
becomes of the same order of magnitude as those distances, this of 
course will no longer be so and unfortunately these very short 
waves are most prominent in the heat motion. 
In his theory of specific heat however Desyk, not withheld by 
this consideration, has applied the ordinary formulae of the theory 
of elasticity to all the modes of motion with which he was 
concerned, down to the shortest waves. Encouraged by his success 
we may avail ourselves of the same simplification in the theory of 
dilatation as has been done already by him and vaN EvErRDINGEN. 
This enables us to continue the calculation of the right hand side of (11). 
$ 7. We shall introduce the two constants of elasticity 4 and 
u, which are also used by Degre and which have been chosen 
in such a way that the potential energy per unit of volume is 
represented by the expression 
uw (ar? + yy? + 22") +44 (ex + Vy + zz)? + $m (a? + y2 + 22’) (12) 
where 
el OS bi, PEs ey 
fe aen ON ay HRD Ae 
We remark that, if y is any one of these six components of 
strain, or a homogeneous linear function of some of them, we may 
write 
PVK AE pa Nn 
This is evident, if we keep in mind that, in the infinitesimal 
expansion determined by g, the quantities §,7,$ are kept constant, 
so that their differential coefficients with respect to the coordinates 
are changed in ratio of 1 to (1+ 4q)7?. 
The modes of vibration of which the heat motion consists, may 
be divided into two‘ groups, that of the longitudinal and that of 
the transverse vibrations. 
Now, if w is an element of volume, the potential energy v, contained 
