1329 
must be regarded as incoherent in phase, products of this kind 
will vanish from the mean value of A,. So we obtain the right 
result, if we put | 
A, = 3 (a, gy 36 a, Us. ln Sif pn d's q's) 
Therefore, according to (9), each harmonic mode of vibration 
contributes its part to the dilatation v—v,, independently of the 
other modes. 
The first of these parts is 
GI é # a, qi) 
for which we may write 
1 0, 1 2 1 I 2 
mak dad PAG a, + $94,9,’) 
g 
or on account of the connection between q and the volume 
0 
aU eG Ardy hk Os Oa) EN dn de (10) 
Now ia, 4,’ is the value of the potential energy u, that belongs 
to the first coordinate during the heat motion in the state considered 
and ta, q,* + $ga/,g,° the value which this potential energy would 
have, if after the increase in volume determined by gq the particles 
had the same deviations determined by g,, from the positions P, P’, P",... 
specified in $ 2. 
Thus we may write for (10) 
du, 
Òv 
To calculate the differential coefficient we must attend only to 
the first coordinate g,, putting O for all the others. 
Further, in performing the differentiation we must imagine that 
in the original volume v the particles have the deviations from their 
positions of equilibrium which, in the real heat motion, correspond 
to the first mode of vibration and that, after an infinitesimal increase 
of the volume they have the same deviations from the new positions 
of equilibrium P, P’, P’,... 
Proceeding in the same way with respect to the other coordinates, 
we obtain 
Me Ow, Ou, Ous 11 
enmet +5) e e . ( ) 
oe CY) 
§ 6. The calculation of the thermal dilatation by means of this 
formula will necessarily be a rather rough one. In the first place 
it is very questionable whether for somewhat high temperatures we 
