1331 
in it and proper to a mode of motion of the first group, is proportional 
to an expression of the form 
A+ 2wyte, | 
while the potential energy v‚ belonging to a mode of the second 
group is proportional to 
uy oo, 
As w changes proportionally to v, we have in virtue of (13) 
dlogv,  ‘dlog (A+-2u) 
0 log v dlogv— 
35 
Òlogv,  d d log u 
1 
3) 
dlog v T dlog v 
and this leads to similar relations for the potential energy 2, u con- 
tained in the whole body. We may write them in the form 
Ou; dlog(A+2u) 
RE TES , . . . . 14 
0 log v | d log v ye 3 oe Cz 
0 dl 
poe = ae | tees cdr Takee,/ ZEEE 
0 log v d log v 
These formulae, of which the first i be used for all the terms 
in (11) that correspond to longitudinal motions and the second for all 
those that refer to transverse motions, also hold for the mean values 
which we have to take on the right hand side of (11). The mean 
values both of w; and of uw, however are each half the total energy, 
and to this latter we must assign, both for the longitudinal and the 
transverse vibrations, the value ¢, which depends on the frequency 
vp in the way specitied in PLanck’s formula. 
§ 8. Let us now first consider the terms on the right hand side 
of (11) that belong to modes of motion with frequencies between 
p and r+». Let N be the total number of these modes, gN 
the number of those in which the vibrations are longitudinal and h/ 
the number of those which consist in transverse vibrations, so that 
g+h=1. To obtain 
du 
dlogv * 
for this group of terms we must multiply (14) and (15) by gN and 
hN respectively and then take the sum, replacing at the same time 
u, and w by their common value 4¢,. We shall also substitute for 
g and A the values that follow from Derpyr’s calculations. He has 
found that the number of the longitudinal and that of the transverse 
modes of motion for which the frequency lies below an arbitrarily 
85 
(16) 
Proceedings Royal Acad. Amsterdam. Vol. XIX. 
