1292 
a given bar with given conditions to be satisfied at the ends. But 
if c, is small enough, the normal vibrations of the equation without 
quadratic term will still have a physical meaning. These vibrations 
may be called quasi-normal vibrations, and the physical meaning of 
the constant c, is to effect a slow exchange of energy between the 
quasi-normal vibrations. 
Now take the solution 
S= Zr (Fe sin ka + Qr cos ke) cos kv (t — ge) . - « (15) 
for a bar with the ends 0 and 22; v being the velocity of propa- 
gation. The force in a point is represented by (7). Using (15) and 
calculating the time-average of S in the point 0, we find 
ues c? 
S= Ek P;? 
4 
The potential energy is expressed by 
2m Cc, 7 : : 
ui = k? (Pi? + Q°) cos? kv (t — or) 
its time-average by 
2% C, 
= ht (Pe? + Qr°) 
Ey == 
Now the mean value of P« is the same as that of Q;; therefore 
we gel 
2% c, ne 
&g = —— 2 k* Px 
4 
re Os arke 0E 
2 2 
är C 2c, av 
& 
As ae is equal to the energy per unit of length, the result 
47 
agrees with (5) and (12). 
4. We can determine the thermal pressure of an isotropic 
solid body in the same way as in 2. For this case, we have for 
the potential energy per unit of volume the expression *) 
EADS BIG WOT EDM IEN a EE 
where the invariants / have the following forms 
er et ee 
1) For the first time indicated by J. Finger, Wiener Sitzungsberichte 103, 163 
(1894), although in a less simple form (l.c. form (55)). Our notation is the one 
of v. Everpincen, l.c p. 11, where no literature is mentioned. Cf. also P. Dunem, 
Recherches sur |’Elasticité, Paris 1906. 
