1298 
state of motion that is to say: integrate that expression with 
respect to a long time 7’ and divide the result by PT?) The 
contribution of the first term can be made as small as one wants 
by taking 7’ larger and larger. So there remains: 
de 08 (fo 
ene tn de bee aeg (2) 
Ox Òz 2 
The average energy is thus the same for all points of the bar if 
au—( i.e, if Hooke’s law is satisfied, and this quite independent 
of the heat-current: 
ren dn 
kB ENEN ae 
Now, there appears to be a connection between the average (3) and 
the same in (2), causing for a— 0 a difference in temperature 
proportional to the magnitude of the heat-current. However, that 
this connection does not exist appears in another way from the 
following considerations. 
3. In direct connection with the treatment published by Degre 
(l.c. § 9) we can examine the influence of deviations of density 
and elastieity on the vibrating movements of our bar. With density 
o and elastic modulus # the equation of motion becomes: 
PEs. Bien ; 
ar = asl al he loge ee nd 
as in this method the terms of higher order are not directly 
taken into account. Further we suppose that g and £ have every- 
where on the bar the constant values 9, and Z, except on a small 
part, from 0 to /, where they are g, + 9, and E,+ £,. Then we 
can calculate in the usual way the secondary waves, appearing when 
primarily a wave Aeir@-%@ runs along the bar, for which 
E 
0 
y= = 
Qo 
is the velocity of propagation. We shall only mention the result. 
From the disturbed little element O there goes in negative direc- 
tion to l a wave 
E E 
iAlp E00, Aiko OPER: vre dn been Gen 
24.0, 
and in positive direction a wave 
o, L,—£,0 
ee BE 
Et SRE gl at (°) 
ovo 
This latter one has to be added to the primary wave. This gives 
