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tant for our purpose: §, always runs in the same direction as §° and &,. 
Where the waves run in opposite directions one has: 
5, hf (u) 5; == (v) 
and ton 6an 
DE a ” at a7] 1 
=—F(P MPL © + A OF, @)) 
du dv 4 
which gives integrated 
RE haces 4 ; : 
Ee nn Bi ij (w) sf 1 (v) EF J. 0 («) f, (v)) + Js (wu) => Js (v) 
Now we imagine &, and §, given for a limited part of the 
bar and therefore restricted to it. Let for t=O the wave &, only 
differ from zero between definite positive limits for w, and §,, also 
between negative limits. Then the waves will meet. If this is 
the only wave motion on the bar for =O, then in &,, f, 
and f, must be zero. Therefore §, will remain zero until the 
waves f, and /, will cover each other. Only for such values of 
«,, for which #, as well is f, — and consequently also their 
derivatives — differ from zero, there is a certain value for §,, and 
when after some time the waves have entirely passed one another, 
§, will be again zero everywhere. So waves in different directions 
run over each other without any exchange of energy. 
From what we have found it follows immediately that the linear 
body will show no fall of temperature even if a heat-current passes 
through it. If namely the heat-motion is dissolved into a great 
number of progressive waves, the presence of a heat-current will 
signify that the waves in one direction have on an average a little 
more energy than those in the opposite direction. The mean energy 
of either of tbe two wave motions however will be the same for 
all points of the bar. The co-operation of the waves running in the 
same direction does not in the least change this, for the arising 
combination-waves run in the same direction. Here we may also 
speak of the co-operation of a wave with itself. That also 
gives a combination-wave in the same direction. Such is the effect 
of the terms we repeatedly neglected. Consequently on our bar there 
is no fall of energy at all, in other words no heat-resistance. 
5. For the present, as for the extension of what we discussed 
in points 3 and + to the three-dimensional problem and the further 
application to a real solid body, we are obliged to limit ourselves to a 
few remarks. Let us first consider the effect of statical deviations such 
as in 3. The first method given there has already been applied by 
Degre in exactly the same manner to a solid, however for longitudinal 
