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§19. First: The results obtained in Problems 1 and 3 are repre- 
sented by Figures 1 and 2 respectively. The results in Problem 2 agree so 
well with those in Problem 1 that a graphical representation of the former 
would not have been distinguishable from one of the latter, i.e. from Figure 
1 (except for the halving of tT, i.e. the doubling of 5 ). 
In each figure the abscissae are the @,and the ordinates are the 
Sk The full lines, originating at ¢=0O and qe=l,2,-+\ @,-2, 
&,-{|, are the world lines of the corresponding "molecules". The dash-dash 
lines represent the main hydrodynamical features, that is, the loci where the 
rigorous, hydrodynamical solution places them according to (A)-(C) in §17: 
the line originating at the lower left corner is the shock, the two lines 
originating at the lower right corner are the front and the back of the 
Riemann rarefaction wave, and the refracted front of the rarefaction wave on 
the shock is also indicated. 
Second: Both figures show that the initial motion, which is a 
family of parallel straight lines in the Q,§ -plane, is significantly modi- 
fied only when the shock or the front of the rarefaction are reached. These 
are the true signals in the sense of (A) in §15; the false signals are at 
thei dines! Sis.a@. or So = @5- a i.e. well ahead of the true ones, and at 
the false signals nothing visible happens. (The numerical material shows 
this in more precise quantitative detail.) Also, the change of direction at 
the shock is rather sudden, while that at the rarefaction is gradual and con- 
tinuous. Summing up, the criterion of (A) in §15 is satisfied, and even the 
details of the compression and the expansion caused by the two walls are those 
which the rigorous, hydrodynamical solution leads one to expect. 
Third: The numerical material shows that in all three problems the 
shocks are followed by oscillations of a more lasting nature than those which 
