340 
28 
accompany the rarefaction. In Problems 1, 2 these are too small to show on 
Figure 1, but in Problem 3 they are considerably greater and Figure 2 shows 
them accordingly. That figure makes it quite clear that the shock, but not 
the rarefaction, is followed by strong "thermic agitation" due to the degrada- 
tion of energy which is caused by the shock alone. 
Fourth: The values of vy obtained in (A), (B) in §17 for the re- 
gions behind the shock and the rarefaction can be compared with the compres- 
sion and the expansion shown on Figures 1, 2. The quantitative agreement is 
excellent. The world lines of individual "molecules" are also in good 
agreement with those obtained from the rigorous, hydrodynamical solution, if 
allowance is made for the post-shock oscillations. 
Fifth: The numerical rate of degradation of energy is, by (39) in 
(B) in §15 together with (D) in §17 and the tT -values of §18, found to be 
(53) : 7A as 1G (Se = 0.00029, 0.00015, 0.00208 
for Problems 1, 2, 3, respectively. 
As discussed in (B) in §15, this is the quantity which provides the 
significant standard of size for the oscillations and trends of the approxi- 
mate energy (38). 
Computations of (38) show that its total oscillationsnever exceed 
the quadruple of (53) in either problem, and that the overall trend of (38) 
is less per unit of § than one-twentieth of (53). This makes the signifi- 
cance of our computing procedure very plausible, and permits an easy spotting 
of computing errors with the help of the oscillations of the approximate 
energy (38). 
§20. A more detailed inspection of the numerical results in Prob- 
lem 3 allows also locating the course of the shock across the rarefaction. 
This is shown by the dash-dot line on Figure 2. It should be noted that 
