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15 
ous motion. When shocks appear, the situation seems considerably less 
favorable. These two remarks suggest themselves immediately: 
First: In the presence of shocks the real motion is not described 
by (15) alone, but by (15) together with (10), (14). Now while (25) is 
clearly an approximation of (15), it is not at all clear whether it is also 
one of (10), (14). 
Second: Actually there are reasons to expect that something must 
go wrong with this latter approximation. Indeed, we saw in §7 that the 
total kinetic plus potential energy is not conserved in hydrodynamics when 
shocks are present, but that it is continuously degraded (i.e. decreasing) 
according to (22). On the other hand the expression (26) represents pre- 
cisely the total kinetic plus potential energy -- no thermic energy 
makes its appearance in this expression, and there is indeed no room for a 
separate thermic energy in such a quasi-molecular model. And since (25) is 
the system of the ordinary (point) mechanical equations of motion belonging 
to the total energy (26), therefore (25) must conserve this energy. 
Thus (25) must conserve (26), while (15) with (10), (14) does not 
conserve the analogue of (26). How then can (25) be an approximation of 
(15) with (10), (14)? 
§11. Nevertheless it is hard to see how (25) can fail to describe 
the equivalents of shocks in certain situations. E.g. let the "beads and 
springs" model of §9 collide with a rigid wall; in this situation the wall 
would send a shock into a compressible substance (in hydrodynamical thecry) , 
and something similar must happen to the "beads and springs" model. The 
boundary conditions which describe this are easy to specify: For t=O 
the substance is in its normal state (i.e. each Vg = i en |, 
hence Rg = &@+ Const. , e.g. xX, = @) and is moving uniformly 
