319 
(which is an identity in the Lagrange-ian form, and was therefore not re=- 
ferred to explicitly in §1, but which is better made use of now), momentum 
and energy give sucoessively 
(10) ete A ie eaies | ee eee 
: 7 Sy waaay 
y 
(11) RD ae eo) 
> TPF GS iY 
Equations (10) are based on the conservation of mass and of momentum alone, 
(11) expresses (with the help of (10)) the conservation of energy. These 
are the familiar equations of Rankine and Hugoniot. 
It is well know that (11) (together with (1), (3) for Pee 
Ss) U, and Pa ee SY ; oe ) necessitates in general that 
S; + iS ; As was shown by C. Duhem, H. Bethe, and H. Weyl, for the 
most important equations of state (1), 
(12) sign (5,-S,) = sign Cp.- pi) sign (tin (10) . 
Now the second law of thermodynamics forbids a decrease of the specific en- 
tropy S along the world line of each elementary volume; i.e, it requires 
(13) Sian Saag ‘= Siam al Ss) 
hence by (13) 
(14) Sign \tiwllod) = sign Cp.- pi) 
Summing up: When a world line crosses a shock, the specific en- 
tropy iS changes, and the second law imposes the additional restriction (14). 
§5. The general motion is thus described by (7), (8) in the re- 
gions where the derivatives are continuous, and by (10), (11) with (14) on 
the discontinuity surfaces. Accordingly the specific entropy S$ is con- 
