320 
stant along each world line while it moves in the regions of continuity, 
but it undergoes a discontinuous increase each time a surface of discontinu- 
ity is crossed. The second law of thermodynamics is automatically fulfilled 
in the former regions, but it excludes 50 per cent of the solutions at the 
latter surfaces. 
This is a most remarkable, and at first sight rather paradoxical 
violation of Hanckel's principle of the "conservation of formal laws", but 
the investigations of W. Rayleigh, G. I. Taylor, and R. Becker on one hand, 
and extensive experimental material on the other, make it impossible to ques- 
tion these conclusions. 
From a mathematical point of view the emergency of shocks, and the 
addition of (10), (11), (14) to (7), (8) represent an extreme complication. 
Without shocks there is usually isentropy, i.e. (8) implies (9), and (7) can 
be written as 
ax >) Ox 
15 CR Pee pee Ree - =e 
ae Ot a P ( = | 
where 
(16) Bail ues ee, oe 
may be considered a known function. Then (15) is a hyperbolic differential 
equation of a familiar type, and can be treated adequately by Riemann's 
classical method of integration. If shocks are present, however, the situ- 
ation changes radically. They act as unknown boundaries for the regions in 
which (7), (8) hold, and along these boundaries (10), (11), (14) must be ful- 
filled. The latter are easily seen to contain twice as many equations as a 
natural boundary condition (on a known boundary) for such a differential 
equation should, and this superdetermination along the unknown boundary 
should lead to its determination. Such problems with a "free boundary" are 
difficult at best, but in the present case an additional difficulty inter- 
