321 
9 
venes. The change of specific entropy S at the shock q = G& (4) depends 
by (11) on the trajectory of the shock, i.e. on the unknow boundary a: a (). 
Now 3 enters explicitly into the differential equation (7). 
Hence we are dealing here with a "free boundary" problem where the coeffi- 
cients of the differential equation themselves depend explicitly on the un- 
known, "free", boundary. 
Problems of this type have never been treated in any generality, 
and appropriate analytical methods to deal with them are entirely unknown. 
Rigorous solutions have only been determined in very special cases, where 
the trajectories of the shocks could be guessed by other means, up to a few 
numerical parameters. Analytical approximative methods (e.g. expansions) 
or numerical ones are also very difficult and restricted to very few special 
cases. Furthermore the approximative numerical procedures do not seem to 
lend themselves for problems of this type to efficient mechanization. 
§6. The idea which will be discussed here is to treat the contin- 
uous case (7), (8) with an approximative, numerical method, and to ignore the 
possibility of shocks, (10), (11), (14). 
In order to diagnose the character and the implications of this 
idea, let us consider a special case of the equations of state (1), (3), 
which is itself of not inconsiderable practical importance. 
Assume that there exists an absolute relation of the form 
(17) p= pay) 5 
where P.(v) is a known, fixed function. That is that the source of (17) 
is not (16) and (9) -- that the differential equation (7) assumes the form 
(15) without (9), i.e. without isentropy. Owing to (3), (17) can hold oniy 
if the caloric equation of state has the form 
(18) = Wi Gd) = Uae + Ves) 
