324 
ie 
(21), (22) must be evaluated with the help of (10), (20). 
§8. In carrying out numerical approximations of the hyperbolic 
differential equation (15), the continuous independent variables By it must 
be replaced by discrete ones. For various reasons it is advantageous to 
carry this out in two successive steps, and to begin by making a alone dis- 
crete. One of these reasons is that @ is really a discrete quantity,which 
was made artificially continuous by the classical transition from the kinetic 
theory to the continuous, hydrodynamical one: the elementary volumes of the 
substance, for which q, is a label, should be naturally discrete entities. 
Thus the label O& is "naturally" discrete, while the coordinates x, tare 
"naturally" continuous. (Note that in this setup the Lagrange-ian form is 
preferable to the Euler-ian, since the latter deals with at only, without 
Ca) Making Q@ discrete, and leaving x,t continuous, has therefore a 
certain physical meaning, and this will turn out to be very helpful presently 
Accordingly, let @-run over a sequence of equidistant values, 
which may as well be normalized so as to be the integers: 
(23) aa= Soyo -2, =. 0, Ks Re Gas 
It is also advantageous to write for x 
\ 
(24) eae (oie ce.) = Xa) 
The hyperbolic (partial) differential equation (15) can now be replaced by 
the approximative system of (total) differential equations 
(25) d* x, 
iene 
Po = elie) mt pe (See = eae 
§9, It is an essential circumstance concerning the equations 
(25) that they are not only mathematical approximations of the rigorous 
Pp 
equation (15), but also the rigorous equations of another physical system, 
