In this case we have to work with the left eigenvectors 29. This changes the characteristic , , and :

We can now solve for the *z*-derivatives of the primitive variables

using the thermodynamic identity

these simplify to

We may write the required themodynamic derivatives in terms of the internal energy per unit mass as

Finally, by using the expression,

we can write the equations for the quasi-conserved variables using the same technique as in section 3.2:

Wed Sep 19 18:16:24 MET DST 2001