This result is interesting because we can think of the first term as the pressure due to energy and the second as the pressure due to the entropy.
What started as a simple picture may be looking confusing now, but remember that underneath all of the new equations that are arising is the Thermodynamic Identity, and the definitions of the other energies from that. Everything else follows from those and can be rederived without much difficulty.
We can utilize yet another mathematical truth, that double derivatives commute, to derive a new set of relations known as the Maxwell Relations.
We know, for example, that = - σ and that = V, by the methodology just discussed. Now, though, we can take the partial derivative of the first equation with respect to p, holding τ constant, to obtain:
Similarly, we can take the partial derivative of the second equation with respect to τ, holding p constant, to obtain:
The mathematical truth useful here is that: