  A large class of semi-Hamiltonian systems of hydrodynamic type is interpreted
as the equations governing families of critical points of functions obeying the
classical linear Darboux equations for conjugate nets.The distinguished role of
the Euler-Poisson-Darboux equations and associated Lauricella-type functions is
emphasised. In particular, it is shown that the classical g-phase Whitham
equations for the KdV and NLS equations are obtained via a g-fold iterated
Darboux-type transformation generated by appropriate Lauricella functions.
