A MIURA OF THE PAINLEVE-I EQUATION AND ITS DISCRETE ANALOGS

We present Miura transformations for the continuous and several discre te Painleve I equations. In the case of the continuous P-I, we use the Hamiltonian formulation of the Painleve equations and show that there exists a Miura transformation between P-I and the binomial, second de gree, equation of Cosgrove SDV. In the case of the discrete P-I's we o btain two different kinds of Miuras. One kind relates a d-P-I to some other d-P-I while the other leads to discrete four-point equations whi ch are the discrete analogs of the derivative of Cosgrove's equation S DV.