  We present a new derivation of the distance-dependent two-point function for
planar Eulerian triangulations and give expressions for more refined generating
functions where we also control hull perimeters. These results are obtained in
the framework of a new recursion relation for slice generating functions and
extend similar results obtained recently for triangulations and
quadrangulations. A number of explicit formulas are given for the statistics of
hull perimeters in infinitely large random planar Eulerian triangulations.
