  In 1899, Hutchinson presented a way to obtain a three-parameter family of
Hessians of cubic surfaces as blowups of Kummer surfaces. We show that this
family consists of those Hessians containing an extra class of conic curves.
Based on this, we find the invariant of a cubic surface C in pentahedral form
that vanishes if its Hessian is in Hutchinson's family, and we give an explicit
map between cubic surfaces in pentahedral form and blowups of Kummer surfaces.
