  In 1978 Lipman presented a proof of the existence of a desingularization for
any excellent surface. The strategy of Lipman's proof is based on the
finiteness of the number H(R) defined as the supreme of the second
Hilbert-Samuel coefficient I, where I range the set of normal m-primary ideals
of a Noetherian complete local ring (R,m). The problem studied in the paper is
the extension of the result of Lipman on H(R) to m-primary ideals I of a
d-dimensional Cohen-Macaulay ring R such that the associated graded ring of R
with respect to I^n is Cohen-Macaulay for n>> 0.
