  We place additional constraints on the three parameters of the dark matter
halo merger rate function recently proposed by Parkinson, Cole & Helly by
utilizing Smoluchowski's coagulation equation, which must be obeyed by any
binary merging process which conserves mass. We find that the constraints from
Smoluchowski's equation are degenerate, limiting to a thin plane in the three
dimensional parameter space. This constraint is consistent with those obtained
from fitting to N-body measures of progenitor mass functions, and provides a
better match to the evolution of the overall dark matter halo mass function,
particularly for the most massive halos. We demonstrate that the proposed
merger rate function does not permit an exact solution of Smoluchowski's
equation and, therefore, the choice of parameters must reflect a compromise
between fitting various parts of the mass function. The techniques described
herein are applicable to more general merger rate functions, which may permit a
more accurate solution of Smoluchowski's equation. The current merger rate
solutions are most probably sufficiently accurate for the vast majority of
applications.
