  We revisit the first type self-similar solutions for ultrarelativistic shock
waves produced by explosions propagating into cold external medium whose
density profile decreases with radius as $\rho\propto r^{-k}$. The first type
solutions proposed by Blandford and McKee (hearafter BM solution) conforms to
the global conservation of energy and applies when $k<4$. They have been found
to be invalid when $k>17/4$ because of the divergence of total energy contained
in the shocked fluids. So far no attention has been paid to the particle
number. We use the BM solution to calculate the total particle number traversed
by the shock and find it diverges when $k>3$. This is inconsistent with the
finite particles in the surrounding medium. We propose a possible solution when
$k>3$ based on the conservation of particle number and discuss its implication
for the second type solutions.
