  We give an alternative proof of the global existence result originally due to
Hidano and Yokoyama for the Cauchy problem for a system of quasi-linear wave
equations in three space dimensions satisfying the weak null condition. The
feature of the new proof lies in that it never uses the Lorentz boost operator
in the energy integral argument. The proof presented here has an advantage over
the former one in that the assumption of compactness of the support of data can
be eliminated and the amount of regularity of data can be lowered in a
straightforward manner. A recent result of Zha for the scalar unknowns is also
refined.
