  We show global existence of small solutions to the Cauchy problem for a
system of quasi-linear wave equations in three space dimensions. The feature of
the system lies in that it satisfies the weak null condition, though we permit
the presence of some quadratic nonlinear terms which do not satisfy the null
condition. Due to the presence of such quadratic terms, the standard argument
no longer works for the proof of global existence. To get over this difficulty,
we extend the ghost weight method of Alinhac so that it works for the system
under consideration. The original theorem of Alinhac for the scalar unknowns is
also refined.
