  For a scalar elliptic self-adjoint operator on a compact manifold without
boundary we have two-term asymptotics for the number of eigenvalues between
zero and lambda when lambda tends to infinity, under an additional dynamical
condition. This is a well known result of J.J.Duistermaat and V.W.Guillemin
from 1975.
  In the case of an elliptic system of first order, the existence of two-term
asymptotics was also established quite early and as in the scalar case Fourier
integral operators have been the crucial tool. The complete computation of the
coefficient of the second term was obtained only in 2013. In the present paper
we simplify that calculation. The main observation is that with the existence
of two-term asymptotics already established, it suffices to study the resolvent
as a pseudodifferential operator in order to identify and compute the second
coefficient.
