  We analyze the long-term evolution of hierarchical triple systems in
Newtonian gravity to second order in the quadrupolar perturbation parameter,
and to sixth order in $\epsilon = a/A$, the ratio of the semimajor axes of the
inner and outer orbits. We apply the ``two-timescale'' method from applied
mathematics to the Lagrange Planetary Equations for the inner and outer orbits,
in which each osculating orbit element is split into an orbit averaged part
that evolves on the long perturbative timescale, and an ``average-free'' part
that is oscillatory in the orbital timescales. Averages over the two orbital
timescales are performed using the well-known secular approximation. We also
incorporate perturbative corrections to the relation between time and the
orbital phases. We place no restrictions on the masses, on the relative orbit
inclinations or on the eccentricities, beyond the requirement that the
quadrupolar parameter and $\epsilon$ both be small. The result is a complete
set of long-timescale evolution equations for the averaged elements of the
inner and outer orbits. At first order in perturbation theory, we obtain the
dotriacontapole contributions explicitly at order $\epsilon^6$. At second order
in perturbation theory, i.e. quadratic in the quadrupole perturbation
amplitude, we find contributions that scale as $\epsilon^{9/2}$ (found in
earlier work), $\epsilon^{5}$, $\epsilon^{11/2}$, and $\epsilon^{6}$. At first
perturbative order and dotriacontapole order, the two averaged semimajor axes
are constant in time (and we prove that this holds to arbitrary multipole
orders); but at second perturbative order, beginning at $O( \epsilon^{5})$,
they are no longer constant. Nevertheless we verify that the total averaged
energy of the system is conserved, and we argue that this behavior is not
incompatible with classical theorems on secular evolution of the semimajor
axes.
