  We analyze the secular evolution of hierarchical triple systems in the
post-Newtonian approximation to general relativity. We expand the Newtonian
three-body equations of motion in powers of the ratio $a/A$, where $a$ and $A$
are the semimajor axis of the inner binary's orbit and of the orbit of the
third body relative to the center of mass of the inner binary, respectively.
The leading order "quadrupole" terms, of order $(a/A)^3$ relative to the
$1/a^2$ acceleration within the inner binary, are responsible for the
well-known Kozai-Lidov oscillations of orbital inclination and eccentricity.
The octupole terms, of order $(a/A)^4$ have been shown to allow the inner orbit
to "flip" from prograde relative to the outer orbit to retrograde and back, and
to permit excursions to very large eccentricities. We carry the expansion of
the equations of motion to hexadecapole order, corresponding to contributions
of order $(a/A)^5$. We also include the leading orbital effects of
post-Newtonian theory, namely the pericenter precessions of the inner and outer
orbits. Using the Lagrange planetary equations for the orbit elements of both
binaries, we average over orbital timescales, obtain the equations for the
secular evolution of the elements through hexadecapole order, and employ them
to analyze cases of astrophysical interest. We find that, for the most part,
the orbital flips found at octupole order are robust against both relativistic
and hexadecapole perturbations. We show that, for equal-mass inner binaries,
where the octupole terms vanish, the hexadecapole contributions can alone
generate orbital flips and excursions to very large eccentricities.
