  We present the secular theory of coplanar $N$-planet system, in the absence
of mean motion resonances between the planets. This theory relies on the
averaging of a perturbation to the two-body problem over the mean longitudes.
We expand the perturbing Hamiltonian in Taylor series with respect to the
ratios of semi-major axes which are considered as small parameters, without
direct restrictions on the eccentricities. Next, we average out the resulting
series term by term. This is possible thanks to a particular but in fact quite
elementary choice of the integration variables. It makes it possible to avoid
Fourier expansions of the perturbing Hamiltonian. We derive high order
expansions of the averaged secular Hamiltonian (here, up to the order of 24)
with respect to the semi-major axes ratio. The resulting secular theory is a
generalization of the octupole theory. The analytical results are compared with
the results of numerical (i.e., practically exact) averaging. We estimate the
convergence radius of the derived expansions, and we propose a further
improvement of the algorithm. As a particular application of the method, we
consider the secular dynamics of three-planet coplanar system. We focus on
stationary solutions in the HD 37124 planetary system.
