H H0 A cos

where cos 0 is one single periodic term from the expansion, chosen for its importance. This solution is accomplished by means of a canonical transformation that removes the cos 0 term. This transformation is then applied to the complete Hamiltonian with the effect that the cos 0 term disappears and, although the argument 0 reappears in the transformed series for R, its coefficient will be of higher order.22

By repeating this process, the dependence of the Hamiltonian on the angular variables l, g, and h can be removed systematically - up to some order of approximation - and then the problem is solved. However, each step of the process requires a hugely laborious algebraic calculation, and Delaunay's theory -which was taken to eighth order in m, e, and i, sixth order in e', and fourth order in a/a' - involved a series expansion of 320 terms and required fifty-seven transformations!23

Delaunay's lunar theory, like that of Hansen's, reduced errors to about 1 arc-second but, in terms of accuracy, it was not the unmitigated success that Grant had described. Newcomb transformed Hansen's theory into a form that allowed term-by-term comparison with that of Delaunay, and he found that the two approaches essentially were equivalent. He demonstrated that, while the work of Hansen and Delaunay had improved the accuracy of the short-period inequalities considerably, the same could not be said for those of long period, which were not represented much better than they had been by Laplace.

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