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They were probably first found by Tisserand (Goroff (1993), Section 2.7). If R = 0, so that we have the two-body problem, it follows immediately from the fact that the Hamiltonian is a function of L alone, that g, h, L, G, and H are all constant, and l = dH/dL = (/u,/a3)1/2 = n. For more details see, for example, Brouwer and Clemence (1961), pp. 541-59. Delaunay's work has a significant place in the history of computer algebra. In 1970, a group at Boeing Scientific Research Laboratories in Seattle tested their computer algebra software on Delaunay's procedure. Remarkably, they found only three errors, two of which were consequences of the first (see Pavelle, Rothstein, and Fitch (1981)).

Hansen's tables might be accurate for the current epoch, but the agreement with ancient observations was poor.24

In 1877, Newcomb became director of the Nautical Almanac Office, and asked Hill for assistance in producing new tables for the Moon and the planets. Hill, a keen admirer of Euler, reintroduced an idea that Euler had first tried in the 1770s by describing the motion of the Moon with respect to a rotating set of axes (in this case the axes rotated with the mean angular speed of the Sun). With certain simplifying assumptions, Hill obtained a differential equation dependent solely on the ratio m of the mean motion of the Sun to that of the Moon, a quantity the numerical value of which was well established. He found a particular solution to this equation in the form of an oval, symmetric with respect to the rotating axes and with the longer axis perpendicular to the direction of the Sun, the first periodic solution to the three-body problem since Lagrange's discovery of special exact solutions in 1772. With the numerical value of m inserted, this solution, known as 'Hill's intermediate orbit' or 'variational curve', represents a good approximate orbit for the Moon which then could be used as the basis for a perturbation analysis in which successive terms decay rapidly.

By means of some ingenious transformations, Hill managed to reduce the resulting perturbation problem to that of solving what is now generally known as 'Hill's equation': x + \$(t)x = 0, where \$ is aperiodic function. Hill expanded \$ as a Fourier series, the coefficients of which were related to an infinite deter-

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