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From the preface to Euler's prize essay on lunar theory (1772). Quoted from Pannekoek (1961) pp. 299-300.

15 Whittaker (1937), Chapter XIII.

In the case of body 1 being the Sun and bodies 2 and 3 are planets, G3 is much smaller than ^,12, and so the right-hand side can be considered small. For the case of bodies 1, 2 and 3 being the Earth, Moon, and Sun, respectively, the right-hand side is again small, but for a different reason. In this case, it is because the perturbation comes from the difference between the effect of the Sun on the Moon and the effect of the Sun on the Earth, these two effects being very nearly equal.

Euler introduced one of the key mathematical ideas that allowed progress to be made in the solution of Eqn (9.3) when the right-hand side was small, namely the approximation of functions by trigonometric series. This was a huge step forward, as it allowed the integrations which needed to be performed to solve these second-order differential equations to be carried out relatively simply. In the expressions for the components of the perturbing forces in the coordinate directions that appear in the differential equations, the distance betweenbodies 2 and 3 appears as an inverse cube. Now, if we ignore the eccentricities in the orbits of the bodies so that r12 and r13 are constant, r-3 = r-3(1 - 2a cos 6 + a2)-3/2 = r-3(1 + a2)-3/2(1-g cos 6)-3/2, where a = r12/r13,6 is the difference between the longitudes of bodies 2 and 3 referred to body 1, and g = 2a/(1 + a2). Using the binomial theorem17 we thus have r-3 = r-3(1 + a2)-3/2(1 + 3 g cos 6 + g2cos2 6 + ■■■). (9.4)

If r12 and r13 are very different in magnitude (as they are for the lunar problem in which r12 is the Earth-Moon distance and r13 is the Earth-Sun distance) then g is much smaller than 1, and only a few terms in this series will yield accurate results. However, in the case of Jupiter and Saturn, g turns out to be about 0.84 and the series converges very slowly. In the case of the Earth and Venus, g & 0.95 and the situation is even worse. Questions of the actual convergence of infinite series like Eqn (9.4), as distinct from the use of such series as approximations, were not addressed seriously in the eighteenth century, though Euler did become concerned about this in his later years. He also suggested that perhaps it might be better to integrate the inverse cube directly using numerical quadrature. His colleague in St Petersburg - Anders Lexell - attempted this, but with little success.

16 Euler also treated the case of non-zero eccentricity.

Discovered by Newton, and independently by James Gregory, in around 1670, though not provided with a rigorous proof until that of Abel in 1826 (Hairer and Wanner (1996), p. 251).

18 In the 1760s, Euler suggested that for some purposes it might be better to approach the whole problem of perturbations by integrating numerically the differential equations using a

Another of Euler's significant contributions was his demonstration that an expansion in powers of cos 0 could be transformed into an expansion in terms of cos n0, r"3 = A + B cos 0 + C cos 20 +----, such series being much more straightforward to integrate. The coefficients A, B, C,... were themselves expressed in terms of infinite series, but Euler showed that they satisfied a two-term recurrence relation, meaning that each coefficient depended on the values of the previous two. He thus only needed to calculate A and B from the series and then the others followed. This new representation could easily be integrated term by term and, moreover, the two integrations that were required improved the convergence of the series, since cos n0 when integrated twice gives -n-2 cos n0. While Euler's procedure still required considerable effort to be implemented, it made the three-body problem appear tractable for the first time.19

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