## Info

Extracts from Newcomb's Researches on the Motion of the Moon (1878) are reprinted in Shapley and Howarth (1929).

The same determinant appears in work done by Adams on the lunar theory in 1868, though this was not published (Barrow-Green (1997), p. 27).

as a theory by which to explain phenomena, was somewhat unsatisfactory. The theory was incredibly complex and provided little insight into the underlying physics; accurate numbers could be churned out, but the qualitative nature of the solutions was lost amid pages and pages of calculations.

### The circular restricted three-body problem

In Hill's radical lunar theory, the intermediate orbit was essentially a solution to what is now known as the 'circular restricted three-body problem'. In this problem, two of the bodies - the Sun and Earth for the lunar theory - called primaries and with masses m1 and m2, are assumed to describe circular orbits around their common centre of gravity, and then the aim is to determine the motion of the third body of negligible mass (the Moon) within the resulting gravitational field. This problem had been studied first by Euler, and significant progress was achieved by Jacobi when he found a new invariant quantity.

Inacoordinate frame rotatingwithangularvelocity w, the equationof motion for the small mass, setting G = 1 for convenience is,

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