Wu h197

where the disturbing forces are represented by

Of course, Eqn (9.7) represents a much more difficult challenge than the equivalent equation with m absent (Eqn (9.1)). However, we can write (reverting to the variable r): h2

— = 1 — c cos(d — a) + sin 6 / m cos 6 dd — cos 6 / m sin 6 dd, ir

where c and a are arbitrary constants, which can be shown by direct differentiation to satisfy Eqn (9.7). This is still an equation for r (since m depends on r), but one that lends itself well to approximation, since m is relatively small.

Clairaut derived expressions for P and Q under the assumption that the Earth-Sun distance is constant, and then he substituted an approximate form for r into the right-hand side of Eqn (9.8). He originally chose, based on the empirical evidence, k

- = 1 — e cos q9, r where k , e, and q are undetermined constants. Thus, he used an approximation that represents a rotating ellipse and in which q should turn out to be slightly less than 1, so that 9 must traverse through slightly more then 360° for the Moon to go from one perigee to the next. This approximation can be justified if the extra terms that are introduced by this process are sufficiently small. In performing the calculations, Clairaut made use of Euler's ideas on the expansion of functions

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