PHILOSOPHICAL TRANSACTIONS. 
XV. — Researches in Physical Astronomy . By J. W. Lubbock, Esq. V.P. and 
Treas. R.S. 
Read May 19, 1831. 
On the Theory of the Moon. 
The method pursued by Clairaut in the solution of this important problem 
of Physical Astronomy, consists in the integration of the differential equations 
furnished by the principles of dynamics, upon the hypothesis that in the 
gravitation of the celestial bodies the force varies inversely as the square of 
the distance, and in which the true longitude of the moon is the independent 
variable ; the time is thus obtained in terms of the true longitude, and by the 
reversion of series the longitude is afterwards obtained in terms of the time, 
which is necessary for the purpose of forming astronomical tables. But while 
on the one hand this method possesses the advantage, that the disturbing func- 
tion can be developed with somewhat greater facility in terms of the true lon- 
gitude of the moon than in terms of the mean longitude, yet on the other 
hand, the differential equations in which the true longitude is the independent 
variable are far more complicated than those in which the time is the inde- 
pendent variable. The latter equations are used in the planetary theory ; so 
that the method of Clairaut has the additional inconvenience, that while the 
lunar theory is a particular case of the problem of the three bodies, one system 
of equations is used in this case, and another in the case of the planets. 
The method of Clairaut has been adopted, however, by Mayer, by Laplace, 
and by M. Damoiseau. The last-mentioned author has arranged his results 
with remarkable clearness, so that any part of his processes may be easily 
verified by any one who does not shrink from this gigantic undertaking ; and 
the immense labour which this method requires, when all sensible quantities 
2 H 
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