PHILOSOPHICAL TRANSACTIONS. 
I. — Researches in Physical Astronomy. By J. W. Lubbock, Esq. V.P. and 
Treas. R.S. 
Read November 17, 1831. 
On the Theory of the Moon. 
In the following paper I have given the developments which are required in 
the Theory of the Moon when the square of the disturbing function is retained. 
These expressions result from the multiplication of series, each consisting of 
many terms ; but they are formed with great facility by means of the second 
Table given in my former paper on the Lunar Theory. 
I have not attempted the numerical calculation of the coefficients of the ine- 
qualities according to the method here explained, at least in the second approxi- 
mation ; but this work, which would tend to perfect the Tables of the Moon, is 
a desideratum in physical astronomy. The calculations will not I think be 
found longer than in the method of Clairaut, nor than those which are required 
in several astronomical problems. The developments which I have given 
ought however to be verified in the first instance, although I have taken great 
pains to ensure their accuracy. 
With respect to the convergence of the expressions, it may be remarked 
that when the same powers of the eccentricities are retained, the results must 
be identical, whichever method be employed. If part of the coefficients of the 
terms already considered due to the higher powers of the eccentricities are 
sensible, it follows that other arguments must be considered in addition to 
those introduced by M, Damoiseau ; and conversely if the arguments which 
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