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XVII. On the Secular P'ariation of the Moons Mean Motion. 
By J. C. Adams, Esq., M.A., F.R.S. 8fc. 
Received and Read June 16, 1853. 
1. In treating a great problem of approximation, such as that presented to us bv 
the investigation of the moon’s motion, experience shows that nothing is more easy 
than to neglect, as insignificant, considerations which ultimately prove to be of the 
greatest importance. One instance of this occurs with reference to the secular 
acceleration of the moon’s mean motion. Although this aeceleration, and the dimi- 
nution of the eccentricity of the earth’s orbit, on which it depends, had been made 
known by observation as separate faets, yet many of the first geometers altogether 
failed to trace any connexion between them, and it was only after making repeated 
attempts to explain the phenomenon by other means, that Laplace himself succeeded 
in referring it to its true cause. 
2. The accurate determination of the amount of the acceleration is a matter of very 
great importance. The effect of an error in any of the periodic inequalities upon 
the moon’s place, is always confined within certain limits, and takes place alternately 
in opposite directions within very moderate intervals of time, whereas the effect of an 
error in the acceleration goes on increasing for an almost indefinite period, so that 
the ealculation of the moon’s place for a very distant epoch, such as that of the 
eclipse of Thales, may be seriously vitiated by it. 
In the Mecanique Celeste, the approximation to the value of the acceleration is 
confined to the principal term, but in the theories of Damoiseau and Plana the 
developments are carried to an immense extent, partieularly in the latter, where the 
multiplier of the change in the square of the eccentricity of the earth’s orbit, which 
occurs in the expression of the secular acceleration, is developed to terms of the 
seventh order. 
As these theories agree in principle, and only differ slightly in the numerical value 
which they assign to the acceleration, and as they passed under the examination of 
Laplace, with especial reference to this subjeer, it might be supposed that at most 
only some small numerical corrections would be required ih order to obtain a very 
exact determination of the amount of this acceleration. 
It has therefore not been without some surprise, that I have lately found that 
Laplace’s explanation of the phenomenon in question is essentially incomplete, and 
that the numerical results of Damoiseau’s and Plana’s theories, with referenee to it, 
consequently require to be very sensibly altered. 
