154 ON THE SECULAR VARIATION OF THE MOON'S MEAN MOTION. [21 



[Abstract^] 



THE author remarks, that in treating a great problem of approximation, 

 such as that presented to us by the investigation of the Moon's motion, 

 experience shews that nothing is more easy than to neglect, on account of 

 their apparent insignificance, 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 acceleration 

 and the diminution of the eccentricity of the Earth's orbit, on which it 

 depends, had been made known by observation as separate facts, yet many 

 of the first geometers altogether failed to trace any connexion between 

 them, and it was not until he had made repeated attempts to explain 

 the phenomenon by other means, that Laplace himself succeeded in referring 

 it to its true cause. 



The accurate determination of the amount of the acceleration is a matter 

 of very great importance. The effect on the Moon's place, of an error in 

 any of the periodic inequalities, is always confined within certain limits, 

 and takes place alternately in opposite directions within very moderate in- 

 tervals of time, whereas the effect of an error in the acceleration goes on 

 increasing for an almost indefinite period, so as to render it impossible to 

 connect observations made at very distant times. 



In the Mecanique Celeste, the approximation to the value of the ac- 

 celeration is confined to the principal term, but in the theories of Damoiseau 

 and Plana, the developments are carried to an immense extent, particularly 

 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 given 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 subject, 

 it might be supposed that only some small numerical rectifications would 

 be required in order to obtain a very exact determination of this value. 



