21] ON THE SECULAR VARIATION OF THE MOON'S MEAN MOTION. 141 



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 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 subject, 

 it might be supposed that at most only some small numerical corrections 

 would be required in 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 reference to it, consequently require to be very sensibly altered. 



3. Laplace's explanation may be briefly stated as follows. He shews 

 that the mean central disturbing force of the Sun, by which the Moon's 

 gravity towards the Earth is diminished, depends not only on the Sun's 

 mean distance, but also on the eccentricity of the Earth's orbit. Now this 

 eccentricity is at present, and for many ages has been, diminishing, while 

 the mean distance remains unaltered. In consequence of this the mean 

 disturbing force is also diminishing, and therefore the Moon's gravity towards 

 the Earth at a given distance is, on the whole, increasing. Also, the area 

 described in a given time by the Moon about the Earth is not affected by 

 this alteration of the central force ; whence it readily follows that the 

 Moon's mean distance from the Earth will be diminished in the same ratio 

 as the force at a given distance is increased, and that the mean angular 

 motion will be increased in double the same ratio. 



4. This is the main principle of Laplace's analytical method, in which 

 he is followed by Damoiseau and Plana ; but it will be observed, that this 

 reasoning supposes that the area described by the Moon in a given time 

 is not permanently altered, or in other words, that the tangential disturbing 

 force produces no permanent effect. On examination, however, it will be 

 found that this is not strictly true, and I will endeavour briefly to point 

 out the manner in which the inequalities of the Moon's motion are modified 

 by a gradual change of the central disturbing force, so as to give rise to 

 such an alteration of the areal velocity. 



