322 



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. 



It has not been, therefore, without surprise, which he has no 

 doubt will be shared by the Society, that the author has lately found 

 that Laplace's explanation of the phenomenon in question is essen- 

 tially incomplete, and that the numerical results of Damoiseau's and 

 Plana's theories, with reference to it, consequently require to be very 

 sensibly altered. 



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

 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 unal- 

 tered. 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 de- 

 scribed 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 the 

 mean angular motion will be increased in double the same ratio. 



This, the author states, 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, he remarks it will be found that 

 this is not strictly true, and he proceeds briefly to point out the 

 manner in which the inequalities of the moon's motion are modified 

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

 such an alteration of the areal velocity. 



As an example, he takes the case of the variation, the most direct 

 effect of the disturbing force. In the ordinary theory, the orbit of 

 the moon, as affected by this inequality only, would be symmetrical 

 with respect to the line of conjunction with the sun, and the areal 

 velocity generated while the moon was moving from quadrature to 

 syzygy, would be exactly destroyed while it was moving from 

 syzygy to quadrature, so that no permanent alteration would be 

 produced. 



In reality, however, the magnitude of the disturbing force by 

 which this inequality is caused, depends in some degree on the ec- 

 centricity of the earth's orbit ; and as this is continually diminishing, 

 the disturbing forces at equal intervals before and after conjunction 

 will not be exactly equal. Hence the orbit will no longer be sym- 

 metrically situated with respect to the line of conjunction, and there- 



