21] ON THE SECULAR VARIATION OF THE MOON'S MEAN MOTION. 155 



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 essentially incom- 

 plete, 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 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 eccen- 

 tricity 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 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 per- 

 manent 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 eccentricity of the 



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