224 Royal Society. 



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- 

 fore the effects of the tangential force before and after conjunction 

 no longer exactly balance each other. 



The other inequalities of the moon's motion will be similarly mo- 

 dified, especially those which depend, more directly, on the eccen- 

 tricity of the earth's orbit, so that each of them will give rise to an 

 uncompensated change of the areal velocity, and all of these must 

 be combined in order to ascertain the total effect. 



Since the distortion of the orbit just pointed out is due to the 

 change of the disturbing force consequent upon a change in the 

 eccentricity of the earth's orbit, and the action of the tangential 

 force permanently to change the rate of description of areas is only 

 brought into play by means of this distortion, it follows that the 

 alteration of the areal velocity will be of the order of the square of 



