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



As an example, I will take 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 of areal velocity 

 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 Earth's orbit, and as this is continually diminishing, the central dis- 

 turbing forces at equal angular distances on opposite sides of conjunction 

 will not be exactly equal. Hence the orbit will no longer be symmetrically 

 situated with respect to the line of conjunction. Now the change of areal 

 velocity produced by the tangential force at any point, depends partly on 

 the value of the radius vector at that point, and consequently the effects 

 of the tangential force before and after conjunction will no longer exactly 

 balance each other. 



The other inequalities of the Moon's motion will be similarly modified, 

 especially those which depend, more directly, on the eccentricity of the 

 Earth's orbit, so that each of them gives rise to an uncompensated change 

 of the areal velocity. 



Since the distortion in the form of the orbit just pointed out is due 

 to the alteration of the disturbing force consequent upon a change in the 

 eccentricity of the Earth's orbit, and it is by virtue of this distortion that 

 the tangential force produces a permanent change in the rate of description 

 of areas, it follows that this alteration of the areal velocity will be of 

 the order of the square of the disturbing force multiplied by the rate of 

 change of the Earth's eccentricity. 



It is evident that the amount of the acceleration of the Moon's mean 

 motion will be directly affected by this alteration of areal velocity. 



5. Having thus briefly indicated the way in which the effect now 

 treated of originates, I will proceed with the analytical investigation of its 

 amount. 



