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



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 symmetrically situated with respect to 

 the line of conjunction, and therefore 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 modified, 

 especially those which depend, more directly, on the eccentricity 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 the disturbing force multiplied by the rate 

 of change of the square of the eccentricity. It is evident that this altera- 

 tion of areal velocity will have a direct effect in changing the acceleration 

 of the Moon's mean motion. 



Having thus briefly indicated the way in which the effect now treated 

 of originates, the author proceeds with the analytical investigation of its 

 amount. In the present communication, however, he proposes to confine his 

 attention to the principal term of the change thus produced in the accele- 

 ration of the Moon's mean motion, deferring to another, though he hopes 

 not a distant opportunity, the fuller treatment of this subject, as well as 

 the determination of the secular variations of the other elements of the 

 Moon's motion, which, arising from the same cause, have also been hitherto 

 overlooked. 



In the usual theory, the reciprocal of the Moon's radius vector is ex- 

 pressed by means of a series of cosines of angles formed by combinations 

 of multiples of the mean angular distance of the Moon from the Sun, of 

 the mean anomalies of the Moon and Sun, and of the Moon's mean distance 

 from the node ; and the Moon's longitude is expressed by means of a series 

 of sines of the same angles, the coefficients of the periodic terms being 

 functions of the ratio of the Sun's mean motion to that of the Moon, of 

 the eccentricities of the two orbits and of their mutual inclination. 



