323 



fore the elfects of the tangential force before and after conjunction 

 no longer exactly balance each other. 



The other inequalities v f 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 

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

 of the eccentricity. It is evident that this alteration of areal velocity 

 will have a direct effect in changing the acceleration of the moon's 

 mean motion. 



Ha\dng thus briefly indicated the way in which the effect now 

 treated of originates, the author proceeds with the analytical inves- 

 tigation 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 acceleration of the moon's mean motion, de- 

 ferring 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 over- 

 looked. 



In the usual theory, the reciprocal of the moon's radius vector is 

 expressed by means of a series of cosines of angles formed by com- 

 binations 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 co- 

 efficients 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 

 tvvo orbits and of their mutual inclination. 



Now, if the eccentricity of the earth's orbit be supposed to remain 

 constant, this is the true f jrm of the expressions for the moon's co- 

 ordinates ; but if that eccentricity be variable, the author shows that 

 the differential equations cannot be satisfied without adding to the 

 expression for the reciprocal of the radius vector, a series of small 

 supplementary terms depending on the sines of the angles whose 

 cosines are already involved in it, and to the expression for the lon- 

 gitude, a series of similar terms depending on the cosines of the same 

 angles ; all the coefficients of these nevv^ terms containing as a factor 

 the differential coefficient of the eccentricity of the earth's orbit 

 taken with respect to the time. 



The author first determines as many of these terms as are neces- 

 sary in the order of approximation to which he restricts himself, and 

 tlien takes them into account in the investigation of the secular ac- 



