Royal Society. 225 



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. 



Having 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 

 two orbits and of their mutual inclination. 



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

 constant, this is the true form 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 new 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 

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

 celeration. The expression which he thus obtains for the first two 

 terms of this acceleration, is 



~ (§» 2 -^>» 4 )yv-E><ft. 



According to Plana, the corresponding expression is 



~ ($ m "-^ mi )f {eh ~ Wi)ndt - 



It will be observed that the coefficient of the second term has been 

 completely altered in consequence of the introduction of the new 

 terms. 



The numerical effect of this alteration is to diminish by V'66 the 

 Phil May. S. 4. Vol. 6. No. 38. Sept. 1853. Q 



