VOL. XLVII.] PHILOSOPHICAL TRANSACTIONS. 143 



the circle of her mean motion Momd, cutting Aa in m and m. Then, because 

 it is a mean motion we seek, generated while the axis ao passes through all its 

 different aspects of the sun ; we may conceive the direct motion already found, 

 of 3° 2' 21-^', to be produced by a constant disturbing force — k, acting on the 

 moon as she revolves in her circular orbit mdotc/; and we have only to inquire 

 how much this force, and its effects, are to be increased, the moon really moving 

 about the same centre x, in the elliptic arc ad ; and how much diminished in the 

 arc Da. 



16. For which purpose, the constant force k is to be increased in the ratio of 

 the mean of the cubes of the moon's distances, in the arc ad, to the cube of 

 TD or CA, and diminished as the mean of the cubes of the distances in d«. Let 

 the forces resulting be k X g and k X h; and these being substituted in the 

 formula, with the arcs "Zdm, or 2T)m, respectively, the sum of the motions found 

 will be the whole mean motion of the apogee, including the correction for the 

 excentricity. 



Now K will be found to be .00557337, and the excentricity tc being .05505, 

 and a the quadrantal arc to radius 1 ; the ratio g, or, which is the same, the 

 sesquiplicate of the time in which the elliptic arc ad is described, to the time in 

 the circular arc dm, that is, {- — —)^, will be 1.110942; and h = ( ^ ~ '^ -)^ 

 = .9001387; hence the whole motion, found as above directed, will be 10962" 

 = 3" 2' 42"; the correction, on account of the excentricity, being only 21". 



Multiply 3° 2' 42" by 1.080853, and the product 3° 17' 28" is the mean mo- 

 tion of the apogee, in a synodical month; exceeding the quantity marked in the 

 tables by no more than 4". 



17. Of the obliquity of the moon's orbit, to the plane of the ecliptic, we take 

 no notice; because though, absolutely speaking, a force in that plane, referred 

 to the moon's orbit, would thence be diminished by about -poVo parts; yet, in 

 the present case, the efl^ect of the obliquity is included in the first determination 

 of the quantity c, from the periodical times of the earth and moon ; all but what 

 belongs to the corrections; and which is only 110". X. 003 = 0". 33, to be 

 subtracted. 



18. The force c is itself the effect of the sun's parallax, and the total effect; 

 excepting only a small difference between his action on the moon, when she is 

 waxing or waning, and when she is in the other half of her orbit; neglected as 

 altogether inconsiderable. 



On the whole, we may conclude, that, in this, as in the other phenomena of 

 the celestial motions, the principles and rules of Sir Isaac Newton are fully con- 

 firmed and verified. 



