VOL. XLIX.] PHILOSOPHICAL TRANSACTION*. 31 



Other motions, by the foregoing propositions these will come out, viz. the an- 

 nual mean precession of the equinoxes generated by the solar force 1 6" 24""; by 

 the lunar 33" 54", the greatest equation of precession by the solar force 1" 23'", 

 by the lunar force l6 45'^' : the nutation of the axis by the solar force 1" 10'", 

 the earth considered as uniformly dense. 



To this part Mr. Walmesley adds some tables for computing these effects in 

 all different circumstances and positions ; but are omitted, as they have been 

 superseded by more recent and correct tables. 



On the Inequalities of the EarthbS Motions. 



Prop. I. To find the Forces of Jupiter and Saturn for Disturbing the Earth's 

 Motions. 



Let s be the sun (fig. 7)) i Jupiter, t the earth revolving in the orbit tot : 

 join SI, IT, st; the latter cutting the moon's orbit hlh in l. Then, by reason- 

 ing after Newton's manner in determining the sun's action on the moon, if si 

 denote the gravitating force of the sun on Jupiter, st will denote the force by 

 which Jupiter depresses the earth towards the sun very nearly ; but the gravity of 

 the sun on Jupiter is to the gravity of Jupiter on the sun, at like distances, from 

 Newton's demonstration, as 1 to 1067 ; and the gravity of Jupiter on the sun is 

 to the gravity of the earth on the sun, as st^ to si"^ : then is the gravity of the 

 earth on the sun to the force of the sun for depressing the moon towards the 

 earth, as st to tl. Uniting these ratios gives the force of Jupiter depressing 

 the earth to the sun, in proportion to the force of the sun depressing the moon to 

 the earth, as st* to si^ X tl X IO67 very nearly, or, because by writing m and 

 w, for the periodic times of the earth and Jupiter, it is st^ : si^ : : m^ -. n-, as w* 

 X ST : n* X TL X 1 067 ; and in this ratio is the force of Jupiter for disturbing 

 the earth's motion, to the sun's force for disturbing the moon's motion. And 

 the latter force being given, there will hence be given the former. 



Also because the gravity of Saturn on the sun is to the gravity of the sun on 

 Saturn, at equal distances, as 3021 is to 1, instead of the number 1067, in the 

 preceding computation, be substituted 302 1 , and Saturn's revolution instead of 

 Jupiter's, then it will show the ratio of Saturn's force on the earth, to the sun's 

 force on the moon. q. e. i. 



Carol. Because the linear errors, arising from different forces, are as those 

 forpes and the squares of the times conjointly ; and the angular errors as those 

 linear ones applied to the radii of the orbits ; it follows that the annual angular 

 errors of the earth with respect to the sun, are to the menstrual angular errors of 

 the moon in respect of the earth, in the ratio compounded of the direct ratio of 

 Jupiter's force on the earth, and of the sun's force on the moon, and the dupli- 

 cate ratio of the periodic times of the earth about the sun, and the moon about 

 the earth conjointly, and of the inverse ratio of the radii st, tl, that is, if h be 



