32 PHILOSOPHICAL TRANSACTIONS. [anNO 1755. 



taken to denote the periodic time of the moon, from what is above demonstrated, 

 as m^ to n^ X A^ X IO67, or as 1 to — - X 1067. fherefore if the errors in 

 a given time for example, in a certain number of years, are to each other as 1 

 to ^ X 1067 ; that is, the inequalities of the earth's motion, are to the inequalities 

 of the moon's motion, in a given time, in a ratio compounded of the duplicate 

 ratio of the earth's periodic time to the periodic time of Jupiter, of the simple 

 ratio of the earth's periodic time about the sun to the moon's periodic time about the 

 earth, and of the ratio of the gravity on Jupiter to the gravity on the sun, conjointly. 

 Therefore, the periodic times being, for Jupiter 4332.514 days, for the earth 

 365.2565, for the moon 27.3215, then the inequalities of the earth's motion by 

 the force of Jupiter, will be to the inequalities of the moon's motion, in a given 

 time, in the ratio of 1 to 1 1229.4. 



For the revolution of Jupiter put now that of Saturn, viz. IO759.275 days ; 

 and for IO67 the number 3021 ; then the inequalities of the earth's motion from 

 the force of Saturn, will be to the inequalities of the moon's motion, in a given 

 time, as 1 to 196076.5 And hence it appears that the force of Saturn is to the 

 force of Jupiter, for disturbing the earth's motion, as 1 to 17.46. 



Frop. 2. To determine the motion of the Nodes and Apses of the Terrestrial 

 Orbit. By the motion of the nodes of the terrestrial orbit, is meant the motion 

 of the line of intersection of the orbits of the earth and Jupiter and Saturn, made 

 in the plane of Jupiter or Saturn. The motion of the moon's nodes in a sidereal 

 year, according to the astronomers, is 19° 20' 32" ; and this motion drawn into 

 100, and diminished in the ratio of 1 to II229.4, by corol. to the preceding 

 prop, is 10' 20'' 5" ; which increased in the ratio of the cosine of the inclina- 

 tion of Jupiter's orbit and the ecliptic, to the cosine of the inclination of the lunar 

 orbit, that is, in the ratio of the cosine of 1° 19' 10" to the cosine of 5° 8'4-, be- 

 comes 10' 22" 26'". This therefore is the regressive motion of the earth's nodes 

 in the plane of Jupiter's orbit in 100 sidereal years, by the force of Jupiter. 

 Then this motion 10' 22" 26'" diminished in the ratio of 1 to 17.46, will produce 

 the motion of the nodes, which in the same time is generated by the force of 

 Saturn, in the plane of his orbit, or even in the plane of Jupiter's orbit nearly, 

 viz. 35' 39"'. Therefore the whole motion of the earth's nodes, from these forces 

 united, in 100 years, in the plane of Jupiter's orbit, is about 10' 53" in ante- 

 cedentia. 



Also in the same manner may be collected the motion of the earth's aphelion : 

 for this motion will be, as far as relates to the force of Jupiter, to the motion of 

 the moon's apogee, in a given time, as I to 11 229.4; therefore if the moon's 

 apogee moves annually 40° 40' 43" in consequentia, the earth's aphelion will 

 move annually 13" 3'" 28'", and in 100 years 2l'44" in consequentia. Further, 

 this motion diminished in the ratio of 1 to 17'46, gives l' \A"\ produced by the 



