VOL. L.] PHILOSOPHICAL TRANSACTIONS. 301 



equator, could be generated in the same time. Let then the satellite be in its 

 greatest declination, or inquadrature with the node, then will sn be a quadrant 

 of a circle, and Nm the measure of the angle Npm or spr, in which case nw, or 

 the horary motion of the node, will be to N77i, that is to the angle spr, as 1 is 

 to m ; but the angle spr, is to double the angle subtended by the versed sine of 

 the arc sp described in the same time by the gravity of the satellite to the pri- 

 mary, that is, to the angle scp which is the horary motion of the satellite about 

 the primary, as the force sr is to the gravity of the satellite to the primary, 

 that is, by corol. to prop. 1, as — ,? to 1, or, because in this case -^ = m, as _f^ 

 to 1 . Hence, by joining the ratios, the horary motion of the node is to the 

 horary motion of the satellite, as -—- to 1 : and if s denote the apparent pe- 

 riodic time of the sun, and l the periodic time of the satellite about its primary, 

 since the horary motion of the satellite is to the horary motion of the sun as s 

 is to L, the horary motion of the node will be to the horary motion of the sun, as 

 — ^ X - to 1 ; and in the same ratio will the annual motion of the node be to 



the sun's annual motion, that is to 36o°. Therefore, if the satellite continued 

 the whole year at its greatest declination from the equator of the primary, the 

 aforesaid force arising from the spheroidical figure of the primary, will generate 

 in the same time a motion of the node = —^~ x - x 3Qo° ; and from what is 

 said above the true annual motion of the node will be the half of this, viz. 



3bcn s o«^c 



--,- X - X 300\ Q.E.I. 



5/* L 



Corol. If computation be made for the moon, assuming the medium incli- 

 nation of her orbit to the terrestrial equator, n will be the co-sine of 23° 28-i-'' ; 

 and putting the earth's semiaxis^^ = 1, the moon's mean distance from the 

 earth's centre will be /= 6o nearly; and hence on the hypothesis that the dif- 

 ference of the semiaxes c-=:'.^±^, it will be -^ x - x 3W — \\^" ; and if 

 it should be c = -pf^, the earth remaining uniformly dense, that motion will 

 be = \5". This will be the annual regressive motion of the moon's nodes in 

 the plane of the terrestrial equator, which reduced to the ecliptic, as will be 

 afterwards taught, for the various position of the nodes, becomes much 

 quicker. 



But this motion of the intersection of the orbits of Jupiter's satellites with 

 the plane of his equator will be much greater ; and it will be computed suffi - 

 ciently accurate by the above formula, at least for the satellites not too near to 

 Jupiter. Thus, for the 4th satellite it will be l = l6^ 16^^ 32"", ^=1, 

 / = 25-299 nearly, and the difference of Jupiter's semiaxes -pV ; and putting 

 the inclination of that satellite's orbit to Jupiter's equator = 3°, n will be the 



