VOL. XLIX.] PHILOSOPHICAL TRANSACTIOXS. 35 



perpendicular to lc, as km X ^ to the arc dh. If therefore n denote the mo- 

 tion of Jupiter's node, while describing the arc dh, the variation of the inclina- 

 tion of the ecliptic to the equator, generated in the same time, will be 

 KM X s X n^ Hence, as — denotes the ratio of the motion of the node to the 



DH DH 



difference between the motions of the node and equinox, and as km is the dif- 

 ference or sum of the cosines of the distances of the points d and h from the 

 equinox, according as the points k and m lie on the same or contrary sides of the 

 centre c, there results the following theorem: " Radius is to the sine of the incli- 

 nation of Jupiter's orbit to the ecliptic, as the difference or sum of the cosines of 

 the distances of the node from the equinox at the beginning and end of a given 

 time, is to a certain sine; then the differences of the motions of the node and 

 equinox, is to the motion of the node, as the sine above found, is to the sine of 

 the variation of the obliquity of the ecliptic. 



Also for the node and inclination of Jupiter's orbit, substitute those of Saturn, 

 then the same theorem will give the variation of the obliquity of the ecliptic 

 generated by Saturn, o. e. i. 



CoROL. 1 . The node d in the figure is the descending node of Jupiter, and 

 the point L the vernal equinox ; hence, and from the reasoning in the prop, it 

 appears, that while the node d and the equinoctial l approach each other, the 

 inclination of the ecliptic to the equator decreases; but increases when the node 

 and equinox recede from each other; or, which comes to the same, while the 

 ascending node of Jupiter's orbit passes from the vernal equinox to the autumnal, 

 the obliquity of the ecliptic diminishes, and when the same node passes from the 

 autumnal to the vernal. equinox, that obliquity increases. 



CoROL. 2. If the points d and h be on different sides of the equinoctial points, 

 i. e. if the node in the time proposed pass through the equinox, it appears from 

 the preceding corollary, that the obliquity of the ecliptic will partly increase and 

 partly decrease; in which case the difference of the increment and decrement 

 will be given by the above theorem ; and there will also be had the sum xjf these, 

 or the whole variation of the obliquity generated in that time, if instead of the 

 difference or sum of the cosines of the distances of the node from the equinox, 

 be substituted in the said theorem the sum of the versed sines of the same dis- 

 tances, as is evident. The same mode of reasoning as in both corollaries will 

 also serve for Saturn. 



Scholium 1 . As there has been much debate among astronomers, both an- 

 cient and modern, whether the obliquity of the ecliptic be constant or variable: 

 but whoever may please to examine the phenomenon by the laws of gravity, he 

 may fully try its investigation by this proposition. Further, as Jupiter's ascend- 

 ing node is now in the sign Cancer, it appears from corol. 1 of this prop, that 



p2 



