34 PHILOSOPHICAL TRANSACTIONS. [anNO 1756. 



longitude with Jupiter's notle have their latitudes increased or diminished. 

 Therefore these fixed stars from the time of Hipparchus, that is, in about 1 QOO 

 years, will have changed their latitudes near 5\ In like manner since all the 

 arcs comprehended between the circles de, d£j and perpendicular to cIe, are as 

 the sines of their distances from the point e, or as the cosines of their distances 

 from Jupiter's node, the increment or decrement of the latitude of any star will 

 be to 15'^ 9", as the cosine of the difference of longitudes of that star, and the 

 nearest node of Jupiter, to radius ; and therefore, having given at once the lon- 

 gitude botli of a star and of Jupiter's nodes, there will be given the variation of 

 the latitude of the star for any time. 



Prop. 3. To determine the Variation of the Obliquity of the Ecliptic arising 

 from the foregoing Forces. — Since, from the preceding proposition, the ecliptic 

 sensibly changes its situation, hence in general it appears that it must also vary 

 its inclination to the equator. In order then to investigate the quantity of this 

 variation, let ved (fig. Q) be the ecliptic; id the orbit of Jupiter cutting the 

 ecliptic in D; ol the equator, and l the equinoctial point. Let de and lv be 

 circular quadrants ; then if in any particle of time the node d be conceived to be 

 transferred by its mean motion to d, the circle dEt described through the points 

 dy E, will represent the situation of the ecliptic after that time. And if on the 

 same there be demitted the perpendiculars d^, rt, the latter v^ will show the va- 

 riation of the obliquity of the ecliptic generated in the same time. Writing 

 therefore s for the sine of the inclination of Jupiter's orbit to the ecliptic, radius 

 being 1 ; then, in the triangle Ddg, Bd : i>g :: 1 : s; but ng : Yt :: 1 : sin. ev ; 

 hence ndivt ::l : s X sin. ev ; and because de = lv, then dl = ev, and there- 

 fore Dd : VI :: 1 : * X sin. dl; and hence it appears that the momentary variation 

 of the obliquity of the ecliptic is as the sine of the distance of Jupiter's node 

 from the equinox. 



Now draw lc to c the centre of the sphere, and on lc the perpendicular dk; 

 then because of the regressive motion both of the node d and of the equinox l, 

 but the equinox with a quicker motion than the node, the points d, l, will mu- 

 tually approach to or recede from each other, with the difference of the velocities. 

 Suppose therefore either of them, as for instance the node d, to move with this 

 difference of velocities, the equinox l continuing immoveable, and let De be a 

 very small arc described by this difference of velocities, also on lc demit the 

 pei-pendicular ek\ then will De : ¥ik :: 1 : dk or sin. dl; hence ndi \l :: De:^ X 

 kA; and the sum of all the variations v/, while the point d with the aforesaid 

 difference of the velocities describes any arc dh, will be to the sum of as many 

 motions of the node d, that is, the variation of the obliquity of the ecliptic 

 generated in any time, will be to the motion of the node, as the sum of all the 

 kA drawn into the sine Sj to the sum of as many arcs De, that is, drawing hm 



