4 



S02 PHILOSOPHICAL TRANSACTIONS. [anNO 1758 



«o-sine of this inclination, and hence —.- x - x 36o° = 34' nearly, the annual 



motion ofthe 4th satellite's nodes in antecedentia in the plane of Jupiter s equator. 



Prop. 3. To Reduce the Motion of the Moon's Nodes, above determined, 



to the Ecliptic. Let nad (fig. 10,) be the equator, age the ecliptic cutting 



the equator in a, e the vernal equinox, a the autumnal, lgn the moon's orbit 

 cutting the ecliptic in g and the equator in n, ld a great circle perpendicular to 

 the equator ; and let dn, ln be quadrants of circles. In a given time by the 

 aforesaid force, let the intersection n be transferred to n, and describe the circle 

 lagn for the situation of the lunar orbit after that time, and cutting the ecliptic 

 in g. Also, for brevity *s sake, call the intersection n the equatorial node, and 

 the intersection g the ecliptic node. Then, drawing not, g^ perpendiculars on 

 the moon's orbit, it is nw : Nm : : 1 : sin. gna, and not : Gd : : \ : sin. lg, also 

 Gd : Gg : : sin. Ng-rf : 1 ; hence uniting the ratios gives Nn : Gg : : sin. Ggd : sin. 



GNA X sin. LG, and therefore g^- = nw X — ' — •: ,- — . Write s for the 



' ^ sin. aga 



sine and t for the cosine of the angle Ggd of the inclination of the moon's orbit 

 to the ecliptic, to radius 1 ; f for the sine and ?j, the co-sine of the arc eg ; p 

 for the sine and q the co-sine of the obliquity of the ecliptic ; then by the 

 resolution of the spherical triangle gan, it will be cos. gna= 7i = qt -\- psu, 

 and hence sin. gna =:-/(! — q'^i'^ — Ipqstu — p^s'^u-;) but 1 may be written 

 for t, and the term p'^s^u^ may be rejected on account of its smallness, s being 

 the sine of 5" 8-^^ ; hence then sin. gna = -/ (pp — Ipqsu) ; also sin. gna : sin. 

 GA or f : : sin. gan or p : sin. gn, therefore sin. gn or cos. lg = 



— ^ — , and sin. lg = w — - — , also sin. gna X sin. l.g =2 bu — qs very nearly. 



sin, GNA p i 2 J J 



Therefore Gg^ = nw X — — - , which is the motion of the lunar nodes in the 

 given time in the plane of the ecliptic : which if that given time be the solar 

 year, Nn will = — .^ X - X 36o°, hence that annual ecliptic motion of the 

 nodes, neglecting any change in the situation of the nodes from any other 

 causes in that time, gives 



?t X or+>« X '^■^ X -^ X 360°, or alsoi|' X "■'^^ X ' X 360° 

 nearly. a. e. i. 



Prop. 4. To Determine the Variation of the Inclination of the Lunar Orbit 

 to the Plane of the Ecliptic arising from the Spheroidal Figure of the Earth. — 

 Let ANH (fig. 1 1) be the equator, ag the ecliptic, and a the point of the au- 

 tumnal equinox: let ngrm be the moon's orbit, cutting the ecliptic in g and the 

 equator in n, in which take the arcs nl, gr, equal to quadrants of circles. Now 

 if the equatorial node n in a particle of time by the aforesaid force be supposed 

 transferred to w, and through l be described the circle wLr, it will show thi« 



