VOL. XLIX.] ' PHILOSOPHICAL TRANSACTIONS. 125 



CoROL. 3. From the proposition in general it follows, that the mean horary 

 regress of the line of intersection of the planes of the terrestrial equator, and of 

 the orbit of any planet revolving about the earth, is as the force of that planet 

 on the terraqueous globe, other things remaining, and the cosine of the inclina- 

 tion of its orbit to the earth's equator conjointly. 



Prop. 1. Prob. To find the Inequality of the Precession of the Equinoxes, 

 which depends on the Various Situation of the Moon's Nodes. — Let sld (fig. 5) 

 be the equator, eafl the ecliptic cutting the equator in l, e the vernal equinox, 

 L the autumnal, gfd the moon's orbit cutting the equator in d and the ecliptic 

 in F, AGS a great circle perpendicular to the equator, and let sd, gd be qua- 

 drants of circles. While the node f describes the horary ecliptic arc f/, the in- 

 tersection D, by the lunar force, will be transferred through the arc d</, and de- 

 scribe the circle sd representing the position of the equator after an hour is 

 elapsed, and cuts the ecliptic in n ; and draw ng, Lr perpendicular to the equator. 

 At that time let b be the sine, and c the cosine, of the inclination of the lunar 

 orbit to the terrestrial equator; h and c being, as before, the sine and cosine of 

 the inclination of the ecliptic, or of the mean inclination of the lunar orbit to 

 the equartor; then (by corol. 3 of the preceding prop.) will the mean horary 

 regress of the intersection of the planes of the equator and ecliptic, produced 

 by the moon's force, be to Df/, the mean horary regress of the intersection of 

 the planes of the equator and lunar orbit, as c to c ; but p/is to the said regress 

 of the intersection of the planes of the equator and ecliptic, as the mean motion 

 of the moon's nodes, to the mean motion of the equinoxes generated by the 

 moon's force; which ratio put as ^ to 1 ; therefore Ff:-Dd::cxk:c; but i>d: 

 Dg :: 1 : b, and d^ : Lr :: 1 : sine of the arc ls, which call k, and let Lr : hn :: /' 

 : 1 ; hence, by composition of ratios, f/: i.n :: b X c X k : b X cXk. 



Through the node p describe the great circular arc fc perpendicular to sl; 

 then, by spherical trigonometry, the cosine fl is to radius 1, as the cotangent 

 FLC to tangent lfc; and sine lfc to sine dfc, as cosine flc to cosine fdc: but 

 as the angle dfc is the sum of the angles dfl and lfe, the sine dfc = sine 

 dfl X cosine lfc + cosine dfl X sine lfc. Hence, writing p and q for the 

 sine and cosine of the angle dfl, viz. of the mean inclination of the moon's 

 orbit to the ecliptic, and v and u for the sine and cosine of the arc ef, viz. of 

 the distance of the node from the vernal equinox, it is then cosine fdl = c = 

 cq + bpu. Also in the triangle fdl, it is b : p :: v : sine dl, therefore is the 



cosine of the arc dl, or sine -of the arc ls, that is, k = -Vb'^— /)V = - x 



B ' B 



bg — cpv. 



Hence therefore is obtained bck = {eg -\- bpu) X {bq — cpu) = bcq^ — 

 (c^ -- Zj^) X pqu — bcp^u^: but by writing 1 for q, and rejecting the term bcp^u^ 



vol. XI. E 



