•28 PHILOSOPHICAL TRANSACTIONS. [aNNO 17551 



nerated by the sun s force, as the tangent of the mean inclination of .the ecliptic 

 to the equator, is to the tangent of the total variation of the same inclination.'* 

 And hence the total variation will come out in the former case = 44"', in the 

 latter 57'', viz. the sun being in the solstices: in other places the variation is in 

 the duplicate ratio of the sine of the sun's distance from the equinox, to radius ; 

 and because the difference between half the total variation and the variation ge- 

 nerated in the time the sun describes any arc lr, is to half the total variation, 

 viz. to 22 ' or 28*'4-, as 2rh* — 1 is to 1, that is, as the co-sine of double the 

 sun's distance from the equinox, to radius ; and therefore, the sun's place being 

 given, there will be given this difference or equation, to be added to the mean 

 inclination of the ecliptic whenever the sun's distance from either equinox is less 

 than 45° : and to be subtracted when that distance is greater. Therefore the 

 inclination of the ecliptic to the equator is greatest when the sun is in the equi- 

 noxes, but least when in the solstices. 



Prop. IV. Proh. To determine the Variation of the Inclination of the Ecliptic, 

 which depends on the Various Situation of the Moon's Nodes. — The same things 

 remaining as in the 2d prop, let now the moon be at k (fig. 5), and describe the 

 arc of a great circl^ kz perpendicular to the equator dzs, and through the point 

 z, the arc zd showing the situation of the equator after the space of an hour ; 

 and let zd cut the moon's orbit in d, and the lines d^, Lr in e and q ; also from 

 the point v of the equator, lv being a quadrant of a ci'-cle, demit v^ perpendi- 

 cular on the arc dz produced. Let p denote the mean horary motion of the 

 equinoxes generated by the moon's force : then by prop. 2 it is p : Dff : : c : c ; 

 and, DS being a quadrant of a circle, from what is demonstrated in prop. 1, it 

 follows that 2j}d : nd:'. 1 : sin.'^ dk ; and hence Bd : ve '.'. 1 : b ; then Be : 

 Lq '. '. sin. DZ : sin. lz, and l^ : \t '. '. sin. lz : cos. lz ; hence, by compounding 

 all these ratios, it is 2/> : vt'.'. c X sin. dz : b X c X sin.^ dk X cos. lz. But 

 the COS. LZ is = sin. dl X sin. dz -\- cos. dl X cos. dz ; hence 2p : vt '.'. c : 



COS Ti 7 



B X c X sm.* DK X sm. dl -|- cos. dl X ;^^ ; but in the spherical triangle 



... ... , COS.dk . COS. DZ , ^ , ^, . 



DKZ it is c : 1 . . cotan. dk or : cotan. dz or —. ; hence at lenerth is 



sin. DK SIO. DZ ^ 



produced 2/> : v/ : : c : b X c x sin. dl X sin.* dk -|- b X c X cos. dl sin. dk 

 X COS. DK. Therefore the sum of all the vt, that is, the sum of all the horary 

 variations of the inclination of the ecliptic, generated in the time of the moon's 

 revolution, the situation of the nodes remaining, is to the sum of as many mo- 

 tions p, as the sum of all the quantities 2b X c X sin. dl X sin.* dk -f- 2b X 

 COS. dl X sin. dk X cos. dk in a circle, to the sum of as many cosines c, that is, 

 as B X c X sin. dl to c. Putting therefore, as before, the mean motion of the 

 nodes, to the mean motion of the. equinoxes generated by the lunar force, as k 

 to' I, the variation of the horary mean inclination of the ecliptic, in a given 



