VOL. XLIX.] PHILOSOPHICAL TRANSACTIONS. 27 



39" 717 

 cession generated by the lunar force is = — j x (be — (c* — b*)pu ), or 



36" 625 



— ~ — X (be — (c^ — b'^)pu ), having no regard to the change of situation of 



the nodes made in that time, the difference between the annual mean and great 



. • -11 u 39".7nlc'- b^)p 39".717 X 2sin. 5°8'4 36".625x 2sin.5°84' 



est precession will be -— — —zt:?—,;- = — : ;:; :r^7r:;-^7r---, or ^. 



^ tan. 2 X 23° 28 i tan. 2 x 23° 28 4 ' tan. 2 x 23° 284.' 



Therefore, " The tangent of double the inclination of the ecliptic to the 

 equator, is to the sine of double the inclination of the lunar orbit to the ecliptic, 

 as the mean annual precession of the equinoxes generated by the lunar force, to 

 the difference between the mean precession and the greatest or least." Hence 

 in the former case that difference is = 6" 37'", in the latter 6" &" ; and there- 

 fore if the total annual precession be stated at 50'' 20"', in that year, in the 

 middle of which the moon's ascending node is near the first degree of Aries, the 

 precession of the equinoxes will be 56'" 57", or 56* 26""' ; but when the node is 

 in the sign Libra, the precession of that year will be 43" 43"", or 44'' 14'". And 

 because the aforesaid difference, in other times, is as the sine of the node's dis- 

 tance from the solstitial points, it will be easily found for any year, the situation 

 of the nodes being given. 



Prop. III. Prob. — To f|nd the Variation of the Inclination; of the Ecliptic to 

 the Equator generated by the Sun's Force. — Every thing remaining as in the 1st 

 proposition, produce the arc ls (fig. 3) to v, till lv be a quadrant of a circle, 

 and demit v* perpendicular to the arc /)N produced ; then will v* be the measure 

 of the horary variation of the inclination of the ecliptic to the equator. But v^: 

 Lr :: ti : pi, and \.r : Ln :: b : 1, and by prop. 1 the horary precession of the 

 equinoxes lw, is to the horary precession when the sun is in the solstices, which 

 call g, as RH* to tr'^: therefore, compounding the ratios, 



6 X TI X RH^ , , , C X RH TH ., . 



vs : e :: : tr% or, because tr = 1, pi = , ti = — , it is 



* PI PM ' PM 



V* ; g- " * - X RH X TH : 1 ; and the sum of all the horary variations vs, while 

 the sun describes the arc lr, is to the sum of as many angles g, as the sum of 

 all the factors RH X th drawn into -, to the sum of as many squares of 1, that 



is, as -^RH^ X - to the arc lr ; and the total variation, by which the inclination 

 of the equator to the ecliptic is diminished, in the sun's progress from the equi- 

 nox to the solstice, is to the sum of the angles g, (which then becomes equal to 

 half the annual precession generated by the solar force, that is, equal to half of 



10^.583 or half of 13^.675) as -- to the arc lv ; and hence the total variation 



.„ , 6 X 10".583 10".583 X tan. 23°28'4- 13".675 x tan. 23''28'4 „ 



Will be = -, or —— ^. Hence arises 



c X 4lv 4lt 4lv 



this theorem : " The motion of the sun, is to the motion of the equinoxes ge- 



E 1 



