26 PHILOSOPHICAL TRANSACTIONS. (^ANNO 175 5. 



on account of the smallness of p, being the sine of 5° 8;'; then is Ln to p/as 

 be — (c* — y^) X pu to b X c X k; and the sum of the motions lw to the sum 

 of the motions f/, while the node p describes the arc ef, as the sum of the 

 quantities be — (c^ — b"^) X pu to the sum of all the b X c X k, that is, as ^ X 

 c ^ EP 4- (c- — b^) X pv to b X c X k X ef: and therefore while the node is 

 passing from the equinox to the solstice the precession of the equinoxes is 



^ + ,^^ ^JL^ 52. and while the node passes from one equinox to the 

 AoxcxAxea' '■ * 



other the precession is -j-. From the former motion taking half the latter, 

 there will remain ^-V- — , ^ ^ ^ for the difference between the true and the 



X C X « X EA 



mean precession, that is, for the greatest equation of precession, viz. when the 

 lunar nodes are in the solstitial points. In other places it appears that this equa- 

 tion is to the greatest equation, as the sine of the nodes distance from the equi- 

 nox is to radius, adding it to the mean precession in the regress of the ascending 

 node, from the vernal equinox to the autumnal, but subtracting it from the au- 

 tumnal to the vernal equinox. It is to be noted also that c^ — b'^ = 2c^ — 1 =z 

 cosine 2 X 23° 28^', and b X c = ± sine 2 X 23'* 284-', and hence 

 c^ _ 6^ ^ !i^. 2 X 23° 28V = tA^^i;^" Therefore ^;,:i^^-4A21^ 



i X c sin. ^ tang Z x 23° 28y 6 X c x A x ea 



(V)o v^ ^ gjjj 5° 8»' 



becomes r — ; and hence comes this following theorem: 



" The tangent of double the inclination of the equator to the ecliptic, is to the 

 sine of double the inclination of the lunar orbit to the ecliptic, as radius to the 

 sine of a certain angle; and then, the mean motion of the nodes, is to the mean 

 motion of the equinoxes generated by the moon's force, as the sine before found 

 to the sine of the greatest equation of the equinoxes." Instead of the sine of 

 double the inclination of the lunar orbit to the ecliptic, in the theorem, may be 

 taken double the sine of the same inclination, since the error hence arising is of 

 no consequence, as may be easily found on trial. But the annual motion of the 

 moon's nodes is IQ" 20^^'; and the annual motion of the equinoxes generated by 

 the lunar force is 3g".7l7 by corol. 1 prop. 1, the ratio of the earth's diameters 

 being = 4ttj whence A is = 1753. And the motion of the equinoxes, when 

 the ratio of the earth's diameters is -j-ff, is 36 ".625, and then k = 1C;01. Hence 

 in the former case the greatest equation of the equinoxes produces ig'" 38" in 

 the latter 18" J 6'". q.e. i. 



Corol. From this proposition, the precession of the equinoxes generated by 

 the lunar force, in a given time, is proportional to the quantity b X c X k, or 

 be — (c^ — b"^) pu : therefore it is greatest when the moon's ascending node is in 

 the beginning of Aries, for then u is = — 1 ; and it is least when that node is 

 in the sine Libra, because w = i in that case. Hence, since the annual pre- 



