VOL. XLIX.] PHILOSOPHICAL TRANSACTIONS. 2Q 



month, will be to the mean horary motion of the nodes p/, as b X c X sin. dl 

 to c X k, that is, because sin. dl = -, andc = cq -{■ hpu, as cpqv -f bp'^vu to 

 c X k, or as pv to k very nearly ; therefore the sum of all the variations of the 

 inclination of the ecliptic, while the moon's node describes the arc ef, is to the 

 motion of the node ef, as the sum of all the pv is to the sum of as many times 

 k, that is, as/) x (1 + 2^) to A X ef ; and the whole variation that changes the 

 inclination of the ecliptic, in the regress of the node from one equinox to an- 

 other, is to the motion of the nodes 1 80°, as 2p to k X el, which therefore is 

 equal to ^-^~— , and which therefore will be easily produced by the following 

 theorem : " The motion of the nodes, is to the motion of the equinoxes pro- 

 duced by the lunar force, as the sine of the inclination of the lunar orbit to the 

 ecliptic, is to the sine of half the whole variation of the inclination of the ecliptic 

 to the equator." 



If the ratio of the earth's diameters be 44-2-, the motion of the moon's nodes 

 will be to the motion of the equinoxes, by the prop, as 1753 to J, and as 1901 

 to 1 if the ratio of the diameter be J-f-f. In the former case the theorem will 

 produce 21^'' 5''' for the whole variation of the inclination of the ecliptic, in the 

 latter case \g" 27" ; generated while the lunar nodes pass from one equinox to 

 another. In places between the equinoxes, the variation will be to the whole 

 variatton, from the demonstration, as 1 -j- m to 2, that is, as the versed sine of 

 the node's distance from the vernal equinox, to the diameter ; or, the difference 

 between half the whole variation and the variation for a given time, is to half the 

 whole variation, viz. to 10''' 32i'", or g" 43^'^'' as the cosine of the node's distance 

 from the vernal equinox, is to radius : and this difference or equation is to be added 

 to the mean inclination of the ecliptic in the node's regress from the summer sol- 

 stice to the winter, but to be subtracted in the other half of the node's revolution, 

 to obtain the true inclination of the ecliptic. And the greatest obliquity of the 

 ecliptic is when the moon's ascending node is in the vernal equinox or beginning 

 of Aries ; but the least, when the same node comes back to the autumnal equinox, 

 or to the sign Libra, q. e, i. 



Prop. V. Prob. To investigate the Inequalities of the Precession of the Equi- 

 noxes, and of the Variation of the Obliquity of the Ecliptic, which depend on the 

 Situation of the Moon's Apogee. — Let the moon describe in the plane of the 

 ecliptic the ellipse apbl (fig. 6), of which c is the centre, t the focus occupied 

 by the earth, ab the greater axis, cd the less semi-axis, tl the common section 

 of the planes of the equator and ecliptic. Let the moon be at p, and draw tp, 

 Tjb, cutting off the sector Tvp described by the moon's horary motion. With 

 the centre t, and radius equal to the greater semi-axis ca, describe the circle 

 HNO, cutting TP and Tp in n and w; and on tl demit the perpendiculars ni, nnij 



