VOL. XLVIII,} PHILOSOPHICAL TRANSACTIONS, 439 



of the lunar orbit, which is completed in each semirevolution of the moon round 

 the earth. 



But whereas the plane of the lunar orbit, which is always inclined to the plane 

 of the ecliptic in an angle of about 5 degrees, never continues in a constant po- 

 sition, like the plane of the ecliptic, so that its pole describes a small circle paral- 

 lel to the ecliptic, at the distance of about 5 degrees fham its pole ; it follows, 

 that the precession, with respect to the lunar orbit, is not the same as with re- 

 spect to the ecliptic ; and that the motion of the pole parallel to the lunar orbit 

 should be referred to the plane of the ecliptic : which is done by resolving the 

 motion of the pole, parallel to the plane of the lunar orbit, into 2 motions, the 

 one parallel to the plane of the ecliptic, and the other perpendicular to it, and in 

 the plane of the solsticial colure. 



The former of these 1 motions gives the precession with respect to the eclip- 

 tic, and has its direction always the same way. The hitter motion has 2 opposite 

 directions, in the 2 semirevolutions of the pole of the lunar orbit round the pole 

 of the ecliptic, and causes an oscillation of the terrestrial axe on the plane of the 

 ecliptic, which is completed in a revolution of the pole of the lunar orbit round 

 the pole of the ecliptic. 



From what has been said, it follows, that there are 5 distinct motions of the 

 pole of the earth ; namely, 2 of precession, which are parallel to the plane of the 

 ecliptic, and 3 of oscillation on the plane of the ecliptic. The 1 of precession are 

 caused, the one by the sun, the other by the moon. That which is caused by 

 the sun is constantly the same, at every quarter of the time of the revolution of 

 the sun round the earth ; that is, every 3 months : that which is caused by the 

 moon, is constantly the same at every quarter of the time of the revolution of 

 the moon round the earth ; that is, about every 7 days. 



Of the 3 motions of oscillation, one is caused by the sun, and is completed in 

 the time of the semirevolution of the sun round the earth, taken from one equi- 

 nox to the following one ; that is, in 6 months. Another is caused by the moon ; 

 and each oscillation is completed in the space of a semirevolution of the moon 

 round the earth ; that is, in 14 days. The third is caused likewise by the moon, 

 and arises from the plane of her orbit being different from the plane of the eclip- 

 tic, and from the pole of the lunar orbit making its revolution about the pole of 

 the ecliptic in about 18-5- years. And this oscillation is completed in the time of 

 the revolution of the pole of the lunar orbit about the pole of the ecliptic ; that 

 is, in about 18-| years. 



It will appear in the memoir, that there is a relation purely geometrical be- 

 tween the quantity of the nutation, during the time of the semirevolution of the 

 pole of the lunar orbit, and the quantity of the precession, caused likewise by the 

 moon in the same time. This lelation is quite independent of the force of the 



