438 PHILOSOPHICAL TRANSACTIONS. [aNNO 1754, 



PM generally to express the instantaneous alteration of the inclination of the 

 earth's axe to the ecliptic for any given declination of the sun. 



The sum of the lines pr is always the same, and has the same sign, or the 

 same direction, during every quarter of the sun's revolution, whether he moves 

 from T to 05, or from 25 to £i, or from £i to Vf, or from VJ to 7"; so that 

 the precession answering to any one quarter of the sun's revolution about the 

 earth, or to 3 months, being known, that multiplied by 4 will be the annual pre- 

 cession ; by 8 will give it for 2 years ; by 1 6 for 4 years, &c. 



Likewise the sum of the lines pm is ever the same for every quarter of the solar 

 revolution ; but it has alternately a contrary sign ; that is, a contrary direction. 

 During the quarter from 7" to 25, the alteration of the inclination of the earth's 

 axe to the ecliptic is positive, and the angle of the inclination increases ; but 

 during the succeeding quarter, or from 25 to ^, the alteration of the inclination 

 is negative, and the angle of the inclination diminishes: and as the diminution 

 from 25 to :^ is equal to the augmentation from y to 25, it follows, that at 

 the end of the semi-revolution the inclination of the earth's axe to the plane of 

 the ecliptic will become again the same, having undergone an oscillation, which 

 is completed in a semirevolution. It is the same, when the sun passes from zii: 

 to T- The angle of the inclination increases from :Ci: to VJ, and decreases from 

 yy to y, where it becomes again the same as it was at iCb. 



And hence the inclination of the earth's axe to the ecliptic may be considered 

 as constant, though subject to this oscillation, and indeed to several others, they 

 being all regular, and performed in regular periods. The earth's inclination to 

 the ecliptic being constant, and the motion of the pole which produces the pre- 

 cession, being always parallel to the plane of the ecliptic, the earth's pole moves 

 in a parallel to the ecliptic, about 23-1- degrees distant from the pole of the eclip- 

 tic, and the terrestrial axe describes a conical surface. To this motion of the 

 terrestrial axe, or pole, is to be ascribed the apparent motion of the stars about the 

 pole of the ecliptic. 



But hitherto we have not considered, that to the precession, thus caused by 

 the sun, we are to add likewise that produced by the moon ; and it remains, that 

 we examine into the motion of the earth's pole, caused by the action of the moon 

 on the redundant matter about the earth's equator. Now all that has been said 

 concerning the sun, is alike applicable to the moon, which we may put in the 

 place of the sun ; the moon's orbit in the place of the ecliptic ; and the time of 

 the moon's revolution round the earth in the place of the revolution of the sun 

 round the earth : and we shall find the motion of the earth's pole parallel to the 

 lunar orbit, which is always the same at every quarter of the time of the revolution 

 of the moon round the earth, and the oscillation of the earth's axe to the plane 



