80 PHILOSOPHICAL TRANSACTIONS. [aNNO 1755. 



and on ta the perpendicular nr. If the moon be supposed to revohe in the 

 circle hno, where it touches it at the place n, the horary precession of the equi- 

 noxes generated by the lunar force, by the demonstration in prop. 1, will be as 

 Ni* ; but that precession increases in the ratio of the force that produces it, and 

 this force is in the inverse triplicate ratio of the moon's distance tp , therefore the 



l'tI 



horary precession is as — j, or as the same quantity — ^ drawn into the constant 



^ ^, ^ . Nl' X N« NI X IW 1 1. 1 CA' + TC X TR , ,, . 



sector TP/), that is, as or as ; but — = • — by the 



^' ' TP TP TP CA X CD* •' 



nature of the ellipse; hence the total precession, generated while the moon re- 



C^ A '" I ■ T C y T R 



volves in her orbit, is as the sum of the quantities ni X i^ X $— in the 



circle; or (because the ambiguous term -| -^ may be rejected, as being 



positive through one half of the circular circumference, and negative in the other 

 hall), as the sum of all the factors ni X iwi in the circle, that is, as the area of 

 the circle hnoh itself: and therefore the precession of the equinoxes, in every 

 revolution of the moon, remains the same in any situation of the apogee. 



The horary variation of the inclination of the ecliptic, if the moon be in n 

 revolving in the circle hno, will be, from what is demonstrated in prop. 3, as 

 NI X Ti : but if the moon be transferred to p, the same variation will be as 



MI X TI NI X TI ^^ ^ 1.U i. • ^'I X TI X N/t NI X na , , 



J—, or as J — X TP/), that is, as ,oras by drawing 



•nq parallel to ti ; and therefore, because of the given parts in the ratio, the va- 

 riation of the inclination of the ecliptic, generated in the time of the moon's 

 revolution, is as the sum of all the factors ni-X nq in the circle, that is, nothing. 



Hence there can be nothing legally collected from the situation of the moon's 

 apogee, to induce any variation, either in the motion of the equinoxes, or in 

 the obliquity of the ecliptic. 



Scholium. From the premises it appears, that the double motions of the 

 earth's poles, by the forces from both the sun and moon separately, unite , the 

 one in a plane parallel to the ecliptic, by which the equinoctial points are con- 

 tinually retracted in antecedentia, and therefore the stars seem to proceed in 

 consequentia ; but the other motion is perpendicular to the plane of the ecliptic, 

 by which the poles of the earth incline and oscillate, by turns approaching to 

 and receding from the pole of the ecliptic, and hence changing the declination 

 of the stars. We deduce the quantity of these motions from the excess of the 

 earth's altitude at the equator above its altitude at the poles, according to a double 

 hypothesis, viz. by which that excess is estimated either the 230th or the 178th 

 part of the whole altitude, hitherto used by the chief mathematicians. But if 

 the known nutation of the earth's axis be assumed, which hitherto has been 

 stated at 1 8" as due to the action of the moon, and there be hence required the 



