VOL. L.] PHILOSOPHICAL TRANSACTIONS. 303 



situation of the moon's orbit after that time; and if on the same be demitted 

 the perpendiculars Nm, Rr, the latter nr will denote the variation in the inclina- 

 tion of the lunar orbit to the ecliptic generated in the same time. Also Nn : not 

 :: 1 : m, and ntw : Rr :: 1 : sin. lr; but because nl = gr, ng = lr; hence, unit- 

 ing the ratios, nw : Rr :: I im X sin. ng: whence it appears that the momentary 

 variation of inclination is proportional to the sine of the distance of the moon's 

 ecliptic node from the equatorial node. On the diameter nm demit the perpen- 

 dicular GK, and oh being the decrement of the arc ng made while the equatorial 

 node N described the arc n??, make hk parallel to gk, then it will be 1 : gk or 

 sin. NG :: gA : k^; and therefore now it will be nu : Rr :: Gh:m X k;^; and hence 

 the sum of all the variations Rr, generated in the time while the ecliptic node g 

 describes the arc mg, will be to the sum of as many motions nw, that is to the 

 motion of the equatorial node n made in the same time, as the sum of all the 

 K^ X m, to the sum of as many arcs gA, that is, as m X mk to mg. Let nh 

 of the node n in the time of a revolution of the node g from the one equinox 

 to the other, then the variation of the inclination generated in the same time, or 

 the total variation, will be -. Hence, since ^^ expresses the ratio of 



MGN ' MGN ^ 



the motion of the equatorial node to the motion of the ecliptic node, there re- 

 sults the following theorem : the motion of the moon's ecliptic node is to the mo- 

 tion of the equatorial node, as double the sine of the mean inclination of the 

 lunar orbit to the equator, to the sine of the total variation in the inclination of 

 the same orbit to the ecliptic. 



In this computation I assumed the mean inclination of the lunar orbit to the 

 equator, viz. 23° 28^5 since in the revolution of the node it is increased as much 

 on the one side as diminished on the other. But the motion of the moon's ecliptic 

 node, is to the motion of her equatorial one, as IQ" 20^' to ll-l'^ or 15'', or as 

 6055 or 4642 to 1, hence by the above theorem the total variation in the incli- 

 nation comes out 27" or 35'', according as the difference of the earth's axis is 

 ■^4^ or T-^. By this quantity then the inclination of the lunar orbit, to the 

 ecliptic will be increased in the moon's ascending node passing from the vernal 

 equinox to the autumnal, and equally diminished in the other half of the node's 

 revolution. In any place g between the equinoxes, the variation of the inclina- 

 tion is to the total variation, as the versed sine of the arc mg is to the diameter, 

 as is evident ; or the difference between half the total variation and the variation 

 sought, is to the said half total variation, as the cosine of the arc mg to radius, 



that is as M — '— to I. a. e. i. 

 P 

 Prop. 5. To Investigate the Motion of the Apses in a Satellite's Orbit nearly 



Circular, so far as arises from the Spheroidal Figure of the Primary Planet. 



By prop. 1, the disturbing force, by which the satellite is urged towards the 



