230 STUDIES ON NEWTON'S LUNAR THEORY. [13 



which will be found to agree closely with that given by Cotes. (See 

 Edleston, Correspondence of Newton and Cotes, p. 98.) Taking '00719, the 

 value given in the second edition of the Principia as an approximation, 

 we find x ='007 18973. 



Cotes's coefficients give x = '00719000. [June, 1873. 



A METHOD OF NEWTON'S FOR FINDING THE MOTION OF THE APSE. 



[" Two lemmas are first established * which give the motion of the apogee in an 

 elliptic orbit of very small eccentricity due to given small disturbing forces acting (1) in 

 the direction of the radius vector, and (2) in the direction perpendicular to it ....... 



Newton assumes that the form of the orbit in which the moon really moves will 

 be related to the form of the oval orbit [which the Variation produces] nearly as an 

 elliptic orbit of small eccentricity with the earth in its focus is related to a circular orbit 

 about the earth in the centre." Catalogue, p. xii.] 



If , + P, Q be the forces in the direction of the radius vector and 



i- 



perpendicular to it, we have the equations for the changes in the elliptic 

 elements, e, OT, of an orbit, 



de In i n \ d h 



-v cos (6 isrj + e sin(0 ra-) -;- = 2 - Q, 



Ctt Ct C Lt 



de . /a > . dw h D h esin(0-CT) 



-57sin(0-r) + eeos(0-Br) , = P--Q- v/r \ ; 



dt ' fit fj, fj. l + ecos(0-cr) 



h D ,,, \.h~- i a \ 2 + ecos(0 nr) 

 whence e -i- = - P cos (0 OT) + - Q sm (0 CT) - 4g -- (, 



dt /j. /A ; 1 +ecos (0 tr) 



a result which becomes identical with Newton's two lemmas, provided that 

 we neglect e in the second member on the right. 



Now we will attempt from this result to find the motion of the Moon's 

 apogee in a way similar to that which was in Newton's mind. 



Let 6 , r be coordinates of the moon moving in an elliptic orbit ; let 

 9, r be the actual coordinates ; and let us assume that 6, r are related 

 to #, r by the equations 



6 = 6, + -5- m 2 sin 20,, r = r t {l- m* cos 2 0,}, 



o 



an assumption which reduces to a known result when the elliptic orbit 

 (r , #) degenerates into a circle. 



For simplicity we will suppose the Sun stationary. 



* [Published in the Catalogue, p. xxvi.] 



