228 STUDIES ON NEWTON'S LUNAR THEORY. [13 



Now assume, as in Newton's Proposition xxvin., that the orbit is an 

 ellipse, with shorter axis in the line of syzygy, that is to say, that its 

 equation is of the form 



1 cos 2 (B-ff) sin 2 (6-6'} I , ia af ^ 



-s = -T7^ \i+-T7l \s--77T-' ^[1+^ + 2x0082(0-0')]. 



r 2 a?(l-xf a-(- - 221 - 



If the line of syzygy were fixed in direction the curvature at the extremity 

 of the minor axis (A) would be 



'1 xY I 1 4ce ~| 



~J I r*i / fi / "I 'VI /"/ / 1 ^_ -V 1 I I 1 I / VI~ I 



^ L ~t~ fty 1-V I 1 tA/ J Cv I L iL/ I | I 1. T^ X/ I _J 



f/0' 



but it we take -r^ = m, 



au 



the curvature at the same apse of the (now revolving) ellipse appears by 

 differentiation of the above value of r, 



1 f <ix ~\ 



at A, the nearer apse , 1 . - T -(l m)- , 



a (I -x)[_ (l+ce)- x ' \ 



Ay\ 



at C, the further apse -,-- , 1 + /1 \s^~ w 



and the former curvature is to the latter as 



: ( 1 + x) [( 1 + xf ( 1 - m) a - + (l-xY (2m - m 5 )], 

 which agrees with Newton's result, p. 401 (Second Ed.), line 4. 



Now the forces upon the Moon at the points A and C, according to 

 Lecture II. p. 8, are 



m'r 



. 

 at A > 



n P- 

 at C, ^ 



m ' r 



m> 



or i = 5 



l + 2w 2 1 + ^m 2 



these are as . _ , 2 2m 2 (1 x) : -r- - r;+m'(l+x) > 



a ratio which Newton gives, not quite correctly (p. 399, line 9) as 



1 2m 2 1 m 2 



* [This is the analytical verification of Newton's "Rationes autem ineundo invenio quod 

 differentia inter curvaturam &c." Prop, xxvm.] 



