XIH.J THE MOTION OF THE APSE, ETC. 63 



we have 



dH 3 309 |~15 9 45 ~| 



~d = 4 + "32" + L~4 " r + 2 m T J C 8 ^ ~ 

 If we write md H = ty, 



this becomes ~r^ a ^ cos 2i|, 



at/ 



where 



3 309 ,15 9 45 



a = m - m TO , b = m~ + - m -\ m , 



4 o 2i 4 2t 8 



and the solution is 



/ /d -4-1) \ , 



tan~ l ( ^/ T- tan \\i \ = 6 Jo? b-t- constant. 



Hence if we denote by -^ the mean rate of change of OT, we have 



de 



3 225 4071 



3 225 4071 

 = m+ - m 



We observe that a + 6 and a ?> are the rates of separation of the Sun 

 from the apse when the Sun and the apse are at quadratures and syzygies 

 with one another, respectively, that is if we take n for the longitude of 

 the apse, or, what is the same thing, if we ignore small terms of short 

 period. Hence the mean rate of separation of the Sun from the apse is 

 a mean proportional between its rates when at quadratures and syzygies 

 respectively with the apse"". 



[* This is the analogue for the case of the apse of Machin and Pemberton's theorem on the 

 motion of the node, inserted in the third edition of the Principia as a scholium to prop, xxxm., 

 lib. m. See some notes by Adams in Brewster's Life of Newton, Appendix xxx.] 



