13. 



STUDIES ON NEWTON'S LUNAR THEORY. 



[Ix appears from an undated fragment among his papers that Adams at one time 

 proposed to give an outline of Newton's methods and results in the Lunar Theory, of 

 which he says " there is no part of Newton's great work which displays more con- 

 spicuously the genius of the author, or better illustrates his manner of working." His 

 attention was directed to the subject in relation to the Portsmouth papers*, and the 

 second investigation below is an analytical parallel to a method of obtaining the motion 

 of the apse, found among those papers and described and, in part, published in the 

 Catalogue (pp. xii, xxvi).] 



ANALYTICAL INTERPRETATION OF NEWTON'S INVESTIGATION OF THE 

 LUNAR INEQUALITY OF THE VARIATION. 



Let a be the actual mean distance of the Moon from the Earth; a(l x] 

 and a ( 1 + x) the least and greatest distances ; of, n, n', m, ju,, r, 6, 6' have 

 the meanings attached to these symbols in the Lectures, f> as sim ; e, e', i are 

 supposed to vanish ; 

 then as in Lecture VI. p. 24, we have 



1 dH 3 dt . . 



Put for ' its approximate value - ; then 

 dd n 



1 dH 3 . . _.,.., ,j A 

 HdJ- -2 - sm2 (*-*) 



dff 



Integrate, considering -ja = m > as ^ 1S ver 7 nearly; 



, H 3 m 2 ,., .,> 



lo ^ = 22-2m COS2 ^-^' 

 or putting h = na?, 



l + - A , cos 2(6- 6')} . 

 41 m ' } 



This agrees exactly with the result of Prop. xxvi. Lib. in. 



* See "A Catalogue of the Portsmouth Collection of Books and Papers written by or 

 belonging to Sir Isaac Newton," Cambridge, 1888. 



292 



