﻿464 A NEW METHOD FOR CORRECTING A PLANET S ORBIT. 



The law of areas gives us 



r r —j— = I: \/p =: a a cos <p . ft, (10) 



and from (7), (9), (10), 



S, = 'Jl + 'ljt^JL+JL,i^ES,p + U^^-l)5x. (11) 



fi ' fi ' fia cos cp ^ ' fi \aa cos f / 



It may be sometimes more convenient to use 



uhz = d'L,-\-tSu+'—^^am^S(f, + i—^ l\8x- (12) 



f^ ' ' '^ ' a cos g) ^ ' \a a cos 9) / 



12. The equation p = {^^) m"' co*^ "P 



gives log ^ = -|- f log k — f log sin 1" — § log /i + 2 log cos cp ; 



and therefore, f log ;u = + § log i- — f log sin 1" — log p + 2 log cos <p ; 



^ — —§d\og p — Stan <pdq,. (13) 



If, as we propose, we consider ;ii as a function of log p and (p, we shall get 



D,ogpZ = — f< (14) 



D, z = — 3 < tan ffi + -^^P— sin E, (15) 



* ^ ' fi a cos 9 ^ ' 



which will be combined with Dl, = = - ; (16) 



D,. = l(-^1'- 1). (17) 



* /i \o a cos (jp / ' 



13. As is well known, ''— 1 -j-ecos (^ — x) ' ^^^^ 

 but we also have, by (6), ?• = r c" , 



<f log r = c? log r + <^ w' , (18') 



d log r , 



by considerations mentioned in § 11. And as, likewise, we can omit the products inter 

 se of the variations of r, among which is r — r, we shall have, near enough for our 

 present purpose, 



d log t rf log r _ 



dt ~ dt ' 



so that d IV = 5 log r — d log r, 



,. , . (? log r 



= S\ogr-Sz^~, 



= 5losr-S.,'-^^^; (18") 



