40 The Rev. Brice Bkonwix on the Theory of rlanetary Dinturhartce. 



d^t-^^^'^dtl^ d? ^^d-T^ * dr' ^'^ dr dr- ^ P ,/,3 J 

 Udt[Tr[^-*Tr)-1'd?}+-h7[7l-^V-''Tr)-'''dr^r 



n, ~1^ h77 V ^ ^ dr) "^ dT^ i • ^^^ 



This has been put under the simplest form. It still contains three terms 

 -vvliich we cannot manage ; but these may be easily eliminated. Neglecting the 

 powers of the disturbing force above the first, and integrating relative to t, add- 



iniT the correction -^, or the value of -^ when the disturbing force is made 

 ^ d-T dr 



to vanish, we have 



J^ ^ ip ^ ^ ^^0^ ^jf ^ ^y suppose. (6) 



aT dx 



Now if we include both the first and second powers of the disturbing force, 

 we have 



which gives us immediately, 



d'P xd4>- d-4, ,„ rd<p dh enpf cd'(t> Cn dipf) [d4> 



And 



d?~'^ Jt It' '''^7l?~JT\d^~''^ (It' ^ dT- J ■ 

 Make 



TT-^d?~'^d? = ^- 



Now as all the terms in the value of X after W are of the second order, 

 and will be multiplied in (5) by quantities of the first order, we may put W 



for -J- in the value of this quantity. Then 



X=W-^,lwf + '!f\^^dt-'^'Ml^Wdt, (7) 



'' h ft, J At 2/4, OT 



and (.5) will become 



