PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



127 



In passing to a second approximation we may neglect the second definite integral, 

 and may calculate the first by the help of the approximate equations (155.) ; which 

 give, in this manner, 



+ iX'{h-TSt)(h-Ps)t'dt 



= -T{{h- Pif + {\- p^Y + c^s - P3)n 



+ 24 {i^'^Pi (^1 - Pi) + ^^^^2 (^2 - Pi) + "^i^a (^3 - Pz) } 



240 



-ix-c,^sp^ + — "''-'' 



240 



945 



(160.) 



We might improve this second approximation in like manner, by calculating a new 

 definite integral C3, with the help of the following more approximate forms for the 

 relations between the varying elements Xj Xg X3 and the initial constants, deduced by 

 our general method : 



^1 — ~ 8;?i ~ g;?! — ~" |x2/{ V^ ■*■ 6 "^ 24 / 2 \^ "^ 12 "•" 60 /' 



^2— ~ 8j92 — gp^— — fj,it V "^ 6 "*" 24 / 2 V* "T" 12 ^ 60 /^ 





A^ — 



y^t V^ "T 6 "*" 24/ ~ 2 \^ "*■ 12 "•" 60/ 



M161.) 



. "^ 6 \^ "^ 60 "^ 40/ ' 

 in which we can only depend on the terms as far as the second order, but which 

 acquire a correctness of the fourth order when cleared of the small divisors, and give 

 then 



'^i=Pi-i^''t (^1 + 4 Pi*) + 4" ^' ^' (^1 + T^iO' 



X2 = ;?2 - f*^ K^2 + -kp^ V + i" ^* ^^ (^2 + T^2 V' 



X3 = ;?3-.2^(e3+-|p3^--F^^')+-B-'''^(^3+T^3<-^^^')- 



(162.) 



