122 



and 



PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



P\- 



— !^ _>?! - 



Ze, 



^2 — — g ^„ — t ' 



(133.) 



^3 - - Si" - "I— + 2 ^ ^ . J 



the latter of these two sets coinciding with (127), and the former set conducting to 

 (128.). 



26. Returning now from this simpler motion to the more complex motion first 

 mentioned, and denoting by Sg the disturbing part or function which must be added 

 to Si in order to make up the whole principal function S of that more complex mo- 

 tion ; we have, by applying our general method, the following rigorous expression 

 for this disturbing function, 



in which we may, approximately, neglect the second definite integral, and calculate 

 the first by the help of the equations of undisturbed motion. In this manner we find, 

 approximately, by (125.) (127-), 



-H2= -^ ^(^e,-^ p,tY -^ {e^^ p^ty^ -^ '^ {e^-^rPzt - ig^Y, (135.) 

 and therefore, by integration, 



So 



- i {^' W+P2^) + "' {Pi-ge,)} ^+ 1,2^^^^ _ ^,2^2 t^, 



(136.) 



or, by (133.), 



S2 = — ^ W + ^1 ^1 + ^1^ + 7}^ + ^2 ^2 + ^2) 



iU^'t 



- V{''3^ + ^3^3 + ^3^+T^('^3 + ^3)^^+^^^^} •• J 



(137.) 



the error being of the fourth order, with respect to the small quantities ^, v. And 

 neglecting this small error, we can deduce, by our general method, approximate forms 

 for the integrals of the equations of disturbed motion, from the corrected function 

 Sj + Sg, as follows : 



""i ~ H + H - ~~t 3" V'^i + "2 ^1/' 



2 - 8^ + S^ - ~~t 3" V''2 + 2" ^2)» 



'^3 = H + s^ = "V-"-T^^-— ('^3+ 2-^3 + T^^V' . 



(138.) 



