PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



121 



25. Suppose now, to illustrate the theory of perturbation, that the constants (ji,, v 

 are small, and that, after separating the expression (111.) for H into the two parts, 



Hi = :|(V + ^2^ + «^3^) + §'>?3. (124.) 



and 



H2 = i{//'f(^i2 + pj2') + ''2^3n, (125.) 



we suppress at first the small part Hg, and so form, by (88.), these other and simpler 

 differential equations of a motion which we shall call undisturbed: 



^ _ ^ __ 



d% 



IT — ^' IT — ^' IT —'~S- 



(126.) 



These new equations have for their rigorous integrals, of the forms (94.) and (95.) 



and 



^1 = ^1 + V\ *, ^2 = ^2+P2^> ^3 = ^3+P3^ — ig^» ' 



^1 = Pl> ^2 =P2> '^3=P3-- g^ 



(127.) 

 (128.) 



and the principal function Sj of the same undisturbed motion is, by (89.), 





=( 



OT 2 + OT ■ 



g^3 



)d. 



p{'+pi +pi 



Px + Vi + Vi 



2 



-^^3 - '^gP3t-\-g'^t'^)dt 

 — §^3)* - gP3^^ + ig'^^> 



> , 



(129.) 



or finally, by (127.), 



^1 — 97 q g^VlS^ ^3) ^ QAg ^ ' 



(130.) 



(131.) 



This function satisfies, as it ought, the following pair of partial differential equa- 

 tions, 



^+4{(ify+(a)V(g)l=-..> 

 ^■+i{(sy+(ify+(it)7=-^^- 



And if by the help of this pair, or in any other way, the form (130.) of this principal 

 function 3i had been found, the integral equations (127.) and (128.) might have been 

 deduced from it, by our general method, as follows : 



ra*, = 



_ S^i __ ^i — gi 



8>)i 



ar, 



8S1 



^^3-8,3 



8 Si ___ 



MDCCCXXXV. 



*i2 -- gg 

 t ' 



t 26 *5 ^ 



R 



. . . . (132.) 



