PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 115 



in which e^ €2 e.^ p^ p^ p^ are the initial values, or values at the time 0, of vi^ fj^ % 

 ^1 "^2 ^3 5 ^^^ S is the definite integral 



pt / 8H , 8H , 8H ,,\ , 



^=Jo V^i8-^ + ^2 8^+^3 8:^3-HJrf/, (85.) 



considered as a function of pjj 1^2 % ^1 ^2 ^3 ^^^ ^- l^e quantity H does not change 

 in the course of the motion, and the function S must satisfy the following pair of 

 partial differential equations of the first order, analogous to the pair (C), 



8S , „/8S 8S 8S \ ^,. - 



"87 + F (^g^, g^, g^, pji, yi2, n,) = U {n,, n-,, %) ; 



, T./8S 8S 8S \ XT, s \ 



8S 

 8^ 



f 



(86.) 



This important function S, which may be called the principal function of the motion, 

 may hence be rigorously expressed under the following form, obtained by reasonings 

 analogous to those of the seventh number of this Essay : 



S = ^1 +/o{- ^ + U (rj„^2,^^) - F (l|.^.||' ^1,^2,^3)} dt ] 



^^l?f^^ ^Si SS 8S, 8S 8S1 \,^ I ^ . 



■^Jo ^VH-H''H-8^'8^-8^''^1''^^'W^^' J 



Si being any arbitrary function of the same quantities v^ ;?2 ^3 ^1 ^2 ^3 ^j so chosen as 

 to vanish with the time. And if this arbitrary function S^ be chosen so as to be a 

 first approximate value of the principal function S, we may neglect, in a second ap- 

 proximation, the second definite integral in (87.)- 



21. A first approximation of this kind can be obtained, whenever, by separating 

 the expression H, (82.), into a predominant and a smaller part, H^ and Hg, and by 

 neglecting the part Hg, we have changed the differential equations (83.) to others, 

 namely, 



1 



yi r ^"-^ 



dt ~~ 8>ji' dt "^ 8»j2' dt ~~ SiJa'J 



and have succeeded in integrating rigorously these simplified equations, belonging to 

 a simpler motion, which may be called the undisturbed motion of the point. For the 

 principal function of such undisturbed motion, namely, the definite integral 



S.=X'(-iS+-2|5 + -3'|f-H,)'^^, (89-) 



considered as a function of f^^ rj2 fj^ ^1 ^2 ^3 ^j will then be an approximate value for the 

 original function of disturbed motion S, which original function corresponds to the 

 more complex differential equations, 



Q 2 



