116 



PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



«?Wi 8 Hi 8 Hg f^^-sTa 8 Hi 



dri 

 ~dt 



dt 



8>3i 



8*)i ' dt "~ 8)j2 8>j2» dt 



8H2 <^^3____^Hj^_^ I 



(90.) 



The function Sj of undisturbed motion must satisfy a pair of partial differential 

 equations of the first order, analogous to the pair (86.) ; and the integrals of undis- 

 turbed motion may be represented thus, 



SSi _ 8S1 1 



(91.) 



Pi 



8 Si 

 1 djji 





while the integrals of disturbed motion may be expressed with equal rigour under the 

 following analogous forms, 



?_Si j_ 8_S2 1 



__ 8S1 8^2 

 ^"1 ~ 8);i "I 8>)i- 



Pl 



8 Si 



_ 8S1 SS2 

 8 Si 



Oi 8 Sg 8 Si 8 Sg 8 Sj 8 Sg 1 



~ 8^ ~ 8^' P2— ~ f^'" 8^' Ps— - jy^ — f^y J 



^3- g;; + 8,3. 



^ 



(92.) 



if S2 denote the rigorous correction of Sj, or the disturbing part of the whole principal 

 function S. And by the foregoing general theory of approximation, this disturbing 

 part or function 83 may be approximately represented by the definite integral (T.), 



S2=-X^2^^' • (93.) 



in calculating which definite integral the equations (91.) may be employed. 

 22. If the integrals of undisturbed motion (91.) have given 

 ^1 = <Pi {t, €\y ^2, ^3, p^, P2, Pi), ^ 



^2 = ^2{ti^l^^2yHi PvP2^Pz)i > (94.) 



^3 = <P3 (^J ^1> ^2> ^3> P\^ P2i Pz)i J 



and 



^1 = -^1 (^^ ^IJ ^2? ^3J /'IJ P2J P^)y'\ 



, «^2 = -+2 (^» ^1> «25 ^3* />1» ;?2J ;^3)j /^ (95.) 



•^i = •^'S (^^ ^1> ^2J ^3J i^D ;'2J ?3)j J 



then the integrals of disturbed motion (92.) may be rigorously transformed as follows, 



^ t J. I ^ ^2 I ^ ^2 I ^ SoX 



^1 = <Pl \ty ^1, e^, ^3, pi + g^, ;?2 + y^^, ;?3 + jfj, 



^2 = <P2 (^. ^1, ^2. ^3,;^1 + g^f, P2 + -g^', i?3 + g^f). > . . . (96.) 

 ^ / J. I ^ So , 8 So ,8 SgX 



^73 = <Pz \U ei, e2, e3, p^ + -g^^ ^2 + i-e^ i^3 + f^). 



