PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



117 



and 



1 = K "^ "^^ V' ''' ^2' ^^'^^ + J7,^P2 + 8^, Pz + nf ), 

 8 So , , / , I ^ So I ^ So , S SoX 



2 = 8^' + ^2 (^. ^1, ^2, ^3./>l + fT,'P2-\' ff^, PZ + 87-)^ > 



S So , , / , I ^ So I ^ So , S So\ 



3 = 8^ + ^3 (^. ^1, ^2. ^3,?l + 8^, 7^2 + 8^, ^^3 + 8^), ^ 



(97.) 



Sg being here the rigorous disturbing function. And the perturbations of position, at 

 any time t, may be approximately expressed by the following formula, 



together with two similar formulae for the perturbations of the two other coordinates, 

 or marks of position ;;2J '73- In these formulae, the coordinates and H2 are supposed 

 to be expressed, by the theory of undisturbed motion, as functions of the time t, and 

 of the constants e^ eg 63 pi p2 Ps- 



23. Again, if the integrals of undisturbed motion have given, by elimination, ex- 

 pressions for these constants, of the forms 



^1 = ^1 + ^1 {^3 ^13 ^23 ^3J ^15 ^2J ^3)3 1 



^2 = ^2 + ^2 (*3 ^\3 ^23 ^33 ^1? «^2J ^^3)^ }- (99). 



), J 



and 



^3 = ^3 + ^3 (^3 ^15 ^25 ^3> ^IJ ^23 ^3); 



J»l = ^1 + "*"l {^3 '7l> ^2J '73> '^IJ ^2> ^3)^ 1 



JO2 = ^2 + "^"2 i^3 ^IJ ??25 '?3J '^IJ ^^2? ^3)? )• 

 J»3 = ^3 + "^3 (^^ ^IJ '?2J ^33 "^13 ^25 ^3) ; J 



and if, for disturbed motion, we establish the definitions 



^1 = 'Jl + ^1 (^J ^^l^ ^2J ^3J '^IJ ^2J ^3)5 T 

 «2 = ^2 + ^2 (^> *?!» '^2? ^3J ^15 ^2J ^3)3 

 «3 = '^S + ^3 {^3 ^\3 %3 ^^SJ ^1> ^^2? ^3)5 



(100.) 



and 



^1 = "^1 + "^1 (^3 ^13 ^2J ^33 ^^15 ^2J ^^3)3 1 

 ^2 = ^2 + ^2 {^3 'Ju »?2> %> ^'li ^2? ^3)5 r* 



^3); J 



(101.) 



(102.) 



^3 = ^3 4- ^3 {^3 ^13 ^25 ^3> '^15 ^^2? ^^3^ 



we shall have, for such disturbed motion, the following rigorous equations, of the 

 forms (94.) and (95.), 



^l = 'P\ {*3 «1) ^2> ^3J ^13 ^23 ^3)3 



'?2 ^^ '^'2 (^3 ^13 ^23 *33 ^13 ^23 ^3)3 



^3 = ^3 ('3 ^13 ^23 *33 ^13 ^23 ^-3)3 J 



;:1 



(103.) 



