106 



PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



which enables us to transform the perturbational terms (M.) (O.) into the following 

 approximate forms : 



and 



82 Sj /»< S Ho 



(U.) 



Arsr. = 2 . 



8 ^ 8 )j . 



I"-^'^^, (V-) 



containing only functions and quantities which may be regarded as given, by the 

 theory of undisturbed motion. 



12. In the same order of approximation, if the variation of the expression (44.) for 

 an undisturbed coordinate ri. be thus denoted, 



i,^=.JlU+l{-^ie + ^ip), . (51.) 



the perturbation of that coordinate niay be expressed as follows : 



' bp 



(W.) 



that is, by (U.), 



8pj»/o 0^1 



Sn. M c 8«. 



A«. 



8 



-«- VS;., 8.,2 "^ 8p, 8., 8^2 ^ • • • "^ a;>3„ ^e.de^JJo ^p, ^^ 





\ . (52.) 



+ 



Besides, the identical equation (47.) gives 



S').- 



8^, 8p, 8.,8^, + 8^, 8.,S^, i- • • • -t- g^^^ g,^g,^^^ ; 



(53.) 



the expression (52.) may therefore be thus abridged. 



^^i 



« 8 7J, c/o 



8H. 



i)i-/o 8^, 



6/^— . . . - 







(X.) 



and shows that instead of the rigorous perturbational terms (M.) we may approxi- 

 mately employ the following. 



A^. = -X^''''> 



(Y.) 



in order to calculate the disturbed configuration at any time t by the rules of undis- 



