PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



123 



and 



Pi 





8S, 8S. 



^2— - g^^' lei — 



8 Si 8 So >jo — ^o . 1 . v^t t 1 1 \ 



or, in the same order of approximation, 



^3 = ^3 + B ^ - Y ^ ^^ "" 4" '^ ^H^3 + 4 ^3 ^ - ]^ ^ ^^) , 



(139.) 



(140.) 



and 



w, 



TtTo 



'CT'i 



= P2-f^'^t{e2 + -QP2t), 



(141.) 



Accordingly, if we develope the rigorous integrals of disturbed motion, (113.) and 

 (114.), as far as the squares (inclusive) of the small quantities /-«< and v, we are con- 

 ducted to these approximate integrals ; and if we develope the rigorous expression 

 (120.) for the principal function of such motion, to the same degree of accuracy, we 

 obtain the sum of the two expressions (130.) and (137.)- 



27. To illustrate still further, in the present example, our general method of suc- 

 cessive approximation, let S3 denote the small unknown correction of the approximate 

 expression (137.)? so that we shall now have, rigorously, for the present disturbed 

 motion, 



S = Si + S2 + S3, (142.) 



Sj and S2 being here determined rigorously by (130.) and (137-). Then, substituting 

 Si + S2 for Si in the general transformation (87.)? we find, rigorously, in the present 

 question. 



«3=-/'i{(g)V(Sf+(||f}'^^ 



+/'i{(l|f+(IS)^+(g)7'^'= 



(143.) 



and if we neglect only terms of the eighth and higher dimensions with respect to 

 the small quantities {/j, v, we may confine ourselves to the first of these two definite 

 integrals, and may employ, in calculating it, the approximate expressions (140.) for 



r2 



