142 PROFESSOR HAMILTON ON A GENERAL METHOD IN DYNAMICS. 



of undisturbed motion ; but in making this choice it became necessary to modify 

 these latter formulae, and to determine a varying orbit essentially distinct in theory 

 (though little differing in practice) from that conceived so beautifully by Lagrange. 

 The orbit which he imagined was more simply connected with the heliocentric mo- 

 tion of a single planet, since it gave, for such heliocentric motion, the velocity as well 

 as the position ; the orbit which we have chosen is perhaps more closely combined 

 with the conception of a multiple system, moving about its common centre of gravity, 

 and influenced in every part by the actions of all the rest. Whichever orbit shall be 

 hereafter adopted by astronomers, they will remember that both are equally fit to 

 represent the celestial appearances, if the numeric elements of either set be suitably 

 determined by observation, and the elements of the other set of orbits be deduced 

 from these by calculation. Meantime mathematicians will judge, whether in sacri- 

 ficing a part of the simplicity of that geometrical conception on which the theories of 

 Lagrange and Poisson are founded, a simplicity of another kind has not been intro- 

 duced, which was wanting in those admirable theories ; by our having succeeded in 

 expressing rigorously the differentials of all our own new varying elements through 

 the coefficients of a single function : whereas it has seemed necessary hitherto to em- 

 ploy one function for the Earth disturbed by Venus, and another function for Venus 

 disturbed by the Earth. 



Integration of the Simplified Equations, which determine the new varying Elements. 



39. The simplified differential equations of varying elements, (S^.), are of the same 

 form as the equations (A.), and may be integrated in a similar manner. If we put, 

 for abridgement, 



and interpret similarly the symbols (^, u, X), &c., we can easily assign the variations 

 of the following 8 combinations, (r, x, v) {^, u, X) {^, k, v) (r, u, X) (r, u, v) (jm., k, X) 

 (r, », X) (^, a, v) ; namely, 



^ (r, «, v) = 2 . w (r S jU; — Tq S |!^Q -|- » 5 <w — «o ^ *'o "f" " ^ ^ ""■ ''o ^ ^) — ^2 ^ ^5 

 S (jU;,ft;,X) = 2 . m (jU^o ^''o — \h\r •\- ca^^it^ — u\k -\-\^v^ — \hv) — Hg S t, 



^ (r, fi;,X) = 2 . w (r ^ |M» — Tq ^ jO/Q -j- ft/Q ^ *o ~" ^ ^ * + ^0 ^ "o ~ ^ ^ ") "~ Hg S ^, I ,^2 \ 



^ (r, a>, v) = 2 . wi (r S jU; — Tq § ^O/Q -f- ft>o ^ *o "~ ** ^ ^ ~l" ** ^ ^ "" "o ^ ^o) ""■ ^2 ^ ^3 

 ^ (fjtj,K,X) =2 .m(^Q^rQ — (jijIt -\- x^&f — Xq^ ofQ -{• Xq'6 Vq — Xlv) — Hg^^, 



^ (r,«,X) =: 2 . m (tI fjb — Tq^ (jbQ -^^ x^ ej — Xq^ Uq -{■ Xq'6 Vq — X S f) — Hg ^ #, 



I {[^,6f, V) = 2 . m{(Jl,QlTQ-^ (Jt.'^T -^ UqIxq— Cj'^X -\- »^X — VqIXq) — H.^^t, 



*o ^0 (^0 "o ''o ^0 being the initial values of the varying elements x X (ji, u r co. If, then, 

 we consider, for example, the first of these 8 combinations (r, x, v), as a function of 



