Page:Annales de mathématiques pures et appliquées, 1815-1816, Tome 6.djvu/277

267

DES ORBITES.

$V=(G'+G_{l}){\tfrac {\operatorname {d} ^{2}n}{\operatorname {d} t^{2}}}-(H'+H_{l}){\tfrac {\operatorname {d} ^{2}m}{\operatorname {d} t^{2}}}+(N'+N_{l}){\tfrac {\operatorname {d} ^{2}g}{\operatorname {d} t^{2}}}-(M'+M_{l}){\tfrac {\operatorname {d} ^{2}h}{\operatorname {d} t^{2}}}$

$-\left\{(M'+M_{l})(H'+H_{l})-(N'+N_{l})(G'+G_{l})\right\}i^{2},$

il viendra

$K=\left\{{\frac {\operatorname {d} m}{\operatorname {d} t}}{\frac {\operatorname {d} ^{2}n}{\operatorname {d} t^{2}}}-{\frac {\operatorname {d} n}{\operatorname {d} t}}{\frac {\operatorname {d} ^{2}m}{\operatorname {d} t^{2}}}\right\}i^{3}+Si^{5}\,;$

$(m'-m_{l})(h_{l}-2h+h')-(n'-n_{l})(g_{l}-2g+g')$

$=2\left\{{\frac {\operatorname {d} m}{\operatorname {d} t}}{\frac {\operatorname {d} ^{2}h}{\operatorname {d} t^{2}}}-{\frac {\operatorname {d} n}{\operatorname {d} t}}{\frac {\operatorname {d} ^{2}g}{\operatorname {d} t^{2}}}\right\}i^{3}+Fi^{5}\,;$

$(n_{l}-2n+n')(g_{l}-2g+g')-(m_{l}-2m+m')(h_{l}-2h+h')$

$=-\left\{{\frac {\operatorname {d} ^{2}m}{\operatorname {d} t^{2}}}{\frac {\operatorname {d} ^{2}h}{\operatorname {d} t^{2}}}-{\frac {\operatorname {d} ^{2}n}{\operatorname {d} t^{2}}}{\frac {\operatorname {d} ^{2}g}{\operatorname {d} t^{2}}}\right\}i^{4}+Vi^{6}.$

En conséquence, on aura (18 et 19)

$\left.{\begin{array}{lc}&Z=-{\frac {2\left\{{\frac {\operatorname {d} m}{\operatorname {d} t}}{\frac {\operatorname {d} ^{2}h}{\operatorname {d} t^{2}}}-{\frac {\operatorname {d} n}{\operatorname {d} t}}{\frac {\operatorname {d} ^{2}g}{\operatorname {d} t^{2}}}\right\}+Fi^{2}}{2\left\{{\frac {\operatorname {d} m}{\operatorname {d} t}}{\frac {\operatorname {d} ^{2}n}{\operatorname {d} t^{2}}}-{\frac {\operatorname {d} n}{\operatorname {d} t}}{\frac {\operatorname {d} ^{2}m}{\operatorname {d} t^{2}}}\right\}+Si^{2}}}\,;\\&2{\frac {\operatorname {d} Z}{\operatorname {d} t}}+\left\{{\tfrac {1}{3}}{\frac {\operatorname {d} ^{3}Z}{\operatorname {d} t^{3}}}+(\zeta '-\zeta _{l})i\right\}i^{2}=+{\frac {\left\{{\frac {\operatorname {d} ^{2}m}{\operatorname {d} t^{2}}}{\frac {\operatorname {d} ^{2}h}{\operatorname {d} t^{2}}}-{\frac {\operatorname {d} ^{2}n}{\operatorname {d} t^{2}}}{\frac {\operatorname {d} ^{2}g}{\operatorname {d} t^{2}}}\right\}-Vi^{2}}{\left\{{\frac {\operatorname {d} m}{\operatorname {d} t}}{\frac {\operatorname {d} ^{2}n}{\operatorname {d} t^{2}}}-{\frac {\operatorname {d} n}{\operatorname {d} t}}{\frac {\operatorname {d} ^{2}m}{\operatorname {d} t^{2}}}\right\}+Si^{2}}}\,;\\\end{array}}\right\}$(22)

résultats sur lesquels nous reviendrons tout à l’heure.

Si présentement on suppose que les observations coïncident, c’est-à-dire, si l’on suppose que l’intervalle de temps $i$ qui sépare l’observation moyenne des deux observations extrêmes est nul, il viendra, en adoptant, pour plus de simplicité les notations de Lagrange

$Z=-{\frac {m'h''-n'g''}{m'n''-n'm''}},\qquad Z'={\frac {m''h''-n''g''}{2(m'n''-n'm'')}}.\qquad$(23)

IV. Si nous n’eussions eu en vue que de parvenir à ces derniers résultats, nous aurions pu les obtenir d’une manière incomparablement plus simple, ainsi que nous l’allons voir ; et il en serait résulté une nouvelle méthode qui, si l’hypothèse qui nous occupe pouvait être admise, paraîtrait devoir l’emporter de beau-