Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/786

aux précédentes, dans lesquelles la caractéristique $\delta$ sera simplement remplacée par la caractéristique $\Delta .$

13. Qu’on ajoute maintenant ensemble les trois équations précédentes multipliées respectivement par $\Delta r,\Delta s,\Delta u,$ et qu’on en retranche la somme des trois pareilles équations, dans lesquelles le $\delta$ est changé en $\Delta ,$ après les avoir multipliées respectivement par $\delta r,\delta s,\delta u,$ on aura cette équation

${\begin{alignedat}{10}&&\Delta r\,d&{\biggl (}{\biggr .}{\frac {d^{2}\mathrm {R} }{dr\,dr'}}&\delta r&+&{\frac {d^{2}\mathrm {R} }{ds\,dr'}}&\delta s&+&{\frac {d^{2}\mathrm {R} }{du\,dr'}}&\delta u&+&&{\frac {d^{2}\mathrm {R} }{dr'^{2}}}&\delta r'&+&&{\frac {d^{2}\mathrm {R} }{dr'ds'}}&\delta s'&+&{\frac {d^{2}\mathrm {R} }{dr'du'}}&\delta u'{\biggl .}{\biggr )}\\&-&\Delta r\ \ &{\biggl (}{\biggr .}{\frac {d^{2}\mathrm {R} }{dr^{2}}}&\delta r&+&{\frac {d^{2}\mathrm {R} }{dr\,ds}}&\delta s&+&{\frac {d^{2}\mathrm {R} }{dr\,du}}&\delta u&+&&{\frac {d^{2}\mathrm {R} }{dr\,dr'}}&\delta r'&+&&{\frac {d^{2}\mathrm {R} }{dr\,ds'}}&\delta s'&+&{\frac {d^{2}\mathrm {R} }{dr\,du'}}&\delta u'{\biggl .}{\biggr )}dt\\&+&\Delta s\,d&{\biggl (}{\biggr .}{\frac {d^{2}\mathrm {R} }{dr\,ds'}}&\delta r&+&{\frac {d^{2}\mathrm {R} }{ds\,ds'}}&\delta s&+&{\frac {d^{2}\mathrm {R} }{du\,ds'}}&\delta u&+&&{\frac {d^{2}\mathrm {R} }{dr'ds'}}&\delta r'&+&&{\frac {d^{2}\mathrm {R} }{ds'^{2}}}&\delta s'&+&{\frac {d^{2}\mathrm {R} }{ds'du'}}&\delta u'{\biggl .}{\biggr )}\\&-&\Delta s\ \ &{\biggl (}{\biggr .}{\frac {d^{2}\mathrm {R} }{dr\,ds}}&\delta r&+&{\frac {d^{2}\mathrm {R} }{ds^{2}}}&\delta s&+&{\frac {d^{2}\mathrm {R} }{ds\,du}}&\delta u&+&&{\frac {d^{2}\mathrm {R} }{ds\,dr'}}&\delta r'&+&&{\frac {d^{2}\mathrm {R} }{ds\,ds'}}&\delta s'&+&{\frac {d^{2}\mathrm {R} }{ds\,du'}}&\delta u'{\biggl .}{\biggr )}dt\\&+&\Delta u\,d&{\biggl (}{\biggr .}{\frac {d^{2}\mathrm {R} }{dr\,du'}}&\delta r&+&{\frac {d^{2}\mathrm {R} }{ds\,du'}}&\delta s&+&{\frac {d^{2}\mathrm {R} }{du\,du'}}&\delta u&+&&{\frac {d^{2}\mathrm {R} }{dr'\,du'}}&\delta r'&+&&{\frac {d^{2}\mathrm {R} }{ds'du'}}&\delta s'&+&{\frac {d^{2}\mathrm {R} }{du'^{2}}}&\delta u'{\biggl .}{\biggr )}\\&-&\Delta u\ \ &{\biggl (}{\biggr .}{\frac {d^{2}\mathrm {R} }{drdu}}&\delta r&+&{\frac {d^{2}\mathrm {R} }{ds\,du}}&\delta s&+&{\frac {d^{2}\mathrm {R} }{du^{2}}}&\delta u&+&&{\frac {d^{2}\mathrm {R} }{du\,dr'}}&\delta r'&+&&{\frac {d^{2}\mathrm {R} }{du\,ds'}}&\delta s'&+&{\frac {d^{2}\mathrm {R} }{du\,du'}}&\delta u'{\biggl .}{\biggr )}dt\\&-&\delta r\,d&{\biggl (}{\biggr .}{\frac {d^{2}\mathrm {R} }{dr\,dr'}}&\Delta r&+&{\frac {d^{2}\mathrm {R} }{ds\,dr'}}&\Delta s&+&{\frac {d^{2}\mathrm {R} }{du\,dr'}}&\Delta u&+&&{\frac {d^{2}\mathrm {R} }{dr'^{2}}}&\Delta r'&+&&{\frac {d^{2}\mathrm {R} }{dr'ds'}}&\Delta s'&+&{\frac {d^{2}\mathrm {R} }{dr'du'}}&\Delta u'{\biggl .}{\biggr )}\\&+&\delta r\ \ &{\biggl (}{\biggr .}{\frac {d^{2}\mathrm {R} }{dr^{2}}}&\Delta r&+&{\frac {d^{2}\mathrm {R} }{dr\,ds}}&\Delta s&+&{\frac {d^{2}\mathrm {R} }{dr\,du}}&\Delta u&+&&{\frac {d^{2}\mathrm {R} }{dr\,dr'}}&\Delta r'&+&&{\frac {d^{2}\mathrm {R} }{dr\,ds'}}&\Delta s'&+&{\frac {d^{2}\mathrm {R} }{dr\,du'}}&\Delta u'{\biggl .}{\biggr )}dt\\&-&\delta s\,d&{\biggl (}{\biggr .}{\frac {d^{2}\mathrm {R} }{dr\,ds'}}&\Delta r&+&{\frac {d^{2}\mathrm {R} }{ds\,ds'}}&\Delta s&+&{\frac {d^{2}\mathrm {R} }{du\,ds'}}&\Delta u&+&&{\frac {d^{2}\mathrm {R} }{dr'ds'}}&\Delta r'&+&&{\frac {d^{2}\mathrm {R} }{ds'^{2}}}&\Delta s'&+&{\frac {d^{2}\mathrm {R} }{ds'du'}}&\Delta u'{\biggl .}{\biggr )}\\&+&\delta s\ \ &{\biggl (}{\biggr .}{\frac {d^{2}\mathrm {R} }{dr\,ds}}&\Delta r&+&{\frac {d^{2}\mathrm {R} }{ds^{2}}}&\Delta s&+&{\frac {d^{2}\mathrm {R} }{ds\,du}}&\Delta u&+&&{\frac {d^{2}\mathrm {R} }{ds\,dr'}}&\Delta r'&+&&{\frac {d^{2}\mathrm {R} }{ds\,ds'}}&\Delta s'&+&{\frac {d^{2}\mathrm {R} }{ds\,du'}}&\Delta u'{\biggl .}{\biggr )}dt\\&-&\delta u\,d&{\biggl (}{\biggr .}{\frac {d^{2}\mathrm {R} }{dr\,du'}}&\Delta r&+&{\frac {d^{2}\mathrm {R} }{ds\,du'}}&\Delta s&+&{\frac {d^{2}\mathrm {R} }{du\,du'}}&\Delta u&+&&{\frac {d^{2}\mathrm {R} }{dr'\,du'}}&\Delta r'&+&&{\frac {d^{2}\mathrm {R} }{ds'du'}}&\Delta s'&+&{\frac {d^{2}\mathrm {R} }{du'^{2}}}&\Delta u'{\biggl .}{\biggr )}\\&+&\delta u\ \ &{\biggl (}{\biggr .}{\frac {d^{2}\mathrm {R} }{drdu}}&\Delta r&+&{\frac {d^{2}\mathrm {R} }{ds\,du}}&\Delta s&+&{\frac {d^{2}\mathrm {R} }{du^{2}}}&\Delta u&+&&{\frac {d^{2}\mathrm {R} }{du\,dr'}}&\Delta r'&+&&{\frac {d^{2}\mathrm {R} }{du\,ds'}}&\Delta s'&+&{\frac {d^{2}\mathrm {R} }{du\,du'}}&\Delta u'{\biggl .}{\biggr )}dt=0.\end{alignedat}}$

14. Et, si l’on exécute les différentiations relatives à la caractéristique $d$ qui se rapporte au temps $t,$ qu’on efface les termes qui se dé-