Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/790

arbitraires, l’équation précédente subsistera toujours, quelques valeurs qu’on leur donne. Supposons d’abord

$\Delta b=0,\quad \Delta c=0,\quad \Delta f=0,\quad \Delta g=0,\quad \Delta h=0,$

ensuite

$\delta a=0,\quad \delta c=0,\quad \delta f=0,\quad \delta g=0,\quad \delta h=0,$

on aura (7)

${\begin{alignedat}{3}\Delta r\ =&{\frac {dr}{da}}\Delta a,\qquad &\Delta s\ =&{\frac {ds}{da}}\Delta a,\qquad &\Delta u\ =&{\frac {du}{da}}\Delta a,\\\Delta r'=&{\frac {dr'}{da}}\Delta a,&\Delta s'=&{\frac {ds'}{da}}\Delta a,&\Delta u'=&{\frac {du'}{da}}\Delta a,\\\delta r\ =&{\frac {dr}{db}}\delta b,&\delta s\ =&{\frac {ds}{db}}\delta b,&\delta u\ =&{\frac {du}{db}}\delta b,\\\delta r'=&{\frac {dr'}{db}}\delta b,&\delta s'=&{\frac {ds'}{db}}\delta b,&\delta u'=&{\frac {du'}{db}}\delta b\,;\end{alignedat}}$

et, ces valeurs étant substituées dans l’équation intégrale ci-dessus, on aura, après avoir effacé le facteur commun $\Delta a\delta b,$

$\left.{\begin{aligned}&{\frac {d^{2}\mathrm {R} }{dr'^{2}}}\left({\frac {dr}{da}}\,{\frac {dr'}{db}}-{\frac {dr}{db}}\,{\frac {dr'}{da}}\right)\\+&{\frac {d^{2}\mathrm {R} }{ds'^{2}}}\left({\frac {ds}{da}}\,{\frac {ds'}{db}}-{\frac {ds}{db}}\,{\frac {ds'}{da}}\right)\\+&{\frac {d^{2}\mathrm {R} }{du'^{2}}}\left({\frac {du}{da}}{\frac {du'}{db}}-{\frac {du}{db}}{\frac {du'}{da}}\right)\\+&{\frac {d^{2}\mathrm {R} }{dr'ds'}}\left({\frac {dr}{da}}\,{\frac {ds'}{db}}+{\frac {ds}{da}}{\frac {dr'}{db}}-{\frac {dr}{db}}\,{\frac {ds'}{da}}-{\frac {ds}{db}}\,{\frac {dr'}{da}}\right)\\+&{\frac {d^{2}\mathrm {R} }{dr'du'}}\left({\frac {dr}{da}}{\frac {du'}{db}}+{\frac {du}{da}}{\frac {dr'}{db}}-{\frac {dr}{db}}{\frac {du'}{da}}-{\frac {du}{db}}{\frac {dr'}{da}}\right)\\+&{\frac {d^{2}\mathrm {R} }{ds'du'}}\left({\frac {ds}{da}}{\frac {du'}{db}}+{\frac {du}{da}}{\frac {ds'}{db}}-{\frac {ds}{db}}{\frac {du'}{da}}-{\frac {du}{db}}{\frac {ds'}{da}}\right)\\+&\left({\frac {d^{2}\mathrm {R} }{ds\,dr'}}-{\frac {d^{2}\mathrm {R} }{dr\,ds'}}\,\right)\left({\frac {dr}{da}}\,{\frac {ds}{db}}-{\frac {dr}{db}}{\frac {ds}{da}}\right)\\+&\left({\frac {d^{2}\mathrm {R} }{du\,dr'}}-{\frac {d^{2}\mathrm {R} }{dr\,du'}}\right)\left({\frac {dr}{da}}{\frac {du}{db}}-{\frac {dr}{db}}{\frac {du}{da}}\right)\\+&\left({\frac {d^{2}\mathrm {R} }{du\,ds'}}-{\frac {d^{2}\mathrm {R} }{ds\,du'}}\right)\left({\frac {ds}{da}}{\frac {du}{db}}-{\frac {ds}{db}}{\frac {du}{da}}\right)\end{aligned}}\right\}=(a,b).$