Page:Joseph Louis de Lagrange - Œuvres, Tome 6.djvu/458

${\begin{aligned}\mathrm {F} =&-{\frac {f+{\cfrac {x}{r}}}{3(1+m)}}+{\frac {{\cfrac {y}{r^{3}}}(x\eta -y\xi )}{1+\mu }}\\&-{\frac {\left(f+{\cfrac {x}{r}}\right)(\xi dy-yd\xi )+{\cfrac {y}{r}}(\eta dy-yd\eta )}{(1+\mu )dt{\sqrt {2h(1+m)}}}}+{\frac {2\mathrm {H} dy}{dt{\sqrt {2h(1+m)}}}},\\\\\mathrm {G} =&-{\frac {\cfrac {y}{r}}{3(1+m)}}-{\frac {{\cfrac {x}{r^{3}}}(x\eta -y\xi )}{1+\mu }}\\&+{\frac {\left(f+{\cfrac {x}{r}}\right)(\xi dx-xd\xi )+{\cfrac {y}{r}}(\eta dx-xd\eta )}{(1+\mu )dt{\sqrt {2h(1+m)}}}}-{\frac {2\mathrm {H} dx}{dt{\sqrt {2h(1+m)}}}},\\\\\mathrm {B} =&{\frac {\zeta dy-yd\zeta }{(1+\mu )dt{\sqrt {2h(1+m)}}}},\\\\\mathrm {C} =&{\frac {xd\zeta -\zeta dx}{(1+\mu )dt{\sqrt {2h(1+m)}}}},\\\\\mathrm {I} =&{\frac {3t}{a}}{\sqrt {\frac {2(1+m)}{a^{3}}}}\mathrm {A} +2{\frac {\left({\cfrac {x\eta -y\xi }{1+\mu }}-{\cfrac {r^{2}\mathrm {G} }{f}}\right)}{\sqrt {a^{3}h}}}+{\frac {y(fx-4h)\mathrm {F} }{2{\sqrt {a^{3}h}}}}.\end{aligned}}$

De plus on aura, par les formules du n^o 37, en y marquant d’un trait les quantités $\delta h,\delta a,\ldots ,\mathrm {X,Y,Z} ,$

${\begin{aligned}d\,\delta h'=&-\mu {\frac {\sqrt {2h}}{1+\mu }}(x\mathrm {Y} '-y\mathrm {X} ')dt,\\d\,\delta a'=&-{\frac {\mu }{1+m}}a^{2}(\mathrm {X} 'dx+\mathrm {Y} 'dy),\\d\,\delta f'=&-{\frac {\mu ydt}{\sqrt {2h(1+m)}}}\left[\left(f+{\frac {x}{r}}\right)\mathrm {X} '+{\frac {y}{r}}\mathrm {Y} '\right]+{\frac {2dyd\,\delta h'}{dt{\sqrt {2h(1+m)}}}},\\d\,\delta g'=&\quad {\frac {\mu xdt}{\sqrt {2h(1+m)}}}\left[\left(f+{\frac {x}{r}}\right)\mathrm {X} '+{\frac {y}{r}}\mathrm {Y} '\right]-{\frac {2dxd\,\delta h'}{dt{\sqrt {2h(1+m)}}}},\\d\,\delta b'=&\quad {\frac {\mu y}{\sqrt {2h(1+m)}}}\mathrm {Z} 'dt,\\d\,\delta c'=&-{\frac {\mu x}{\sqrt {2h(1+m)}}}\mathrm {Z} 'dt,\\d\,\delta i'=&3t{\sqrt {\frac {2(1+m)}{a^{3}}}}{\frac {d\,\delta a'}{a}}-{\frac {2r^{2}d\,\delta g'}{f{\sqrt {a^{3}h}}}}+{\frac {y(fx-4h)d\,\delta f'}{2{\sqrt {ah^{3}}}}}.\end{aligned}}$

Donc, si l’on différentie les valeurs de $\delta h,\delta a,\ldots$ données ci-dessus, et