Page:Joseph Louis de Lagrange - Œuvres, Tome 5.djvu/80

En différentiant de même la valeur de $\mathrm {V} ''$ par rapport à $r,s,u,$ on aura

${\begin{aligned}{\frac {\delta \mathrm {V} ''}{\delta r}}=&{\frac {3f(1+x)^{2}}{p'^{5}}}\left[-2(\mathrm {C-B} )r+2\mathrm {H} -4\mathrm {G} u-4\mathrm {F} s\right]\\&-{\frac {3f(1+x)y}{p'^{5}}}\left(\mathrm {B-C-4H} r-4\mathrm {G} s+4\mathrm {F} u\right)\\&+{\frac {3f(1+x)z}{p'^{5}}}\left[2(\mathrm {B-A} )u-2\mathrm {H} s-\mathrm {G} r-\mathrm {F} \right]\\&-{\frac {3fy^{2}}{2p'^{5}}}\left[-2(\mathrm {B-C} )r-2\mathrm {H} +4\mathrm {G} u+4\mathrm {F} s\right]\\&+{\frac {3fyz}{p'^{5}}}\left[2(\mathrm {A-C} )s+2\mathrm {H} u+\mathrm {G} -\mathrm {F} r\right],\\\\{\frac {\delta \mathrm {V} ''}{\delta s}}=&{\frac {3f(1+x)^{2}}{2p'^{5}}}\left[-8(\mathrm {C-A} )s+4\mathrm {H} u+4\mathrm {G} -4\mathrm {F} r\right]\\&-{\frac {3f(1+x)y}{p'^{5}}}\left[2(\mathrm {B+C-2A} )u-4\mathrm {H} s-4\mathrm {G} r-2\mathrm {F} \right]\\&+{\frac {3f(1+x)z}{p'^{5}}}\left[2(\mathrm {A-C} )-2\mathrm {H} r-16\mathrm {G} s-6\mathrm {F} u\right]\\&+{\frac {3fy^{2}}{2p'^{5}}}(4\mathrm {H} u+4\mathrm {F} r)\\&+{\frac {3fz^{2}}{2p'^{5}}}\left[-8(\mathrm {A-C} )s-8\mathrm {H} u-4\mathrm {G} \right]\\&+{\frac {3fyz}{p'^{5}}}\left[2(\mathrm {A-C} )r+2\mathrm {H} -6\mathrm {G} u-4\mathrm {F} s\right],\\\\{\frac {\delta \mathrm {V} ''}{\delta u}}=&{\frac {3f(1+x)^{2}}{2p'^{5}}}(4\mathrm {H} s-4\mathrm {G} r)\\&-{\frac {3f(1+x)y}{p'^{5}}}\left[2(\mathrm {B+C-2A} )s-4\mathrm {H} u-2\mathrm {G} +4\mathrm {F} r\right]\\&+{\frac {3f(1+x)z}{p'^{5}}}\left[2(\mathrm {B-A} )r+2\mathrm {H} -4\mathrm {G} u-6\mathrm {F} s\right]\\&+{\frac {3fy^{2}}{2p'^{5}}}\left[-8(\mathrm {B-A} )u+4\mathrm {H} s+4\mathrm {G} r+4\mathrm {F} \right]\\&+{\frac {3fz^{2}}{2p'^{5}}}\left[-8(\mathrm {A-B} )u-8\mathrm {H} s-4\mathrm {F} \right]\\&+{\frac {3fyz}{p'^{5}}}\left[2(\mathrm {A-B} )+2\mathrm {H} r-6\mathrm {G} s-16\mathrm {F} u\right].\end{aligned}}$