Page:Annales de mathématiques pures et appliquées, 1829-1830, Tome 20.djvu/280

et $Q\,;$ si $x$, et $y$ deviennent respectivement $x+g$ et $y+h,\ P$ et $Q$ deviendront respectivement (48)

${\begin{alignedat}{2}P+{\frac {\operatorname {d} P}{\operatorname {d} x}}{\frac {g}{1}}&+{\frac {\operatorname {d} ^{2}P}{\operatorname {d} x^{2}}}{\frac {g^{2}}{1.2}}+\ldots &Q+{\frac {\operatorname {d} Q}{\operatorname {d} x}}{\frac {g}{1}}&+{\frac {\operatorname {d} ^{2}Q}{\operatorname {d} x^{2}}}{\frac {g^{2}}{1.2}}+\ldots &\\\\{\frac {\operatorname {d} P}{\operatorname {d} y}}{\frac {h}{1}}&+2{\frac {\operatorname {d} ^{2}P}{\operatorname {d} x\operatorname {d} y}}{\frac {gh}{1.2}}+\ldots &{\frac {\operatorname {d} Q}{\operatorname {d} y}}{\frac {h}{1}}&+2{\frac {\operatorname {d} ^{2}Q}{\operatorname {d} x\operatorname {d} y}}{\frac {gh}{1.2}}+\ldots &\\\\&+{\frac {\operatorname {d} ^{2}P}{\operatorname {d} y^{2}}}{\frac {h^{2}}{1.2}}+\ldots &&+{\frac {\operatorname {d} ^{2}Q}{\operatorname {d} y^{2}}}{\frac {h^{2}}{1.2}}+\ldots \,;&\end{alignedat}}$

de sorte qu’en dénotant par $G$ et $H$ les accroissement respectifs de $P$ et $Q,$ on aura

$G={\frac {\operatorname {d} P}{\operatorname {d} x}}{\frac {g}{1}}+{\frac {\operatorname {d} P}{\operatorname {d} y}}{\frac {h}{1}}+{\frac {\operatorname {d} ^{2}P}{\operatorname {d} x^{2}}}{\frac {g^{2}}{1.2}}+2{\frac {\operatorname {d} ^{2}P}{\operatorname {d} x\operatorname {d} y}}{\frac {gh}{1.2}}+{\frac {\operatorname {d} ^{2}P}{\operatorname {d} y^{2}}}{\frac {h^{2}}{1.2}}+\ldots$

$H={\frac {\operatorname {d} Q}{\operatorname {d} x}}{\frac {g}{1}}+{\frac {\operatorname {d} Q}{\operatorname {d} y}}{\frac {h}{1}}+{\frac {\operatorname {d} ^{2}Q}{\operatorname {d} x^{2}}}{\frac {g^{2}}{1.2}}+2{\frac {\operatorname {d} ^{2}Q}{\operatorname {d} x\operatorname {d} y}}{\frac {gh}{1.2}}+{\frac {\operatorname {d} ^{2}P}{\operatorname {d} y^{2}}}{\frac {h^{2}}{1.2}}+\ldots \,;$

mais, $P$ et $Q$ devenant respectivement $P+F$ et $Q+H,\ S$ doit devenir (48)

${\begin{alignedat}{1}S+{\frac {\operatorname {d} S}{\operatorname {d} P}}{\frac {G}{1}}&+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} P^{2}}}{\frac {G^{2}}{1.2}}+\ldots \\\\+{\frac {\operatorname {d} S}{\operatorname {d} Q}}{\frac {H}{1}}&+2{\frac {\operatorname {d} ^{2}S}{\operatorname {d} P\operatorname {d} Q}}{\frac {GH}{1.2}}+\ldots \\\\&+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} Q^{2}}}{\frac {H^{2}}{1.2}}+\ldots \end{alignedat}}$

ou, en mettant pour $G$ et $H$ leurs valeurs, développant et ordonnant,

$S+\left({\frac {\operatorname {d} S}{\operatorname {d} P}}{\frac {\operatorname {d} P}{\operatorname {d} x}}+{\frac {\operatorname {d} S}{\operatorname {d} Q}}{\frac {\operatorname {d} Q}{\operatorname {d} x}}\right){\frac {g}{1}}$