Page:Annales de mathématiques pures et appliquées, 1829-1830, Tome 20.djvu/269

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c’est-à-dire que, lorsqu’on différence successivement une fonction de plusieurs variables, par rapport à deux d’entre elles, on parvient toujours au même résultat final, quel que soit l’ordre suivant lequel on fait succéder les différentiations l’une à l’autre.

Supposons encore que, dans la fonction $\operatorname {f} (x+g,y+h,z,\ldots ),\ z$ se change en $z+k,\ k$ étant également une constante quelconque, alors, en vertu de la formule (45),

\left.{\begin{aligned}&S\\\\&{\frac {\operatorname {d} S}{\operatorname {d} x}}\\\\&{\frac {\operatorname {d} S}{\operatorname {d} y}}\\\\&{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}\\\\&{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}\\\\&{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}\\\\&{\frac {\operatorname {d} ^{3}S}{\operatorname {d} x^{3}}}\\\\&{\frac {\operatorname {d} ^{3}S}{\operatorname {d} x^{2}\operatorname {d} y}}\\\\&{\frac {\operatorname {d} ^{3}S}{\operatorname {d} x\operatorname {d} y^{2}}}\\\\&{\frac {\operatorname {d} ^{3}S}{\operatorname {d} y^{3}}}\\&\ldots \end{aligned}}\right\}{\text{deviendront}}\left\{{\begin{aligned}&S+{\frac {\operatorname {d} S}{\operatorname {d} z}}{\frac {k}{1}}+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}{\frac {k^{2}}{1.2}}+{\frac {\operatorname {d} ^{3}S}{\operatorname {d} z^{3}}}{\frac {k^{3}}{1.2.3}}+\ldots \\\\&{\frac {\operatorname {d} S}{\operatorname {d} x}}+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} z}}{\frac {k}{1}}+{\frac {\operatorname {d} ^{3}S}{\operatorname {d} x\operatorname {d} z^{2}}}{\frac {k^{2}}{1.2}}+\ldots \\\\&{\frac {\operatorname {d} S}{\operatorname {d} y}}+{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}}{\frac {k}{1}}+{\frac {\operatorname {d} ^{3}S}{\operatorname {d} y\operatorname {d} z^{2}}}{\frac {k^{2}}{1.2}}+\ldots \\\\&{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}}+{\frac {\operatorname {d} ^{3}S}{\operatorname {d} x^{2}\operatorname {d} z}}{\frac {k}{1}}+\ldots \\\\&{\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}+{\frac {\operatorname {d} ^{3}S}{\operatorname {d} x\operatorname {d} y\operatorname {d} z}}{\frac {k}{1}}+\ldots \\\\&{\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}+{\frac {\operatorname {d} ^{3}S}{\operatorname {d} y^{2}\operatorname {d} z}}{\frac {k}{1}}+\ldots \\\\&{\frac {\operatorname {d} ^{3}S}{\operatorname {d} x^{3}}}+\ldots \\\\&{\frac {\operatorname {d} ^{3}S}{\operatorname {d} x^{2}\operatorname {d} y}}+\ldots \\\\&{\frac {\operatorname {d} ^{3}S}{\operatorname {d} x\operatorname {d} y^{2}}}+\ldots \\\\&{\frac {\operatorname {d} ^{3}S}{\operatorname {d} y^{3}}}+\ldots \\&\ldots \ldots \ldots \ldots \end{aligned}}\right.