Page:Annales de mathématiques pures et appliquées, 1829-1830, Tome 20.djvu/273

${\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}},\quad {\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}},\quad {\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}\,;$

dans le troisième, quatre ; savoir,

${\frac {\operatorname {d} ^{3}S}{\operatorname {d} x^{3}}},\quad {\frac {\operatorname {d} ^{3}S}{\operatorname {d} x^{2}\operatorname {d} y}},\quad {\frac {\operatorname {d} ^{3}S}{\operatorname {d} x\operatorname {d} y^{2}}},\quad {\frac {\operatorname {d} ^{3}S}{\operatorname {d} y^{3}}}\,;$

et ainsi de suite, c’est-à-dire, en général, $n+1$ pour n.^ième ordre, savoir ;

${\frac {\operatorname {d} ^{n}S}{\operatorname {d} x^{n}}},\ {\frac {\operatorname {d} ^{n}S}{\operatorname {d} x^{n-1}\operatorname {d} y}},\ {\frac {\operatorname {d} ^{n}S}{\operatorname {d} x^{n-2}\operatorname {d} y^{2}}},\ldots {\frac {\operatorname {d} ^{n}S}{\operatorname {d} x^{2}\operatorname {d} y^{n-2}}},\ {\frac {\operatorname {d} ^{n}S}{\operatorname {d} x\operatorname {d} y^{n-1}}},\ {\frac {\operatorname {d} ^{n}S}{\operatorname {d} y^{n}}}.$

S’il s’agit d’une fonction de trois variables, elle aura, dans le premier ordre, trois coefficiens différentiels, savoir :

${\frac {\operatorname {d} S}{\operatorname {d} x}},\qquad {\frac {\operatorname {d} S}{\operatorname {d} y}}$

${\frac {\operatorname {d} S}{\operatorname {d} z}}\,;$

dans le second, six ; savoir :

${\frac {\operatorname {d} ^{2}S}{\operatorname {d} x^{2}}},\qquad {\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}},\qquad {\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}},$

${\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} z}},\qquad {\frac {\operatorname {d} ^{2}S}{\operatorname {d} y\operatorname {d} z}},$

${\frac {\operatorname {d} ^{2}S}{\operatorname {d} z^{2}}}\,;$

dans le troisième, dix ; savoir :