Page:Annales de mathématiques pures et appliquées, 1819-1820, Tome 10.djvu/259

251

DES FONCTIONS.

${\begin{aligned}&{\frac {\operatorname {d} U}{\operatorname {d} a}}={\frac {\operatorname {d} U}{\operatorname {d} x}}{\frac {\operatorname {d} x}{\operatorname {d} a}}+{\frac {\operatorname {d} U}{\operatorname {d} x_{1}}}{\frac {\operatorname {d} x_{1}}{\operatorname {d} a}},\\\\&{\frac {\operatorname {d} U}{\operatorname {d} b}}={\frac {\operatorname {d} U}{\operatorname {d} x}}{\frac {\operatorname {d} x}{\operatorname {d} b}}+{\frac {\operatorname {d} U}{\operatorname {d} x_{1}}}{\frac {\operatorname {d} x_{1}}{\operatorname {d} b}},\\\\&{\frac {\operatorname {d} U}{\operatorname {d} a_{1}}}={\frac {\operatorname {d} U}{\operatorname {d} x}}{\frac {\operatorname {d} x}{\operatorname {d} a_{1}}}+{\frac {\operatorname {d} U}{\operatorname {d} x_{1}}}{\frac {\operatorname {d} x_{1}}{\operatorname {d} a_{1}}},\\\\&{\frac {\operatorname {d} U}{\operatorname {d} b_{1}}}={\frac {\operatorname {d} U}{\operatorname {d} x}}{\frac {\operatorname {d} x}{\operatorname {d} b_{1}}}+{\frac {\operatorname {d} U}{\operatorname {d} x_{1}}}{\frac {\operatorname {d} x_{1}}{\operatorname {d} b_{1}}},\end{aligned}}$

nous aurons

$\left.{\begin{aligned}&{\frac {\operatorname {d} U}{\operatorname {d} b}}=z{\frac {\operatorname {d} U}{\operatorname {d} a}},\\\\&{\frac {\operatorname {d} U}{\operatorname {d} b_{1}}}=z_{1}{\frac {\operatorname {d} U}{\operatorname {d} a_{1}}}.\end{aligned}}\right\}\quad$(14)

Enfin, en suivant la marche qui nous a conduit des équations (4, 8) à la formule (11), nous parviendrons aux suivantes,

$\left.{\begin{aligned}&{\frac {\operatorname {d} ^{m}U}{\operatorname {d} b^{m}}}={\frac {\operatorname {d} ^{m-1}.z^{m}{\frac {\operatorname {d} U}{\operatorname {d} a}}}{\operatorname {d} a^{m-1}}},\\\\&{\frac {\operatorname {d} ^{m_{1}}U}{\operatorname {d} b_{1}^{m_{1}}}}={\frac {\operatorname {d} ^{m_{1}-1}.z_{1}^{m_{1}}{\frac {\operatorname {d} U}{\operatorname {d} a_{1}}}}{\operatorname {d} a_{1}^{m_{1}-1}}}.\end{aligned}}\right\}\quad$(15)

Pour compléter cette recherche, il faut parvenir à l’expression de

${\frac {\operatorname {d} ^{m+m_{1}}U}{\operatorname {d} b^{m}\operatorname {d} b_{1}^{m_{1}}}}.$

Pour cela, soit