Page:Annales de mathématiques pures et appliquées, 1823-1824, Tome 14.djvu/332

Considérons, dans ce développement, la colonne dont le rang est $i+2,$ laquelle est, comme on le voit,

${\frac {\operatorname {d} y_{i}}{\operatorname {d} x}}{\frac {\operatorname {d} V}{\operatorname {d} y_{i}}}-{\frac {\operatorname {d} \left({\frac {\operatorname {d} y_{i}}{\operatorname {d} x_{1}}}{\frac {\operatorname {d} V}{\operatorname {d} y_{i}}}\right)}{\operatorname {d} t}}+{\frac {\operatorname {d} ^{2}\left({\frac {\operatorname {d} y_{i}}{\operatorname {d} x_{2}}}{\frac {\operatorname {d} V}{\operatorname {d} y_{i}}}\right)}{\operatorname {d} t^{2}}}-\ldots \pm {\frac {\operatorname {d} ^{i+1}\left({\frac {\operatorname {d} V}{\operatorname {d} x_{i+1}}}\right)}{\operatorname {d} t^{i+1}}}\,;$

en la développant et ordonnant suivant les différentielles successives de ${\frac {\operatorname {d} V}{\operatorname {d} y_{i}}}\,;$ on pourra lui donner la forme suivante :

${\begin{array}{r|r|r|r}{\frac {\operatorname {d} y_{i}}{\operatorname {d} x}}&{\frac {\operatorname {d} V}{\operatorname {d} y_{i}}}-{\frac {\operatorname {d} y_{i}}{\operatorname {d} x_{1}}}&{\frac {\operatorname {d} \left({\frac {\operatorname {d} V}{\operatorname {d} y_{i}}}\right)}{\operatorname {d} t}}+{\frac {\operatorname {d} y_{i}}{\operatorname {d} x_{2}}}&{\frac {\operatorname {d} ^{2}\left({\frac {\operatorname {d} V}{\operatorname {d} y_{i}}}\right)}{\operatorname {d} t^{2}}}-\ldots \pm {\frac {\operatorname {d} y_{i}}{\operatorname {d} x_{i}}}{\frac {\operatorname {d} ^{i}\left({\frac {\operatorname {d} V}{\operatorname {d} y_{i}}}\right)}{\operatorname {d} t^{i}}}\\\\-{\frac {\operatorname {d} \left({\frac {\operatorname {d} y_{i}}{\operatorname {d} x_{1}}}\right)}{\operatorname {d} t}}&+2{\frac {\operatorname {d} \left({\frac {\operatorname {d} y_{i}}{\operatorname {d} x_{2}}}\right)}{\operatorname {d} t}}&-3{\frac {\operatorname {d} \left({\frac {\operatorname {d} y_{i}}{\operatorname {d} x_{3}}}\right)}{\operatorname {d} t}}&+\ldots \\\\+{\frac {\operatorname {d} ^{2}\left({\frac {\operatorname {d} y_{i}}{\operatorname {d} x_{2}}}\right)}{\operatorname {d} t^{2}}}&-3{\frac {\operatorname {d} ^{2}\left({\frac {\operatorname {d} y_{i}}{\operatorname {d} x_{3}}}\right)}{\operatorname {d} t^{2}}}&+\ldots \ldots &\\\\-{\frac {\operatorname {d} ^{3}\left({\frac {\operatorname {d} y_{i}}{\operatorname {d} x_{3}}}\right)}{\operatorname {d} t^{3}}}&+\ldots \ldots &\pm {\frac {i}{1}}.{\frac {i-1}{2}}{\frac {\operatorname {d} ^{i-2}\left({\frac {\operatorname {d} y_{i}}{\operatorname {d} x_{i}}}\right)}{\operatorname {d} t^{i-2}}}&\\\\+\ldots \ldots &\pm i{\frac {\operatorname {d} ^{i-1}\left({\frac {\operatorname {d} y_{i}}{\operatorname {d} x_{i}}}\right)}{\operatorname {d} t^{i-1}}}&&\\\\\pm {\frac {\operatorname {d} ^{i}\left({\frac {\operatorname {d} y_{i}}{\operatorname {d} x_{i}}}\right)}{\operatorname {d} t^{i}}}&&&\end{array}}$

Or, les coefficiens des termes,

${\frac {\operatorname {d} V}{\operatorname {d} x_{i}}},\quad {\frac {\operatorname {d} \left({\frac {\operatorname {d} V}{\operatorname {d} y_{i}}}\right)}{\operatorname {d} t}},\quad {\frac {\operatorname {d} ^{2}\left({\frac {\operatorname {d} V}{\operatorname {d} y_{i}}}\right)}{\operatorname {d} t^{2}}},\ldots {\frac {\operatorname {d} ^{i-1}\left({\frac {\operatorname {d} V}{\operatorname {d} y_{i}}}\right)}{\operatorname {d} t^{i-1}}},$