Page:Annales de mathématiques pures et appliquées, 1823-1824, Tome 14.djvu/242

 Les corrections sont expliquées en page de discussion

${\begin{array}{rrr}\operatorname {d} '{\frac {\operatorname {d} u}{\operatorname {d} a}}={\frac {\operatorname {dd} 'u}{\operatorname {d} a}},&\operatorname {d} '{\frac {\operatorname {d} u}{\operatorname {d} b}}={\frac {\operatorname {dd} 'u}{\operatorname {d} b}},&\operatorname {d} '{\frac {\operatorname {d} u}{\operatorname {d} c}}={\frac {\operatorname {dd} 'u}{\operatorname {d} c}},\\\\{\frac {\operatorname {d} ^{2}u}{\operatorname {d} a\operatorname {d} b}}={\frac {\operatorname {d} ^{2}u}{\operatorname {d} b\operatorname {d} a}},&{\frac {\operatorname {d} ^{2}u}{\operatorname {d} a\operatorname {d} c}}={\frac {\operatorname {d} ^{2}u}{\operatorname {d} c\operatorname {d} a}},&{\frac {\operatorname {d} ^{2}u}{\operatorname {d} b\operatorname {d} c}}={\frac {\operatorname {d} ^{2}u}{\operatorname {d} c\operatorname {d} b}}\,;\end{array}}$

mais il faudrait bien se garder de croire que l’on a de même

$\operatorname {d} '{\frac {\operatorname {d} u}{\operatorname {d} t}}={\frac {\operatorname {dd} 'u}{\operatorname {d} t}},\ \operatorname {d} '{\frac {\operatorname {d} u}{\operatorname {d} x}}={\frac {\operatorname {dd} 'u}{\operatorname {d} x}},\ \operatorname {d} '{\frac {\operatorname {d} u}{\operatorname {d} y}}={\frac {\operatorname {dd} 'u}{\operatorname {d} y}},\ \operatorname {d} '{\frac {\operatorname {d} u}{\operatorname {d} z}}={\frac {\operatorname {dd} 'u}{\operatorname {d} z}},$

En effet, en vertu du n.^o 4, on a, par exemple,

$\operatorname {d} '{\frac {\operatorname {d} u}{\operatorname {d} t}}={\frac {\operatorname {d} ^{2}u}{\operatorname {d} t^{2}}}+{\frac {\operatorname {d} ^{2}u}{\operatorname {d} x\operatorname {d} t}}\operatorname {d} 'x+{\frac {\operatorname {d} ^{2}u}{\operatorname {d} y\operatorname {d} t}}\operatorname {d} 'y+{\frac {\operatorname {d} ^{2}u}{\operatorname {d} z\operatorname {d} t}}\operatorname {d} 'z,$

tandis que l’équation identique

$\operatorname {d} 'u={\frac {\operatorname {d} u}{\operatorname {d} t}}+{\frac {\operatorname {d} u}{\operatorname {d} x}}\operatorname {d} 'x+{\frac {\operatorname {d} u}{\operatorname {d} y}}\operatorname {d} 'y+{\frac {\operatorname {d} u}{\operatorname {d} z}}\operatorname {d} 'z,$

différentiée par rapport à $t,$ en regardant $x,y,z$ comme constans donne

${\frac {\operatorname {dd} 'u}{\operatorname {d} t}}={\frac {\operatorname {d} ^{2}u}{\operatorname {d} t^{2}}}+{\frac {\operatorname {d} ^{2}u}{\operatorname {d} t\operatorname {d} x}}\operatorname {d} 'x+{\frac {\operatorname {d} ^{2}u}{\operatorname {d} t\operatorname {d} y}}\operatorname {d} 'y+{\frac {\operatorname {d} ^{2}u}{\operatorname {d} t\operatorname {d} z}}\operatorname {d} 'z$

$+{\frac {\operatorname {d} u}{\operatorname {d} x}}{\frac {\operatorname {dd} 'x}{\operatorname {d} t}}+{\frac {\operatorname {d} u}{\operatorname {d} y}}{\frac {\operatorname {dd} 'y}{\operatorname {d} t}}+{\frac {\operatorname {d} u}{\operatorname {d} z}}{\frac {\operatorname {dd} 'z}{\operatorname {d} t}},$

et par suite

${\frac {\operatorname {dd} 'u}{\operatorname {d} t}}=\operatorname {d} '{\frac {\operatorname {d} u}{\operatorname {d} t}}+{\frac {\operatorname {d} u}{\operatorname {d} x}}{\frac {\operatorname {dd} 'x}{\operatorname {d} t}}+{\frac {\operatorname {d} u}{\operatorname {d} y}}{\frac {\operatorname {dd} 'y}{\operatorname {d} t}}+{\frac {\operatorname {d} u}{\operatorname {d} z}}{\frac {\operatorname {dd} 'z}{\operatorname {d} t}},$

et non pas simplement