Page:Annales de mathématiques pures et appliquées, 1821-1822, Tome 12.djvu/27

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ET DES SURFACES COURBES.

${\frac {\operatorname {d} v}{\operatorname {d} x}}={\frac {\operatorname {d} t}{\operatorname {d} x}}{\frac {\operatorname {d} z}{\operatorname {d} x}}+{\frac {\operatorname {d} u}{\operatorname {d} x}}{\frac {\operatorname {d} z}{\operatorname {d} y}},\qquad {\frac {\operatorname {d} v}{\operatorname {d} y}}={\frac {\operatorname {d} t}{\operatorname {d} y}}{\frac {\operatorname {d} z}{\operatorname {d} x}}+{\frac {\operatorname {d} u}{\operatorname {d} y}}{\frac {\operatorname {d} z}{\operatorname {d} y}}.$

En posant

${\frac {\operatorname {d} z}{\operatorname {d} x}}=p\quad {\frac {\operatorname {d} z}{\operatorname {d} y}}=q,\quad {\frac {\operatorname {d} v}{\operatorname {d} t}}=p'\quad {\frac {\operatorname {d} v}{\operatorname {d} u}}=q'.$

ces équations deviennent d’abord

${\frac {\operatorname {d} v}{\operatorname {d} x}}=p{\frac {\operatorname {d} t}{\operatorname {d} x}}+q{\frac {\operatorname {d} u}{\operatorname {d} x}},\qquad {\frac {\operatorname {d} v}{\operatorname {d} y}}=p{\frac {\operatorname {d} t}{\operatorname {d} y}}+q{\frac {\operatorname {d} u}{\operatorname {d} y}},$

on a d’un autre côté

$\operatorname {d} v=p'\operatorname {d} t+q'\operatorname {d} u=p'\left({\frac {\operatorname {d} t}{\operatorname {d} x}}\operatorname {d} x+{\frac {\operatorname {d} t}{\operatorname {d} y}}\operatorname {d} y\right)+q'\left({\frac {\operatorname {d} u}{\operatorname {d} x}}\operatorname {d} x+{\frac {\operatorname {d} u}{\operatorname {d} y}}\operatorname {d} y\right)$

c’est à-dire,

$\operatorname {d} v=\left(p'{\frac {\operatorname {d} t}{\operatorname {d} x}}+q'{\frac {\operatorname {d} u}{\operatorname {d} x}}\right)\operatorname {d} x+\left(p'{\frac {\operatorname {d} t}{\operatorname {d} y}}+q'{\frac {\operatorname {d} u}{\operatorname {d} y}}\right)\operatorname {d} y,$

d’où

${\frac {\operatorname {d} v}{\operatorname {d} x}}=p'{\frac {\operatorname {d} t}{\operatorname {d} x}}+q'{\frac {\operatorname {d} u}{\operatorname {d} x}},\qquad {\frac {\operatorname {d} v}{\operatorname {d} y}}=p'{\frac {\operatorname {d} t}{\operatorname {d} y}}+q'{\frac {\operatorname {d} u}{\operatorname {d} y}}\,;$

valeurs qui, comparées à celles ci-dessus, s’accocdent à donner

$p=p',\qquad q=q',$

c’est à-dire,

${\frac {\operatorname {d} z}{\operatorname {d} x}}={\frac {\operatorname {d} v}{\operatorname {d} t}},\qquad {\frac {\operatorname {d} z}{\operatorname {d} y}}={\frac {\operatorname {d} v}{\operatorname {d} u}}\,;\qquad$(4)

or, ces coefficiens fixent la direction des plans tangens aux points