Page:Annales de mathématiques pures et appliquées, 1822-1823, Tome 13.djvu/26

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INTÉGRALES

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

et, si $X$ et $Y$ sont d’autres fonctions de $z,$

$X',X'',X''',\ldots ,\quad Y',Y'',Y''',\ldots$

seront pareillement les symboles respectifs de

${\frac {\operatorname {d} X}{\operatorname {d} z}},\quad {\frac {\operatorname {d} ^{2}X}{\operatorname {d} z^{2}}},\quad {\frac {\operatorname {d} ^{3}X}{\operatorname {d} z^{3}}},\ldots {\frac {\operatorname {d} Y}{\operatorname {d} z}},\quad {\frac {\operatorname {d} ^{2}Y}{\operatorname {d} z^{2}}},\quad {\frac {\operatorname {d} ^{3}Y}{\operatorname {d} z^{3}}},\ldots .$

Nous ne recourrons ainsi aux notations du calcul différentiel ordinaire que lorsqu’il s’agira de représenter des coefficiens différentiels partiels. Ainsi

$\left({\frac {\operatorname {d} V}{\operatorname {d} x}}\right),\left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right),\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right),\ldots \left({\frac {\operatorname {d} V}{\operatorname {d} y}}\right),\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right),\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right),\ldots$

seront les coefficiens différentiels partiels que l’on obtient pour la fonction $V,$ en n’y considérant successivement que

$x,x',x'',\ldots ,\quad y,y',y'',\ldots$

comme variables. En conséquence, les expressions

$\left({\frac {\operatorname {d} V}{\operatorname {d} x}}\right)',\left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right)',\left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)',\ldots \left({\frac {\operatorname {d} V}{\operatorname {d} y}}\right)',\left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)',\left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)',\ldots$

seront la même chose que

${\frac {\operatorname {d} \left({\frac {\operatorname {d} V}{\operatorname {d} x}}\right)}{\operatorname {d} z}},{\frac {\operatorname {d} \left({\frac {\operatorname {d} V}{\operatorname {d} x'}}\right)}{\operatorname {d} z}},{\frac {\operatorname {d} \left({\frac {\operatorname {d} V}{\operatorname {d} x''}}\right)}{\operatorname {d} z}},\ldots {\frac {\operatorname {d} \left({\frac {\operatorname {d} V}{\operatorname {d} y}}\right)}{\operatorname {d} z}},{\frac {\operatorname {d} \left({\frac {\operatorname {d} V}{\operatorname {d} y'}}\right)}{\operatorname {d} z}},{\frac {\operatorname {d} \left({\frac {\operatorname {d} V}{\operatorname {d} y''}}\right)}{\operatorname {d} z}},\ldots$

Pareillement, les expressions