Page:Annales de mathématiques pures et appliquées, 1819-1820, Tome 10.djvu/261

253

DES FONCTIONS.

${\frac {\operatorname {d} _{1}^{m}.z^{m}{\frac {\operatorname {d} U}{\operatorname {d} a}}}{\operatorname {d} b_{1}m_{1}}}=m\operatorname {d} .{\frac {\frac {\operatorname {d} _{1}^{m}.z^{m-1}U}{\operatorname {d} b_{1}^{m_{1}}}}{\operatorname {d} b}}-(m-1)\operatorname {d} .{\frac {\frac {\operatorname {d} _{1}^{m}.z^{m}U}{\operatorname {d} b_{1}^{m_{1}}}}{\operatorname {d} a}}\,;$

mais, en vertu des équations (15), on a

${\frac {\operatorname {d} _{1}^{m}.z^{m-1}U}{\operatorname {d} b_{1}^{m_{1}}}}={\frac {\operatorname {d} ^{m_{1}-1}.z_{1}^{m_{1}}{\frac {\operatorname {d} .z^{m-1}U}{\operatorname {d} a_{1}}}}{\operatorname {d} a_{1}^{m_{1}-1}}},$

et encore

${\frac {\operatorname {d} _{1}^{m}.z^{m}U}{\operatorname {d} b_{1}^{m_{1}}}}={\frac {\operatorname {d} ^{m_{1}-1}.z_{1}^{m_{1}}{\frac {\operatorname {d} .z^{m}U}{\operatorname {d} a_{1}}}}{\operatorname {d} a_{1}^{m_{1}-1}}}\,;$

ainsi l’on aura

${\frac {\operatorname {d} _{1}^{m}.z^{m}{\frac {\operatorname {d} U}{\operatorname {d} a}}}{\operatorname {d} b_{1}^{m_{1}}}}={\frac {\operatorname {d} ^{m_{1}-1}\left(m{\frac {\operatorname {d} .z_{1}^{m_{1}}{\frac {\operatorname {d} .z^{m-1}U}{\operatorname {d} a_{1}}}}{\operatorname {d} b}}-(m-1){\frac {\operatorname {d} .z_{1}^{m_{1}}{\frac {\operatorname {d} .z^{m}U}{\operatorname {d} a_{1}}}}{\operatorname {d} a}}\right)}{\operatorname {d} a_{1}^{m_{1}-1}}}\,;$

en observant que la quantité dans les parenthèses peut être mise sous cette forme

$z^{m}z_{1}^{m_{1}}{\frac {\operatorname {d} ^{2}U}{\operatorname {d} a\operatorname {d} a_{1}}}+z_{1}^{m_{1}}{\frac {\operatorname {d} .z^{m}}{\operatorname {d} a_{1}}}{\frac {\operatorname {d} U}{\operatorname {d} a}}+z^{m}{\frac {\operatorname {d} .z_{1}^{m_{1}}}{\operatorname {d} a}}{\frac {\operatorname {d} U}{\operatorname {d} a_{1}}},$

il viendra finalement

${\frac {\operatorname {d} _{1}^{m}.z^{m}{\frac {\operatorname {d} U}{\operatorname {d} a}}}{\operatorname {d} b_{1}^{m_{1}}}}={\frac {\operatorname {d} ^{m_{1}-1}\left(z^{m}z_{1}^{m_{1}}{\frac {\operatorname {d} ^{2}U}{\operatorname {d} a\operatorname {d} a_{1}}}+z_{1}^{m_{1}}{\frac {\operatorname {d} .z^{m}}{\operatorname {d} a_{1}}}{\frac {\operatorname {d} U}{\operatorname {d} a}}+z^{m}{\frac {\operatorname {d} .z_{1}^{m_{1}}}{\operatorname {d} a}}{\frac {\operatorname {d} U}{\operatorname {d} a_{1}}}\right)}{\operatorname {d} a_{1}^{m_{1}}}}\,;$

mais on a

${\frac {\operatorname {d} ^{m+m_{1}}U}{\operatorname {d} b^{m}\operatorname {d} b_{1}^{m_{1}}}}={\frac {\operatorname {d} ^{m+m_{1}-1}.z^{m}{\frac {\operatorname {d} U}{\operatorname {d} a}}}{\operatorname {d} a^{m-1}\operatorname {d} b_{1}^{m_{1}}}}\,;$

partant