Page:Annales de mathématiques pures et appliquées, 1822-1823, Tome 13.djvu/55

51

INDÉTERMINÉES.

${\begin{aligned}KZ&=KZ,\\L{\frac {\operatorname {d} Z}{\operatorname {d} x}}&=-{\frac {\operatorname {d} L}{\operatorname {d} x}}Z+{\frac {\operatorname {d} (LZ)}{\operatorname {d} x}},\\\\M{\frac {\operatorname {d} Z}{\operatorname {d} y}}&=-{\frac {\operatorname {d} M}{\operatorname {d} y}}Z+{\frac {\operatorname {d} (MZ)}{\operatorname {d} y}},\\\\N{\frac {\operatorname {d} ^{2}Z}{\operatorname {d} x^{2}}}&=+{\frac {\operatorname {d} ^{2}N}{\operatorname {d} x^{2}}}Z-{\frac {\operatorname {d} \left({\frac {\operatorname {d} N}{\operatorname {d} x}}Z-N{\frac {\operatorname {d} Z}{\operatorname {d} x}}\right)}{\operatorname {d} x}},\\\\O{\frac {\operatorname {d} ^{2}Z}{\operatorname {d} x\operatorname {d} y}}&=+{\frac {\operatorname {d} ^{2}O}{\operatorname {d} x\operatorname {d} y}}Z-{\frac {\operatorname {d} \left({\frac {\operatorname {d} O}{\operatorname {d} x}}Z\right)}{\operatorname {d} x}}-{\frac {\operatorname {d} \left({\frac {\operatorname {d} O}{\operatorname {d} x}}Z\right)}{\operatorname {d} y}}+{\frac {\operatorname {d} ^{2}(OZ)}{\operatorname {d} x\operatorname {d} y}},\\\\P{\frac {\operatorname {d} ^{2}Z}{\operatorname {d} y^{2}}}&=+{\frac {\operatorname {d} ^{2}P}{\operatorname {d} x^{2}}}Z-{\frac {\operatorname {d} \left({\frac {\operatorname {d} P}{\operatorname {d} y}}Z-P{\frac {\operatorname {d} Z}{\operatorname {d} y}}\right)}{\operatorname {d} y}},\\\\Q{\frac {\operatorname {d} ^{3}Z}{\operatorname {d} x^{3}}}&=-{\frac {\operatorname {d} ^{3}Q}{\operatorname {d} x^{3}}}Z+{\frac {\operatorname {d} \left({\frac {\operatorname {d} ^{2}Q}{\operatorname {d} x^{2}}}Z-{\frac {\operatorname {d} Q}{\operatorname {d} x}}{\frac {\operatorname {d} Z}{\operatorname {d} x}}+Q{\frac {\operatorname {d} ^{2}Z}{\operatorname {d} x^{2}}}\right)}{\operatorname {d} x}},\\\\R{\frac {\operatorname {d} ^{3}Z}{\operatorname {d} x^{2}\operatorname {d} y}}&=-{\frac {\operatorname {d} ^{3}R}{\operatorname {d} x^{2}\operatorname {d} y}}Z+{\frac {\operatorname {d} \left({\frac {\operatorname {d} ^{2}R}{\operatorname {d} x\operatorname {d} y}}Z-{\frac {\operatorname {d} R}{\operatorname {d} y}}{\frac {\operatorname {d} Z}{\operatorname {d} x}}\right)}{\operatorname {d} x}}+{\frac {\operatorname {d} \left({\frac {\operatorname {d} ^{2}R}{\operatorname {d} x^{2}}}Z\right)}{\operatorname {d} y}},\\\\&\qquad \qquad -{\frac {\operatorname {d} ^{2}\left({\frac {\operatorname {d} R}{\operatorname {d} x}}Z-R{\frac {\operatorname {d} Z}{\operatorname {d} x}}\right)}{\operatorname {d} x\operatorname {d} y}},\\\\S{\frac {\operatorname {d} ^{3}Z}{\operatorname {d} x\operatorname {d} y^{2}}}&=-{\frac {\operatorname {d} ^{3}S}{\operatorname {d} x\operatorname {d} y^{2}}}Z+{\frac {\operatorname {d} \left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} y^{2}}}Z\right)}{\operatorname {d} x}}+{\frac {\operatorname {d} \left({\frac {\operatorname {d} ^{2}S}{\operatorname {d} x\operatorname {d} y}}Z-{\frac {\operatorname {d} S}{\operatorname {d} x}}{\frac {\operatorname {d} Z}{\operatorname {d} y}}\right)}{\operatorname {d} y}},\\\\&\qquad \qquad -{\frac {\operatorname {d} ^{2}\left({\frac {\operatorname {d} S}{\operatorname {d} y}}Z-S{\frac {\operatorname {d} Z}{\operatorname {d} y}}\right)}{\operatorname {d} x\operatorname {d} y}},\\\\T{\frac {\operatorname {d} ^{3}Z}{\operatorname {d} y^{3}}}&=-{\frac {\operatorname {d} ^{3}T}{\operatorname {d} y^{3}}}Z+{\frac {\operatorname {d} \left({\frac {\operatorname {d} ^{2}T}{\operatorname {d} y^{2}}}Z-{\frac {\operatorname {d} T}{\operatorname {d} y}}{\frac {\operatorname {d} Z}{\operatorname {d} y}}+T{\frac {\operatorname {d} ^{2}Z}{\operatorname {d} y^{2}}}\right)}{\operatorname {d} y}},\end{aligned}}$