Page:Annales de mathématiques pures et appliquées, 1826-1827, Tome 17.djvu/326

${\begin{aligned}&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&{\frac {\operatorname {d} ^{n}u'}{\operatorname {d} \alpha ^{n}}}+(1-\alpha ){\frac {\operatorname {d} ^{n}P}{\operatorname {d} \alpha ^{n}}}-3{\frac {\operatorname {d} ^{n-1}P}{\operatorname {d} \alpha ^{n-1}}}=0\,;\end{aligned}}$

d’où

$\left.{\begin{aligned}&P={\frac {\operatorname {d} u'}{\operatorname {d} \alpha }}+{\frac {1-\alpha }{1}}.{\frac {\operatorname {d} P}{\operatorname {d} \alpha }},\\\\&{\frac {\operatorname {d} P}{\operatorname {d} \alpha }}={\frac {1}{2}}{\frac {\operatorname {d} ^{2}u'}{\operatorname {d} \alpha ^{2}}}+{\frac {1-\alpha }{2}}.{\frac {\operatorname {d} ^{2}P}{\operatorname {d} \alpha ^{2}}},\\\\&{\frac {\operatorname {d} ^{2}P}{\operatorname {d} \alpha ^{2}}}={\frac {1}{3}}{\frac {\operatorname {d} ^{3}u'}{\operatorname {d} \alpha ^{3}}}+{\frac {1-\alpha }{3}}.{\frac {\operatorname {d} ^{3}P}{\operatorname {d} \alpha ^{3}}},\\&\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \\&{\frac {\operatorname {d} ^{n-1}P}{\operatorname {d} \alpha ^{n-1}}}={\frac {1}{n}}{\frac {\operatorname {d} ^{n}u'}{\operatorname {d} \alpha ^{n}}}+{\frac {1-\alpha }{n}}.{\frac {\operatorname {d} ^{n}P}{\operatorname {d} \alpha ^{n}}}\,;\end{aligned}}\right\}\quad$(A)

et par conséquent

${\begin{aligned}U&=u'+(1-\alpha )P\\&=u'+{\frac {1-\alpha }{1}}{\frac {\operatorname {d} u'}{\operatorname {d} \alpha }}+{\frac {(1-\alpha )^{2}}{1}}.{\frac {\operatorname {d} P}{\operatorname {d} \alpha }}\\\\&=u'+{\frac {1-\alpha }{1}}{\frac {\operatorname {d} u'}{\operatorname {d} \alpha }}+{\frac {(1-\alpha )^{2}}{1.2}}{\frac {\operatorname {d} ^{2}u'}{\operatorname {d} \alpha ^{2}}}+{\frac {(1-\alpha )^{3}}{1.2}}.{\frac {\operatorname {d} ^{2}P}{\operatorname {d} \alpha ^{2}}}\\\\&=u'+{\frac {1-\alpha }{1}}{\frac {\operatorname {d} u'}{\operatorname {d} \alpha }}+{\frac {(1-\alpha )^{2}}{1.2}}{\frac {\operatorname {d} ^{2}u'}{\operatorname {d} \alpha ^{2}}}+{\frac {(1-\alpha )^{3}}{1.2.3}}.{\frac {\operatorname {d} ^{3}u'}{\operatorname {d} \alpha ^{3}}}+{\frac {(1-\alpha )^{4}}{1.2.3}}{\frac {\operatorname {d} ^{3}P}{\operatorname {d} \alpha ^{3}}},\end{aligned}}$

et, en général