Page:Annales de mathématiques pures et appliquées, 1815-1816, Tome 6.djvu/94

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(49)$\left\{{\begin{aligned}&B=\phi b,\\1&B_{1}=c^{-1}.\operatorname {D} \phi b.b_{1},\\2&B_{2}=c^{-2}.\operatorname {D} \phi b.b_{2}+c^{-2}.\operatorname {DD} \phi b.b_{1}^{2}\\&\quad \qquad \qquad \qquad +\operatorname {D} \left(c^{-2}\right).\operatorname {D} \phi b.c_{1}b_{1}\\3&B_{3}=c^{-3}.\operatorname {D} \phi b.b_{3}+c^{-3}.\operatorname {DD} \phi b.2b_{1}b_{2}+c^{-3}.{\frac {\operatorname {D} ^{2}\operatorname {D} \phi b.b_{1}^{3}}{2}}.b_{1}^{3}\\&\qquad \qquad +\operatorname {D} \left(c^{-3}\right).\operatorname {D} \phi b\left(c_{1}b_{2}+c_{2}b_{1}\right)+\operatorname {D} \left(c^{-3}\right).\operatorname {DD} \phi b.c_{1}b_{1}^{2}\\&\quad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad +{\frac {\operatorname {D} ^{2}(c^{-3}}{2}}\operatorname {D} \phi b.c_{1}^{2}b_{1}\\4&B_{4}=c^{-4}.\operatorname {D} \phi b.b_{4}+c^{-4}.\operatorname {DD} \phi b\left(2b_{1}b_{3}+b_{2}^{2}\right)\\&\qquad \qquad \qquad \qquad \qquad \qquad +c^{-4}.{\frac {\operatorname {D^{2}D} \phi b}{2}}.3b_{1}^{2}b_{2}\\&\qquad +\operatorname {D} \left(c^{-4}\right).\operatorname {D} \phi b\left(c_{1}b_{3}+c_{2}b_{2}+c_{3}b_{1}\right)\\&\qquad \qquad \qquad \qquad \qquad \qquad +\operatorname {D} \left(c^{-4}\right).\operatorname {DD} \phi \left(2c_{1}b_{2}+c_{2}b_{1}^{2}\right)\\&\qquad \qquad \qquad \qquad \qquad \qquad +{\frac {\operatorname {D} ^{2}(c^{-2}}{2}}.\operatorname {D} \phi b\left(c_{1}^{2}b_{2}+2c_{1}c_{2}b_{1}\right)\\&\qquad +c^{-4}.{\frac {\operatorname {D^{3}D} \phi b}{6}}.b_{1}^{4}\\&\qquad +\operatorname {D} \left(c^{-4}\right).{\frac {\operatorname {D^{2}D} \phi b}{2}}.c_{1}b_{1}^{3}\\&\qquad +{\frac {\operatorname {D} ^{2}\left(c^{-4}\right)}{2}}.\operatorname {DD} \phi b.c_{1}^{2}b_{1}^{2}\\&\qquad +{\frac {\operatorname {D} ^{3}\left(c^{-4}\right)}{6}}.\operatorname {D} \phi b.c_{1}^{3}b_{1}\\5&B_{5}=c^{-5}.\operatorname {D} \phi b.b_{5}+c^{-5}.\operatorname {DD} \phi b\left(2b_{1}b_{4}+2b_{2}b_{3}\right)\\&\qquad \qquad \qquad \qquad \qquad +c^{-5}.{\frac {\operatorname {D^{2}D} \phi b}{2}}.\left(3b_{1}^{2}b_{3}+3b_{1}b_{2}^{2}\right)\\&+\operatorname {D} \left(c^{-5}\right).\operatorname {D} \phi b\left(c_{1}b_{4}+c_{2}b_{3}+c_{3}b_{2}+c_{4}b_{1}\right)\\&\qquad \qquad +\operatorname {D} \left(c^{-5}\right).\operatorname {DD} \phi b.\left(2c_{1}b_{1}b_{3}+c_{1}b_{2}^{2}+2c_{2}b_{1}b_{2}+c_{3}b_{1}^{2}\right)\\&\qquad \qquad +{\frac {\operatorname {D} ^{2}\left(c^{-5}\right)}{2}}.\operatorname {D} \phi b.\left(c_{1}^{2}b_{3}+2c_{1}c_{2}b_{2}+2c_{1}c_{3}b_{1}+c_{2}^{2}b_{1}\right)\\&+c^{-5}.{\frac {\operatorname {D^{3}D} \phi b}{2}}.4b_{1}^{3}b_{2}+\qquad \qquad +c^{-5}.{\frac {\operatorname {D} ^{4}\operatorname {D} \phi b}{24}}.b_{1}^{5}\\&+\operatorname {D} \left(c^{-5}\right).{\frac {\operatorname {D^{2}D} \phi b}{2}}(3c_{1}b_{1}^{2}+c_{2}b_{1}^{3})+\operatorname {D} \left(c^{-5}\right).{\frac {\operatorname {D^{3}D} \phi b}{6}}.c_{1}b_{1}^{4}\\\end{aligned}}\right.$