Page:Annales de mathématiques pures et appliquées, 1815-1816, Tome 6.djvu/95

89

DES DÉRIVATIONS.

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

et ainsi de suite.

Si l’on effectue les dérivations des puissances négatives de $c,$ qui ne sont qu’indiquées, ainsi que celles de $\operatorname {D} \phi b,$ et qu’on ordonne selon les dérivées de $\phi b,$ on obtient

${\text{(50)}}\left\{{\begin{aligned}&B=\phi b\\&1B_{1}=\operatorname {D} \phi b.c^{-1}.b_{1}\\&{\begin{aligned}&2B_{2}=\operatorname {D} \phi b\\\\\end{aligned}}\left|{\begin{aligned}&c^{-2}.b_{2}+2{\frac {\operatorname {D} ^{2}\phi b}{2}}.c^{-2}.b_{1}^{2}\\&-2c^{-3}.b_{1}c_{2}\\\end{aligned}}\right.\\&{\begin{aligned}&3B_{3}=\operatorname {D} \phi b\\\\\\\end{aligned}}\left|{\begin{aligned}&c^{-3}.b_{3}\qquad +2{\frac {\operatorname {D} ^{2}\phi b}{2}}\\&-3c^{-4}\left(b_{1}c_{2}+b_{2}c_{1}\right)\\&+{\frac {3.4}{2}}.c^{-5}.b_{1}c_{1}^{2}\\\end{aligned}}\right.\left|{\begin{aligned}&c^{-3}.2b_{1}b_{2}+3{\frac {\operatorname {D} ^{3}\phi b}{6}}.c^{-3}.b_{1}^{3}\\&-3c^{-4}.b_{1}^{2}c_{1}\\\\\end{aligned}}\right.\\&{\begin{aligned}&4B_{4}=\operatorname {D} \phi b\\\\\\\\\end{aligned}}\left|{\begin{aligned}&c^{-4}.b_{4}+\qquad \qquad 2{\frac {\operatorname {D} ^{2}\phi b}{2}}\\&-4c^{-5}.\left(b_{1}c_{3}+b_{2}c_{2}+b_{3}c_{1}\right)\\&+{\frac {4.5}{2}}.c^{-6}\left(2b_{1}c_{1}c_{2}+b_{2}c_{1}^{2}\right)\\&-{\frac {4.5.6}{2.3}}.c^{-7}.b_{1}c_{1}^{3}\\\end{aligned}}\right.\left|{\begin{aligned}&c^{-4}\left(2b_{1}b_{3}+b_{2}^{2}\right)\\&-4c^{-5}\left(b_{1}^{2}c_{2}+2b_{1}b_{2}c_{1}\right)\\&+{\frac {4.5}{2}}.c^{-6}.b_{1}^{2}c_{1}^{2}\\\\\end{aligned}}\right.\\&{\begin{aligned}&+3{\frac {\operatorname {D} ^{3}\phi b}{6}}\\\\\end{aligned}}\left|{\begin{aligned}&c^{-4}.3b_{1}^{2}b_{2}+4{\frac {\operatorname {D} ^{4}\phi b}{24}}.c^{-4}.b_{1}^{4}\\&-4c^{-5}.b_{1}^{3}c_{1}\\\end{aligned}}\right.\\\end{aligned}}\right.$