Page:Annales de mathématiques pures et appliquées, 1815-1816, Tome 6.djvu/276

266

RECHERCHE

$n_{l}-2n+n'={\frac {\operatorname {d} ^{2}n}{\operatorname {d} t^{2}}}i^{2}+(N'+N_{l})i^{4}\,;$

${\begin{array}{rlll}m'-m_{l}&=2{\frac {\operatorname {d} m}{\operatorname {d} t}}i&+{\frac {\operatorname {d} ^{3}m}{\operatorname {d} t^{3}}}{\frac {i^{3}}{3}}&+(M'-M_{l})i^{4},\\n'-n_{l}&=2{\frac {\operatorname {d} n}{\operatorname {d} t}}i&+{\frac {\operatorname {d} ^{3}n}{\operatorname {d} t^{3}}}{\frac {i^{3}}{3}}&+(N'-N_{l})i^{4},\\Z'-Z_{l}&=2{\frac {\operatorname {d} Z}{\operatorname {d} t}}i&+{\frac {\operatorname {d} ^{3}Z}{\operatorname {d} t^{3}}}{\frac {i^{3}}{3}}&+(wx'-wx_{l})i^{4},\\\end{array}}$

${\begin{array}{rllll}m-m_{l}&={\frac {\operatorname {d} m}{\operatorname {d} t}}{\frac {i}{1}}&-{\frac {\operatorname {d} ^{2}m}{\operatorname {d} t^{2}}}{\frac {i^{2}}{1.2}}&+{\frac {\operatorname {d} ^{3}m}{\operatorname {d} t^{3}}}{\frac {i^{3}}{1.2.3}}&-M_{l}i^{4},\\n-n_{l}&={\frac {\operatorname {d} n}{\operatorname {d} t}}{\frac {i}{1}}&-{\frac {\operatorname {d} ^{2}n}{\operatorname {d} t^{2}}}{\frac {i^{2}}{1.2}}&+{\frac {\operatorname {d} ^{3}n}{\operatorname {d} t^{3}}}{\frac {i^{3}}{1.2.3}}&-N_{l}i^{4},\\m'-m&={\frac {\operatorname {d} m}{\operatorname {d} t}}{\frac {i}{1}}&+{\frac {\operatorname {d} ^{2}m}{\operatorname {d} t^{2}}}{\frac {i^{2}}{1.2}}&+{\frac {\operatorname {d} ^{3}m}{\operatorname {d} t^{3}}}{\frac {i^{3}}{1.2.3}}&+M'i^{4},\\n'-n&={\frac {\operatorname {d} n}{\operatorname {d} t}}{\frac {i}{1}}&+{\frac {\operatorname {d} ^{2}n}{\operatorname {d} t^{2}}}{\frac {i^{2}}{1.2}}&+{\frac {\operatorname {d} ^{3}n}{\operatorname {d} t^{3}}}{\frac {i^{3}}{1.2.3}}&+N'i^{4},\\\end{array}}$

d’après quoi, en posant pour abréger

$S=(N'+N_{l}){\frac {\operatorname {d} m}{\operatorname {d} t}}-(M'+M_{l}){\frac {\operatorname {d} n}{\operatorname {d} t}}-{\tfrac {1}{6}}\left({\frac {\operatorname {d} ^{2}m}{\operatorname {d} t^{2}}}{\frac {\operatorname {d} ^{3}n}{\operatorname {d} t^{3}}}-{\frac {\operatorname {d} ^{2}n}{\operatorname {d} t^{2}}}{\frac {\operatorname {d} ^{3}m}{\operatorname {d} t^{3}}}\right)$

$-{\tfrac {1}{2}}\left\{(N'-N_{l}){\frac {\operatorname {d} ^{2}m}{\operatorname {d} t^{2}}}-(M'-M_{l}){\frac {\operatorname {d} ^{2}n}{\operatorname {d} t^{2}}}\right\}i$

$+{\tfrac {1}{6}}\left\{(N'+N_{l}){\frac {\operatorname {d} ^{3}m}{\operatorname {d} t^{3}}}-(M'+M_{l}){\frac {\operatorname {d} ^{3}n}{\operatorname {d} t^{3}}}\right\}i^{2}-(M_{l}N'-M'N_{l})i^{3}\,;$

$F=2(H'+H_{l}){\frac {\operatorname {d} m}{\operatorname {d} t}}-2(G'+G_{l}){\frac {\operatorname {d} n}{\operatorname {d} t}}+{\tfrac {1}{3}}\left({\frac {\operatorname {d} ^{3}m}{\operatorname {d} t^{3}}}{\frac {\operatorname {d} ^{h}n}{\operatorname {d} t^{2}}}-{\frac {\operatorname {d} ^{3}n}{\operatorname {d} t^{3}}}{\frac {\operatorname {d} ^{2}g}{\operatorname {d} t^{2}}}\right)$

$+\left\{(M'-M_{l}){\frac {\operatorname {d} ^{2}h}{\operatorname {d} t^{2}}}-(N'-N_{l}){\frac {\operatorname {d} ^{2}g}{\operatorname {d} t^{2}}}\right\}i+{\tfrac {1}{3}}\left\{(H'+H_{l}){\frac {\operatorname {d} ^{3}m}{\operatorname {d} t^{3}}}-(G'+G_{l}){\frac {\operatorname {d} ^{3}n}{\operatorname {d} t^{3}}}\right\}i^{2}$

$+\left\{(M'-M_{l})(H'+H_{l})-(N'-N_{l})(G'+G_{l})\right\}i^{3}\,;$