Page:Annales de mathématiques pures et appliquées, 1815-1816, Tome 6.djvu/275

265

DES ORBITES.

Présentement, $g,h,m,n,Z$ étant des fonctions de $t,$ lesquelles deviennent $g_{l},h_{l},m_{l},n_{l},Z_{l}$ ou $g',h',m',n',Z'$ lorsque $t$ devient $t-i$ ou $t+i,$ il s’ensuit qu’on doit avoir

${\begin{array}{rccccc}g_{l}&=g&-{\frac {\operatorname {d} g}{\operatorname {d} t}}{\frac {i}{1}}&+{\frac {\operatorname {d} ^{2}g}{\operatorname {d} t^{2}}}{\frac {i^{2}}{1.2}}&-{\frac {\operatorname {d} ^{3}g}{\operatorname {d} t^{3}}}{\frac {i^{3}}{1.2.3}}&+G_{l}i^{4},\\g'&=g&+{\frac {\operatorname {d} g}{\operatorname {d} t}}{\frac {i}{1}}&+{\frac {\operatorname {d} ^{2}g}{\operatorname {d} t^{2}}}{\frac {i^{2}}{1.2}}&+{\frac {\operatorname {d} ^{3}g}{\operatorname {d} t^{3}}}{\frac {i^{3}}{1.2.3}}&+G'i^{4}\,;\\h_{l}&=h&-{\frac {\operatorname {d} h}{\operatorname {d} t}}{\frac {i}{1}}&+{\frac {\operatorname {d} ^{2}h}{\operatorname {d} t^{2}}}{\frac {i^{2}}{1.2}}&-{\frac {\operatorname {d} ^{3}h}{\operatorname {d} t^{3}}}{\frac {i^{3}}{1.2.3}}&+H_{l}i^{4},\\h'&=h&+{\frac {\operatorname {d} h}{\operatorname {d} t}}{\frac {i}{1}}&+{\frac {\operatorname {d} ^{2}h}{\operatorname {d} t^{2}}}{\frac {i^{2}}{1.2}}&+{\frac {\operatorname {d} ^{3}h}{\operatorname {d} t^{3}}}{\frac {i^{3}}{1.2.3}}&+H'i^{4}\,;\\m_{l}&=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_{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_{l}&=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_{l}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}\,;\\Z_{l}&=Z&-{\frac {\operatorname {d} Z}{\operatorname {d} t}}{\frac {i}{1}}&+{\frac {\operatorname {d} ^{2}Z}{\operatorname {d} t^{2}}}{\frac {i^{2}}{1.2}}&-{\frac {\operatorname {d} ^{3}Z}{\operatorname {d} t^{3}}}{\frac {i^{3}}{1.2.3}}&+\zeta _{l}i^{4},\\Z'&=Z&-{\frac {\operatorname {d} Z}{\operatorname {d} t}}{\frac {i}{1}}&+{\frac {\operatorname {d} ^{2}Z}{\operatorname {d} t^{2}}}{\frac {i^{2}}{1.2}}&+{\frac {\operatorname {d} ^{3}Z}{\operatorname {d} t^{3}}}{\frac {i^{3}}{1.2.3}}&+\zeta 'i^{4}\,;\\\end{array}}$

$G_{l},G',H_{l},H',M_{l},M',N_{l},N',\zeta _{l},\zeta '$ étant des fonctions de $t$ qui ne deviennent pas nulles lorsque $i$ est zéro.

On déduit de tout cela

${\begin{array}{cl}g_{l}-2g+g'&={\frac {\operatorname {d} ^{2}g}{\operatorname {d} t^{2}}}i^{2}+(G'+G_{l})i^{4}\,;\\h_{l}-2h+h'&={\frac {\operatorname {d} ^{2}h}{\operatorname {d} t^{2}}}i^{2}+(H'+H_{l})i^{4}\,;\\m_{l}-2m+m'&={\frac {\operatorname {d} ^{2}m}{\operatorname {d} t^{2}}}i^{2}+(M'+M_{l})i^{4}.\\\end{array}}$