Page:Annales de mathématiques pures et appliquées, 1810-1811, Tome 1.djvu/92

88

FORMULES.

${\begin{array}{llllr}{\tfrac {25}{2153}}&+{\tfrac {1}{3}}\left({\tfrac {25}{2153}}\right)^{3}&+{\tfrac {1}{5}}\left({\tfrac {25}{2153}}\right)^{5}&+{\text{etc.}}&=\mathrm {S} _{1}\\{\tfrac {2}{359}}&+{\tfrac {1}{3}}\left({\tfrac {2}{359}}\right)^{3}&+{\tfrac {1}{5}}\left({\tfrac {2}{359}}\right)^{5}&+{\text{etc.}}&=\mathrm {S} _{2}\\{\tfrac {1}{337}}&+{\tfrac {1}{3}}\left({\tfrac {1}{337}}\right)^{3}&+{\tfrac {1}{5}}\left({\tfrac {1}{337}}\right)^{5}&+{\text{etc.}}&=\mathrm {S} _{3}\\{\tfrac {200}{117449}}&+{\tfrac {1}{3}}\left({\tfrac {200}{117449}}\right)^{3}&+{\tfrac {1}{5}}\left({\tfrac {200}{117449}}\right)^{5}&+{\text{etc.}}&=\mathrm {S} _{4}\\{\tfrac {1}{2311}}&+{\tfrac {1}{3}}\left({\tfrac {1}{2311}}\right)^{3}&+{\tfrac {1}{5}}\left({\tfrac {1}{2311}}\right)^{5}&+{\text{etc.}}&=\mathrm {S} _{5}\\{\tfrac {25}{121993}}&+{\tfrac {1}{3}}\left({\tfrac {25}{121993}}\right)^{3}&+{\tfrac {1}{5}}\left({\tfrac {25}{121993}}\right)^{5}&+{\text{etc.}}&=\mathrm {S} _{6}\\{\tfrac {1}{28799}}&+{\tfrac {1}{3}}\left({\tfrac {1}{28799}}\right)^{3}&+{\tfrac {1}{5}}\left({\tfrac {1}{28799}}\right)^{5}&+{\text{etc.}}&=\mathrm {S} _{7}\\{\tfrac {1}{46817}}&+{\tfrac {1}{3}}\left({\tfrac {1}{46817}}\right)^{3}&+{\tfrac {1}{5}}\left({\tfrac {1}{46817}}\right)^{5}&+{\text{etc.}}&=\mathrm {S} _{8}\\\end{array}}$

nous aurons :

${\begin{array}{lllllllr}&-3\mathrm {L} _{2}&+2\mathrm {L} _{3}&-\mathrm {L} _{7}&+2\mathrm {L} _{11}&-\mathrm {L} _{19}&\cdots &=2\mathrm {MS} _{1}\\&-\mathrm {L} _{3}&-\mathrm {L} _{7}&-\mathrm {L} _{1}7&+2\mathrm {L} _{19}&\cdots &\cdots &=2\mathrm {MS} _{2}\\&-3\mathrm {L} _{2}&-\mathrm {L} _{3}&-\mathrm {L} _{7}&+2\mathrm {L} _{13}&\cdots &\cdots &=2\mathrm {MS} _{3}\\&-\mathrm {L} _{3}&+6\mathrm {L} _{7}&-2\mathrm {L} _{11}&-\mathrm {L} _{17}&-\mathrm {L} _{19}&\cdots &=2\mathrm {MS} _{4}\\&\quad 2\mathrm {L} _{2}&-\mathrm {L} _{3}&-\mathrm {L} _{5}&-\mathrm {L} _{7}&-\mathrm {L} _{11}&+2\mathrm {L} _{17}&=2\mathrm {MS} _{5}\\&-3\mathrm {L} _{2}&-2\mathrm {L} _{3}&-\mathrm {L} _{7}&-2\mathrm {L} _{11}r&+2\mathrm {L} _{13}&+2\mathrm {L} _{19}&=2\mathrm {MS} _{6}\\&\quad 6\mathrm {L} _{2}&+2\mathrm {L} _{3}&+2\mathrm {L} _{5}&-\mathrm {L} _{7}&-2\mathrm {L} _{11}&-\mathrm {L} _{17}&=2\mathrm {MS} _{7}\\&-4\mathrm {L} _{2}&+4\mathrm {L} _{3}&-\mathrm {L} _{7}&-\mathrm {L} _{11}&+2\mathrm {L} _{17}&-\mathrm {L} _{19}&=2\mathrm {MS} _{8}\\\end{array}}$

équations qui, étant résolues suivant les règles ordinaires, donnent :

${\begin{array}{llllllll}\mathrm {L_{2}} &=\mathrm {M} \left(\ \ 94\mathrm {S} _{1}-\ \ 40\mathrm {S} _{2}-\ \ 74\mathrm {S} _{3}+\ \ 10\mathrm {S} _{4}+\ \ 28\mathrm {S} _{5}+\ \ 74\mathrm {S} _{6}+\ \ 14\mathrm {S} _{7}-\ \ 36\mathrm {S} _{8}\right)\\\mathrm {L_{3}} &=\mathrm {M} \left(148\mathrm {S} _{1}-\ \ 62\mathrm {S} _{2}-116\mathrm {S} _{3}+\ \ 16\mathrm {S} _{4}+\ \ 44\mathrm {S} _{5}+116\mathrm {S} _{6}+\ \ 22\mathrm {S} _{7}-\ \ 56\mathrm {S} _{8}\right)\\\mathrm {L_{5}} &=\mathrm {M} \left(226\mathrm {S} _{1}-103\mathrm {S} _{2}-183\mathrm {S} _{3}+\ \ 22\mathrm {S} _{4}+\ \ 64\mathrm {S} _{5}+183\mathrm {S} _{6}+\ \ 33\mathrm {S} _{7}-\ \ 88\mathrm {S} _{8}\right)\\\mathrm {L_{7}} &=\mathrm {M} \left(266\mathrm {S} _{1}-115\mathrm {S} _{2}-211\mathrm {S} _{3}+\ \ 28\mathrm {S} _{4}+\ \ 78\mathrm {S} _{5}+211\mathrm {S} _{6}+\ \ 39\mathrm {S} _{7}-102\mathrm {S} _{8}\right)\\\mathrm {L_{11}} &=\mathrm {M} \left(328\mathrm {S} _{1}-142\mathrm {S} _{2}-260\mathrm {S} _{3}+\ \ 34\mathrm {S} _{4}+\ \ 96\mathrm {S} _{5}+260\mathrm {S} _{6}+\ \ 48\mathrm {S} _{7}-126\mathrm {S} _{8}\right)\\\mathrm {L_{13}} &=\mathrm {M} \left(348\mathrm {S} _{1}-{\tfrac {297}{2}}\mathrm {S} _{2}-{\tfrac {547}{2}}\mathrm {S} _{3}+\ \ 37\mathrm {S} _{4}+103\mathrm {S} _{5}+{\tfrac {549}{2}}\mathrm {S} _{6}+{\tfrac {103}{2}}\mathrm {S} _{7}-133\mathrm {S} _{8}\right)\\\mathrm {L_{17}} &=\mathrm {M} \left(390\mathrm {S} _{1}-171\mathrm {S} _{2}-311\mathrm {S} _{3}+\ \ 40\mathrm {S} _{4}+114\mathrm {S} _{5}+311\mathrm {S} _{6}+\ \ 57\mathrm {S} _{7}-150\mathrm {S} _{8}\right)\\\mathrm {L_{19}} &=\mathrm {M} \left(402\mathrm {S} _{1}-173\mathrm {S} _{2}-319\mathrm {S} _{3}+\ \ 42\mathrm {S} _{4}+118\mathrm {S} _{5}+319\mathrm {S} _{6}+\ \ 59\mathrm {S} _{7}-154\mathrm {S} _{8}\right)\\\end{array}}$