Page:Mémoires de l’Académie des sciences, Tome 4.djvu/647

$\mathrm {P_{3}\,Q_{3}\,R_{3}\,S_{3}\,T_{3}} \ldots ;$ ainsi de suite. Il résulte de tout ce qui précède, que les valeurs de $a,\,b,\,c,\,d,\,e\ldots ,$ déduites des équations

${\begin{aligned}a+2\ \ b+3\;c+4\ \ d+5\ \ e&+{\text{etc.=A}}\\a+2^{3}b+3^{3}c+4^{3}d+5^{3}e&+{\text{etc.=B}}\\a+2^{5}b+3^{5}c+4^{5}d+5^{5}e&+{\text{etc.=C}}\\a+2^{7}b+3^{7}c+4^{7}d+5^{7}e&+{\text{etc.=D}}\\a+2^{9}b+3^{9}c+4^{9}d+5^{9}e&+{\text{etc.=E}}\\\end{aligned}}$

etc.

sont exprimées ainsi :

${\begin{aligned}{\frac {1}{2}}a=\mathrm {A} &-\mathrm {B} \left({\frac {\pi ^{2}}{2.3}}-{\frac {1}{1^{2}}}\right)+\mathrm {C} \left({\frac {\pi ^{4}}{2.3.4.5}}-{\frac {1}{1^{2}}}{\frac {\pi ^{2}}{2.3}}+{\frac {1}{1^{4}}}\right)\\&-\mathrm {D} \left({\frac {\pi ^{6}}{2.3.4.5.6.7}}-{\frac {1}{1^{2}}}{\frac {\pi ^{4}}{2.3.4.5}}+{\frac {1}{1^{4}}}{\frac {\pi ^{2}}{2.3}}-{\frac {1}{1^{6}}}\right)\\&+\mathrm {E} \left({\frac {\pi ^{8}}{2.3.4.5.6.7.8.9}}-{\frac {1}{1^{2}}}{\frac {\pi ^{6}}{2.3.4.5.6.7}}+{\frac {1}{1^{4}}}{\frac {\pi ^{4}}{2.3.4.5}}-{\frac {1}{1^{6}}}{\frac {\pi ^{2}}{2.3}}+{\frac {1}{1^{8}}}\right)-{\text{etc}}.\\\\-{\frac {1}{2}}2b=\mathrm {A} &-\mathrm {B} \left({\frac {\pi ^{2}}{2.3}}-{\frac {1}{2^{2}}}\right)+\mathrm {C} \left({\frac {\pi ^{4}}{2.3.4.5}}-{\frac {1}{2^{2}}}{\frac {\pi ^{2}}{2.3}}+{\frac {1}{2^{4}}}\right)\\&-\mathrm {D} \left({\frac {\pi ^{6}}{2.3.4.5.6.7}}-{\frac {1}{2^{2}}}{\frac {\pi ^{4}}{2.3.4.5}}+{\frac {1}{2^{4}}}{\frac {\pi ^{2}}{2.3}}-{\frac {1}{2^{6}}}\right)\\&+\mathrm {E} \left({\frac {\pi ^{8}}{2.3.4.5.6.7.8.9}}-{\frac {1}{2^{2}}}{\frac {\pi ^{6}}{2.3.4.5.6.7}}+{\frac {1}{2^{4}}}{\frac {\pi ^{4}}{2.3.4.5}}-{\frac {1}{2^{6}}}{\frac {\pi ^{2}}{2.3}}+{\frac {1}{2^{8}}}\right)-{\text{etc}}.\\\\{\frac {1}{2}}3c=\mathrm {A} &-\mathrm {B} \left({\frac {\pi ^{2}}{2.3}}-{\frac {1}{3^{2}}}\right)+\mathrm {C} \left({\frac {\pi ^{4}}{2.3.4.5}}-{\frac {1}{3^{2}}}{\frac {\pi ^{2}}{2.3}}+{\frac {1}{3^{4}}}\right)\\&-\mathrm {D} \left({\frac {\pi ^{6}}{2.3.4.5.6.7}}-{\frac {1}{3^{2}}}{\frac {\pi ^{4}}{2.3.4.5}}+{\frac {1}{3^{4}}}{\frac {\pi ^{2}}{2.3}}-{\frac {1}{3^{6}}}\right)\\&+\mathrm {E} \left({\frac {\pi ^{8}}{2.3.4.5.6.7.8.9}}-{\frac {1}{3^{2}}}{\frac {\pi ^{6}}{2.3.4.5.6.7}}+{\frac {1}{3^{4}}}{\frac {\pi ^{4}}{2.3.4.5}}-{\frac {1}{3^{6}}}{\frac {\pi ^{2}}{2.3}}+{\frac {1}{3^{8}}}\right)-{\text{etc}}.\end{aligned}}$