Page:Annales de mathématiques pures et appliquées, 1814-1815, Tome 5.djvu/281

275

D’ASTRONOMIE.

${\begin{array}{lrrr}H'''=+4-8x^{2}&+xy&-5y^{2}\\&+9x^{3}y&+4x^{2}y^{2}&+xy^{3}\\&&-7x^{4}y^{2}&-x^{3}y^{3}\\&&&+2x^{5}y^{3}\,;\\\end{array}}$

on aura finalement

${\frac {\operatorname {d} z'''}{\operatorname {d} \lambda }}={\frac {r^{4}v}{\left(pu^{3}-qr^{2}\right)^{5}}}\left(p^{2}u^{6}{\frac {F'''}{r}}-pqr^{2}u^{3}G'''+q^{2}r^{3}H'''\right).$

158. Mettant ensemble les exposions différentielles des trois numéros précédens, on trouve

${\frac {2\pi }{pq}}.{\frac {\operatorname {d} ^{3}t}{\operatorname {d} \lambda ^{3}}}={\tfrac {v}{r\left(pu^{3}-qr^{2}\right)^{5}}}\left\{{\begin{aligned}&p^{4}u^{12}F'\\+&p^{3}qu^{9}r^{2}(F''-G')\\+&p^{2}q^{2}u^{6}r^{4}(F'''-G''+H')\\+&pq^{3}u^{3}r^{6}(-G'''+H'')\\+&q^{4}r^{8}H'''.\\\end{aligned}}\right\}$

Or, on trouve, après les réductions

${\begin{array}{rl}F'=&+10\\&-2x^{2}-31xy-26y^{2}\\&\quad \quad +3x^{3}y+43x^{2}y^{2}+76xy^{3}\\&\qquad \qquad \qquad -x^{4}y^{2}-30x^{3}y^{3}-84x^{2}y^{4}\\&\qquad \qquad \qquad \qquad \qquad \qquad +8x^{4}y^{4}+42x^{3}y^{5}-8x^{4}y^{6}\\F''-G'=&-10\\&+2x^{2}-8xy+14y^{2}\\&\qquad \qquad +38x^{2}y^{2}-10xy^{3}\\&\qquad \qquad -2x^{4}y^{2}-24x^{3}y^{3}-18x^{2}y^{4}\\&\qquad \qquad \qquad \qquad \qquad \qquad +4x^{4}y^{4}+18x^{3}y^{5}-4x^{4}y^{6},\\\end{array}}$