Page:Annales de mathématiques pures et appliquées, 1814-1815, Tome 5.djvu/279

273

D’ASTRONOMIE.

$z'={\frac {u^{7}vS'}{\left(pu^{3}-qr^{2}\right)^{3}}}?$

On a ici

${\begin{array}{lrl}m=7,&S'=&5y-x^{2}y-6xy^{2}+2x^{2}y^{3},\\\varepsilon =0,&M'=&-2xy-6y^{2}+4xy^{3},\\n=3,&N'=&5-x^{2}-12xy+6x^{2}y^{2},\\\end{array}}$

d’où il résulte

${\begin{array}{rrl}{\frac {F'}{r}}=10-2x^{2}&-21xy-26y^{2}&\\&+x^{3}y+22x^{2}y^{2}&+50xy^{3}\\&&-8x^{3}y^{3}-34x^{2}y^{4}+8x^{3}y^{5},\\{\frac {G'}{r}}=10-2x^{2}&+2xy-2y^{2}&\\&-2x^{3}y-8x^{2}y^{2}&-12xy^{3}\\&&+4x^{3}y^{3}+14x^{2}y^{4}-4x^{3}y^{5},\\H'=\qquad \qquad &-37xy+24y^{2}&\\&-19x^{3}y+49x^{2}y^{2}&-26xy^{3}\\&\qquad \qquad -3x^{4}y^{2}&-30x^{3}y^{3}+8x^{2}y^{4}\\&&\qquad \qquad +6x^{4}y^{4}.\\\end{array}}$

En suite de quoi on aura finalement

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

156. Exemple III. On demande la différentielle de

$z''={\frac {u^{4}r^{2}vs''}{\left(pu^{3}-qr^{2}\right)^{3}}}?$

On a ici $m=4,\ s''=4x-2y,$

${\begin{array}{rrl}&\varepsilon =2,&M''=4,\\&n=2,&N''=-2\,;\\\end{array}}$

donc