Page:Annales de mathématiques pures et appliquées, 1819-1820, Tome 10.djvu/45

41

DES FONCTIONS POLYNOMES.

${\frac {\operatorname {d} u}{\operatorname {d} x}}=\Sigma \left\{mx^{m-1}z^{n}\left(a_{m,n}{\frac {\operatorname {d} u}{\operatorname {d} y}}+b_{m,n}{\frac {\operatorname {d} u}{\operatorname {d} y'}}\right)\right\},\qquad$(1)

${\frac {\operatorname {d} u}{\operatorname {d} z}}=\Sigma \left\{nx^{m}z^{n-1}\left(a_{m,n}{\frac {\operatorname {d} u}{\operatorname {d} y}}+b_{m,n}{\frac {\operatorname {d} u}{\operatorname {d} y'}}\right)\right\},\qquad$(2)

${\frac {\operatorname {d} u}{\operatorname {d} a_{p,q}}}=x^{p}z^{q}{\frac {\operatorname {d} u}{\operatorname {d} y}},\qquad$(3)

${\frac {\operatorname {d} u}{\operatorname {d} b_{p,q}}}=x^{p}z^{q}{\frac {\operatorname {d} u}{\operatorname {d} y'}}.\qquad$(4)

En vertu de ces deux dernières équations, on peut changer (1) et (2) en celles-ci

${\frac {\operatorname {d} u}{\operatorname {d} x}}=\Sigma \left\{m\left(a_{m,n}{\frac {\operatorname {d} u}{\operatorname {d} a_{m-1,n}}}+b_{m,n}{\frac {\operatorname {d} u}{\operatorname {d} b_{m-1,n}}}\right)\right\},\qquad$(5)

${\frac {\operatorname {d} u}{\operatorname {d} z}}=\Sigma \left\{n\left(a_{m,n}{\frac {\operatorname {d} u}{\operatorname {d} a_{m,n-1}}}+b_{m,n}{\frac {\operatorname {d} u}{\operatorname {d} b_{m,n-1}}}\right)\right\}.\qquad$(6)

Mettant, dans l’équation (3), $p+h$ et $q+k,$ au lieu de $p$ et $q,$ elle deviendra

${\frac {\operatorname {d} u}{\operatorname {d} a_{p+h,q+k}}}=x^{p+h}z^{q+k}{\frac {\operatorname {d} u}{\operatorname {d} y}},$

qui, comparé avec l’équation (3), donne

${\frac {\operatorname {d} u}{\operatorname {d} a_{p+h,q+k}}}=x^{h}z^{k}{\frac {\operatorname {d} u}{\operatorname {d} a_{p,q}}}\,;\qquad$(7)

et on tirera de même de l’équation (4)

${\frac {\operatorname {d} u}{\operatorname {d} b_{p+h,q+k}}}=x^{h}z^{k}{\frac {\operatorname {d} u}{\operatorname {d} a_{p,q}}}.\qquad$(8)