Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 9.djvu/45

on tirera

$0={\frac {\partial ^{2}z^{(1)}}{\partial \varpi \partial \theta }}+m^{(1)}{\frac {\partial z^{(1)}}{\partial \varpi }}+n{\frac {\partial z^{(1)}}{\partial \theta }}+l^{(1)}z^{(1)}.$

Soit

$z^{(2)}={\frac {\partial z^{(1)}}{\partial \theta }}+m^{(1)}z^{(1)},$

et l’on aura

$0={\frac {\partial z^{(2)}}{\partial \varpi }}+nz^{(2)}+z^{(1)}\left(l^{(1)}-{\frac {\partial m^{(1)}}{\partial \varpi }}-nm^{(1)}\right)\,;$

d’où, en faisant

$\mu ^{(1)}=l^{(1)}-{\frac {\partial m^{(1)}}{\partial \varpi }}-nm^{(1)},\qquad m^{(2)}=m^{(1)}-{\frac {\cfrac {\partial \mu ^{(1)}}{\partial \theta }}{\mu ^{(1)}}}$

et

$l^{(2)}=\mu ^{(1)}+nm^{(1)}-n{\frac {\cfrac {\partial \mu ^{(1)}}{\partial \theta }}{\mu ^{(1)}}}+{\frac {\partial n}{\partial \theta }},$

on tirera

$0={\frac {\partial ^{2}z^{(2)}}{\partial \varpi \partial \theta }}+m^{(2)}{\frac {\partial z^{(2)}}{\partial \varpi }}+n{\frac {\partial z^{(2)}}{\partial \theta }}+l^{(2)}z^{(2)}.$

Soit

$z^{(3)}={\frac {\partial z^{(2)}}{\partial \theta }}+m^{(2)}z^{(2)},$

et l’on aura

$0={\frac {\partial z^{(3)}}{\partial \varpi }}+nz^{(3)}+z^{(2)}\left(l^{(2)}-{\frac {\partial m^{(2)}}{\partial \varpi }}-nm^{(2)}\right)\,;$

d’où, en faisant

$\mu ^{(2)}=l^{(2)}-{\frac {\partial m^{(2)}}{\partial \varpi }}-nm^{(2)},\qquad m^{(3)}=m^{(2)}-{\frac {\cfrac {\partial \mu ^{(2)}}{\partial \theta }}{\mu ^{(2)}}}$

et

$l^{(3)}=\mu ^{(2)}+nm^{(2)}-n{\frac {\cfrac {\partial \mu ^{(2)}}{\partial \theta }}{\mu ^{(2)}}}+{\frac {\partial n}{\partial \theta }},$

on tirera

$0={\frac {\partial ^{2}z^{(3)}}{\partial \varpi \partial \theta }}+m^{(3)}{\frac {\partial z^{(3)}}{\partial \varpi }}+n{\frac {\partial z^{(3)}}{\partial \theta }}+l^{(3)}z^{(3)}.$

En continuant ainsi, on aura

$0={\frac {\partial z^{(r-1)}}{\partial \varpi }}+nz^{(r-1)}+z^{(r-2)}\left(l^{(r-2)}-{\frac {\partial m^{(r-2)}}{\partial \varpi }}-nm^{(r-2)}\right)\,;$