Page:Laplace - Œuvres complètes, Gauthier-Villars, 1878, tome 9.djvu/48

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que l’on fasse

{\frac {\partial \mathrm {T} }{\partial \theta }}-\mathrm {T} {\frac {\cfrac {\partial \mu }{\partial \theta }}{\mu }}+m\mathrm {T} =\mathrm {T} ^{(1)},

m-{\frac {\cfrac {\partial \mu }{\partial \theta }}{\mu }}=m^{(1)},

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

on aura

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)}+\mathrm {T} ^{(1)}\,;

on fera ensuite

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)+\mathrm {T} ^{(1)},

d’où l’on tirera, par la méthode précédente, l’équation

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)}+\mathrm {T} ^{(2)}\,;

en continuant d’opérer ainsi, on parviendra aux équations suivantes :

{\begin{aligned}0\quad =&{\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)+\mathrm {T} ^{(r-2)},\\0\quad =&{\frac {\partial ^{2}z^{(r-1)}}{\partial \varpi \partial \theta }}+m^{(r-1)}{\frac {\partial z^{(r-1)}}{\partial \varpi }}+n{\frac {\partial z^{(r-1)}}{\partial \theta }}+l^{(r-1)}z^{(r-1)}+\mathrm {T} ^{(r-1)},\\z^{(r)}=&{\frac {\partial z^{(r-1)}}{\partial \theta }}+m^{(r-1)}z^{(r-1)},\\0\quad =&{\frac {\partial z^{(r)}}{\partial \varpi }}+nz^{(r)}+z^{(r-1)}\left(l^{(r-1)}-{\frac {\partial m^{(r-1)}}{\partial \varpi }}-nm^{(r-1)}\right)+\mathrm {T} ^{(r-1)}.\end{aligned}}

Maintenant, puisque l’expression de $z$ se termine, on a

0=l^{(r-1)}-{\frac {\partial m^{(r-1)}}{\partial \varpi }}-nm^{(r-1)}\,;