Page:Joseph Louis de Lagrange - Œuvres, Tome 12.djvu/249

241

SECONDE PARTIE. — SECTION IX.

sorte que l’on ait

$p^{2}+q^{2}+r^{2}=x^{2}+y^{2}+z^{2}\quad {\text{et}}\quad \mathrm {T} ={\frac {\alpha x^{2}+\beta y^{2}+\gamma z^{2}}{2}}\,;$

or, en faisant varier successivement $p,\,q,\,r,$ on aura, à cause que $x,\,y,\,z$ sont fonctions de ces variables,

${\begin{aligned}{\frac {\partial \mathrm {T} }{\partial p}}=&\alpha x{\frac {\partial x}{\partial p}}+\beta y{\frac {\partial y}{\partial p}}+\gamma z{\frac {\partial z}{\partial p}},\\{\frac {\partial \mathrm {T} }{\partial q}}=&\alpha x{\frac {\partial x}{\partial q}}+\beta y{\frac {\partial y}{\partial q}}+\gamma z{\frac {\partial z}{\partial q}},\\{\frac {\partial \mathrm {T} }{\partial r}}=&\alpha x{\frac {\partial x}{\partial r}}+\beta y{\frac {\partial y}{\partial r}}+\gamma z{\frac {\partial z}{\partial r}}\,;\end{aligned}}$

mais

$x=p'p+q'q+r'r,\qquad y=p''p+q''q+r''r,\qquad z=p'''p+q'''q+r'''r,$

comme on l’a déjà vu plus haut ; donc

${\frac {\partial x}{\partial p}}=p',\qquad {\frac {\partial x}{\partial q}}=q',\quad {\frac {\partial x}{\partial r}}=r',\qquad {\frac {\partial y}{\partial p}}=p'',\quad {\frac {\partial y}{\partial q}}=q'',\qquad \ldots \,;$

substituant ces valeurs, on aura donc

${\begin{aligned}{\frac {\partial \mathrm {t} }{\partial p}}=&p'\alpha x+p''\beta y+p'''\gamma z,\\{\frac {\partial \mathrm {t} }{\partial q}}=&q'\alpha x+q''\beta y+q'''\gamma z,\\{\frac {\partial \mathrm {t} }{\partial r}}=&r'\alpha x+r''\beta y+r'''\gamma z,\\\end{aligned}}$

De sorte qu’en vertu des équations de condition entre les coefficients $p',\,q',\,r',\,p'',\ldots ,$ on aura

$\left({\frac {\partial \mathrm {T} }{\partial p}}\right)^{2}+\left({\frac {\partial \mathrm {T} }{\partial q}}\right)^{2}+\left({\frac {\partial \mathrm {T} }{\partial r}}\right)^{2}=\alpha ^{2}x^{2}+\beta ^{2}y^{2}+\gamma ^{2}z^{2}$

et

$\left(d{\frac {\partial \mathrm {T} }{\partial p}}\right)^{2}+\left(d{\frac {\partial \mathrm {T} }{\partial q}}\right)^{2}+\left(d{\frac {\partial \mathrm {T} }{\partial r}}\right)^{2}=\alpha ^{2}dx^{2}+\beta ^{2}dy^{2}+\gamma ^{2}dz^{2}.$

Par conséquent, les trois équations finales de l’article 24 se rédui-