Page:Joseph Louis de Lagrange - Œuvres, Tome 14.djvu/243

et

$\varphi (x)=\varphi (t)+\mathrm {P} \varphi '(t)+d\left[\mathrm {Q} \varphi '(t)\right]+d^{2}\left[\mathrm {R} \varphi '(t)\right]+d^{3}\left[\mathrm {S} \varphi '(t)\right]+\ldots ,$

il sera

$\mathrm {P} +d\mathrm {Q} +d^{2}\mathrm {R} +d^{3}\mathrm {S} +\ldots =\varphi (t)+\mathrm {P} \varphi '(t)+d\left[\mathrm {Q} \varphi '(t)\right]+\ldots ,$

donc

$\mathrm {P} =\varphi (t),\quad d\mathrm {Q} =\mathrm {P} \varphi '(t),\quad d^{2}\mathrm {R} =d\left[\mathrm {Q} \varphi '(t)\right],\quad d^{3}\mathrm {S} =d^{2}\left[\mathrm {R} \varphi '(t)\right],\quad \ldots$

ou

$\mathrm {P} =\varphi (t),\qquad \mathrm {Q} ={\frac {1}{2}}\varphi ^{2}(t),\qquad \mathrm {R} ={\frac {1}{1.2.3}}\varphi ^{3}(t),\qquad \ldots .$

Troisième démonstration. Par les princpes d’Analyses

$\psi (t)=\psi (x)-\varphi (x)\psi '(x)+\varphi ^{2}(x){\frac {d\,\psi '(x)}{1.2\,dx}}-\varphi ^{3}(x){\frac {d^{2}\psi '(x)}{1.2.3\,dx^{2}}}+\ldots \,;$

de même

$\varphi (t)\psi '(t)=\varphi (x)\psi '(x)-\varphi (x){\frac {d\left[\varphi (x)\psi '(x)\right]}{dx}}+\varphi ^{2}(x){\frac {d^{2}\left[\varphi (x)\psi '(x)\right]}{1.2\,dx^{2}}}-\ldots \,;$

donc il sera

$\left.{\begin{aligned}&\psi (t)-\psi (x)\\+&\varphi (t)\psi '(t)\\+&{\frac {d\,\varphi ^{2}(t)\psi '(t)}{1.2\,dt}}\\+&{\frac {d^{2}\varphi ^{3}(t)\psi '(t)}{1.2.3\,dt^{2}}}\\+&{\frac {d^{3}\varphi ^{4}(t)\psi '(t)}{1.2.3.4\,dt^{3}}}\end{aligned}}\right\}=$

$\left\{{\begin{alignedat}{4}-\varphi (x)\psi '(x)+\varphi ^{2}(x)&{\frac {d\psi '(x)}{1.2\,dx}}&-\varphi ^{3}(x)&{\frac {d^{2}\psi '(x)}{1.2.3\,dx^{2}}}&+\varphi ^{4}(x)&{\frac {d^{3}\psi '(x)}{1.2.3.4\,dx^{3}}}&-\ldots \\+\varphi (x)\psi '(x)-\varphi (x)\;&{\frac {d[\varphi (x)\psi '(x)}{dx}}&+\varphi ^{2}(x)&{\frac {d^{2}\left[\varphi (x)\psi '(x)\right]}{1.2.3\,dx^{2}}}&-\varphi ^{3}(x)&{\frac {d^{3}\left[\varphi (x)\psi '(x)\right]}{1.2.3\,dx^{3}}}&+\ldots \\+&{\frac {d[\varphi ^{2}(x)\psi '(x)}{1.2\,dx}}&-\varphi (x)\;&{\frac {d^{2}\left[\varphi ^{2}(x)\psi '(x)\right]}{1.2\,dx^{2}}}&+\varphi ^{2}(x)&{\frac {d^{3}\left[\varphi ^{2}(x)\psi '(x)\right]}{1.2.1.2\,dx^{3}}}&-\ldots \\&&+&{\frac {d^{2}\left[\varphi ^{3}(x)\psi '(x)\right]}{1.2.3\,dx^{2}}}&-\varphi (x)\;&{\frac {d^{3}\left[\varphi ^{3}(x)\psi '(x)\right]}{1.2.3\,dx^{3}}}&+\ldots \\&&&&+&{\frac {d^{3}\left[\varphi ^{4}(x)\psi '(x)\right]}{1.2.3.4\,dx^{3}}}&-\ldots \end{alignedat}}\right.$

Or, du côté droit, tous les membres placés dans les mêmes lignes verticales se détruisent, parce qu’il est en général

$y^{m}a^{m-1}z-my^{m-1}d^{m-1}yz+{\frac {m(m-1)}{1.2}}y^{m-2}d^{m-1}y^{2}z'\ldots \pm d^{m-1}y^{m}z=0.$