Page:Joseph Louis de Lagrange - Œuvres, Tome 14.djvu/154

adeoque

${\begin{aligned}\delta \int \mathrm {Z} =&\int \mathrm {N} \delta y+\int \mathrm {P} d\,\delta y+\int \mathrm {Q} d^{2}\delta y+\ldots \\&+\int \mathrm {L} \int (\mathrm {N} )\delta y+\int \mathrm {L} \int (\mathrm {P} )d\,\delta y+\ldots .\end{aligned}}$

Sit $\int \mathrm {L} ,$ posito $x=a,\ =\mathrm {H}$ et $\mathrm {H-\int L=V} \,;$ erit per § IV

$\delta \int \mathrm {Z}$ (posito $x=a$)

$=\int \mathrm {\left[N+(N)V\right]} \delta y+\int \mathrm {\left[P+(P)V\right]} d\,\delta y+\int \mathrm {\left[Q+(Q)V\right]} d^{2}\delta y\ldots ,$

unde, ut supra, erit pro maximo minimove,

$\mathrm {N+(N)V} -d\mathrm {\left[P+(P)V\right]} +d^{2}\mathrm {\left[Q+(Q)V\right]} -\ldots =0.$

Eodem modo operandum pro formula prop. IV, cap. III ; et in universum quæcumque et quotcunque integralia involvantur ; hujus itaque analysin brevitatis gratia omitto ; et progrediar ad formulam prop. V ejusdem Cap., quae mira facilitate etiam resolvitur.

Sit 3^o

${\begin{aligned}\delta \mathrm {Z} =&\mathrm {L} \delta \pi +\mathrm {N} \delta y+\mathrm {P} \delta \,dy+\mathrm {Q} \delta \,d^{2}y+\ldots ,\\\pi =&\int \mathrm {Z} \,;\qquad \delta (\mathrm {Z} )=(\mathrm {L} )\delta \pi +(\mathrm {N} )\delta y+(\mathrm {P} )\delta \,dy+\ldots ,\end{aligned}}$

seu, eliminando $(z),$

$d\,\delta \pi =(\mathrm {L} )\delta \pi +(\mathrm {N} )\delta y+(\mathrm {P} )d\,\delta y+\ldots \,;$

sit, brevitatis gratia,

$(\mathrm {N} )\delta y+(\mathrm {P} )d\,\delta y+\ldots =\mathrm {V} ,$

erit æqualis

$d\,\delta \pi -(\mathrm {L} )\delta \pi =\mathrm {V} \,;$

unde per regulas cognitas integrando habebimus

$\delta \pi =e^{\int (\mathrm {L} )}\int \mathrm {V} e^{-\int (\mathrm {L} )}$

unde fiet

${\begin{aligned}\delta \int \mathrm {Z} =&\int \mathrm {N} \delta y+\int \mathrm {P} d\,\delta y+\ldots \\&+\int e^{\int (\mathrm {L} )}\mathrm {L} \int e^{-\int (\mathrm {L} )}(\mathrm {N} )\delta y+\int e^{\int (\mathrm {L} )}\mathrm {L} \int e^{-\int (\mathrm {L} )}(\mathrm {P} )\delta y+\ldots \,;\end{aligned}}$

quapropter, si $\int e^{\int (\mathrm {L} )}\mathrm {L} ,$ posito $x=a,$ abeat in $\mathrm {H} ,$ et $\mathrm {H} -\int e^{\int (\mathrm {L} )}\mathrm {L}$ in $\mathrm {V} ,$ habebimus operando ut supra

$\mathrm {N+(N)} e^{-\int (\mathrm {L} )}\mathrm {V} -d\left[\mathrm {P+(P)} e^{-\int (\mathrm {L} )}\mathrm {V} \right]+\ldots =0,$