Page:Joseph Louis de Lagrange - Œuvres, Tome 14.djvu/153

Problemata.

Invenire œquationem inter $x$ et $y,$ ut pro dato ipsius $x$ valore puto $x=a,$ formula hœc $\int \mathrm {Z}$ maximum minimumve valorem obtineat.

Resolutiones.

Sit 1^o

$\delta \mathrm {Z} =\mathrm {N} \delta y+\mathrm {P} \delta \,dy+\mathrm {Q} \delta \,d^{2}y+\mathrm {R} \delta \,d^{3}y+\ldots$

($x$ enim in hac differentia ponitur constans § I). Igitur quoniam differentia duorum totorum sequalis est summae differentiarum omnium partium, adeoque $\delta \int \mathrm {Z} =\int \delta \mathrm {Z} ,$ erit

${\begin{aligned}\delta \int \mathrm {Z} &=\int \mathrm {N} \delta y+\int \mathrm {P} \delta \,dy+\int \mathrm {Q} \delta \,d^{2}y+\ldots \\&=\int \mathrm {N} \delta y+\int \mathrm {P} d\,\delta y+\int \mathrm {Q} d^{2}\,\delta y+\ldots \quad {\text{(§ II)}}\\&=\int \mathrm {N} \delta y+\mathrm {P} \delta y-\int d\mathrm {P} \,\delta y+\mathrm {Q} d\,\delta y-d\mathrm {Q} \delta y+\int d^{2}\mathrm {Q} \delta y\ldots \ {\text{(§ III)}}\end{aligned}}$

unde

$\delta \int \mathrm {Z} =\int \left(\mathrm {N} -d\mathrm {P} +d^{2}\mathrm {Q} -\ldots \right)\delta y+(\mathrm {P} -d\mathrm {Q} +\ldots )\delta y+(\mathrm {Q} *\ldots )d\,\delta y+\ldots$

seu, ponendo $y,$ qui respondet $x=a,$ cum nonnullis sequentibus invariabilem, unde totidem habentur puncta per quæ curva invenienda transire debet, erit $\delta y=0,\ d\delta y=0,\ d^{2}\delta y=0,\ \ldots ,$ unde tandem

$\delta \int \mathrm {Z} =\int \left(\mathrm {N} -d\mathrm {P} +d^{2}\mathrm {Q} -d^{3}\mathrm {R} \ldots \right)\delta y\,;$

adeoque ex methodo maximorum, et minimorum communi

$\mathrm {N} -d\mathrm {P} +d^{2}\mathrm {Q} -d^{3}\mathrm {R} \ldots =0,$

dabit æquationem quæsitam (9).

Sit 2^o

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

et

$\delta (\mathrm {Z} )=(\mathrm {N} )\delta y+(\mathrm {P} )\delta \,dy+(\mathrm {Q} )\delta \,d^{2}y+\ldots$

unde

$\delta \pi =\int (\mathrm {N} )\delta y+\int (\mathrm {P} )\delta \,dy+\ldots$