Problemata.
Invenire œquationem inter
et
ut pro dato ipsius
valore puto
formula hœc
maximum minimumve valorem obtineat.
Resolutiones.
Sit 1o
![{\displaystyle \delta \mathrm {Z} =\mathrm {N} \delta y+\mathrm {P} \delta \,dy+\mathrm {Q} \delta \,d^{2}y+\mathrm {R} \delta \,d^{3}y+\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/24766efa11288525881a882acdc2e76c339711d8)
(
enim in hac differentia ponitur constans § I). Igitur quoniam differentia duorum totorum sequalis est summae differentiarum omnium partium, adeoque
erit
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30917b7e52441d662bfecf72a60c56044fd6e720)
unde
![{\displaystyle \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 }](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e1ecc374f4b9b2da4735e568027433c474f65a2)
seu, ponendo
qui respondet
cum nonnullis sequentibus invariabilem, unde totidem habentur puncta per quæ curva invenienda transire debet, erit
unde tandem
![{\displaystyle \delta \int \mathrm {Z} =\int \left(\mathrm {N} -d\mathrm {P} +d^{2}\mathrm {Q} -d^{3}\mathrm {R} \ldots \right)\delta y\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4128080978ea950d5bcfc5a2774f1df3a6c09d7f)
adeoque ex methodo maximorum, et minimorum communi
![{\displaystyle \mathrm {N} -d\mathrm {P} +d^{2}\mathrm {Q} -d^{3}\mathrm {R} \ldots =0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fc14c1c3d4459de63597c6030c414295b9b4373)
dabit æquationem quæsitam (9).
Sit 2o
![{\displaystyle \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} )\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6360ddb4f7a750b8121f4f02fe0322da26427480)
et
![{\displaystyle \delta (\mathrm {Z} )=(\mathrm {N} )\delta y+(\mathrm {P} )\delta \,dy+(\mathrm {Q} )\delta \,d^{2}y+\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b6c4fe40a824788ed34823ac55d2bbe84a831af)
unde
![{\displaystyle \delta \pi =\int (\mathrm {N} )\delta y+\int (\mathrm {P} )\delta \,dy+\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/06585da45d0ac6a72c25bfc924d03ca853d5bd46)