adeoque
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d9579c698cdc57df603ec7318de59742d73337a6)
Sit
posito
et
erit per § IV
![{\displaystyle \delta \int \mathrm {Z} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4f7f735bef27077704927326463ea0c5199c136)
(posito
![{\displaystyle x=a}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aaae23950e96a955ab5b07015a168fd931d4d82b)
)
![{\displaystyle =\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 ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a2ecd3baaad74547599e7f4c3d8ff8bb43ba8bf)
unde, ut supra, erit pro maximo minimove,
![{\displaystyle \mathrm {N+(N)V} -d\mathrm {\left[P+(P)V\right]} +d^{2}\mathrm {\left[Q+(Q)V\right]} -\ldots =0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98e73f625c8be020ea31ae4451be807921ac045b)
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 3o
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b096c50d7aa326aa9a193454d7f114d94e5a1162)
seu, eliminando ![{\displaystyle (z),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3998217c1d6348b96b3cc8a3dbbaad9066a0eb6c)
![{\displaystyle d\,\delta \pi =(\mathrm {L} )\delta \pi +(\mathrm {N} )\delta y+(\mathrm {P} )d\,\delta y+\ldots \,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d6fc4e7112bf4bced22a3e856e55b67c95fbe8e5)
sit, brevitatis gratia,
![{\displaystyle (\mathrm {N} )\delta y+(\mathrm {P} )d\,\delta y+\ldots =\mathrm {V} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/55486bfa30f021ca0ad532f5738487ec6eb2bd67)
erit æqualis
![{\displaystyle d\,\delta \pi -(\mathrm {L} )\delta \pi =\mathrm {V} \,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/960d4c8fa2e4c29fdd5ff91d1d7515ff5aff2f2d)
unde per regulas cognitas integrando habebimus
![{\displaystyle \delta \pi =e^{\int (\mathrm {L} )}\int \mathrm {V} e^{-\int (\mathrm {L} )}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84b969e7d0f2942d8610331fff031dbb0d375f1a)
unde fiet
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c842acdf75c0bd4672d2dde3d95b7718a0df8d26)
quapropter, si
posito
abeat in
et
in
habebimus operando ut supra
![{\displaystyle \mathrm {N+(N)} e^{-\int (\mathrm {L} )}\mathrm {V} -d\left[\mathrm {P+(P)} e^{-\int (\mathrm {L} )}\mathrm {V} \right]+\ldots =0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18aaaab87e4f6f99c1ffb66c1e609f7be66b899c)