ensuite l’équation (K) étant multipliée par
et ensuite intégrée en regardant
comme constante, donnera, à cause de
![{\displaystyle \mathrm {\frac {P^{2}X}{T^{2}}} +\int {\frac {d\left(\mathrm {\cfrac {Q^{2}Y}{T^{2}}} \right)}{dy}}\mathrm {\frac {P}{Q}} dx=2\Phi (\mathrm {Z} )+\Pi (y)\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/522a7686c53c94cc7e41dc6e7cb45fe28e39ea34)
mais
![{\displaystyle {\frac {d\left(\mathrm {\cfrac {Q^{2}Y}{T^{2}}} \right)}{dy}}={\frac {1}{\mathrm {T} ^{2}}}{\frac {d\mathrm {Q^{2}Y} }{dy}}-2\mathrm {\frac {Q^{2}Y}{T^{3}}} {\frac {d\mathrm {T} }{dy}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/007b875bb1c6b00dc77da89c426250408112de42)
donc on aura
![{\displaystyle {\begin{aligned}\int {\frac {d\left(\mathrm {\cfrac {Q^{2}Y}{T^{2}}} \right)}{dy}}\mathrm {\frac {P}{Q}} dx=&{\frac {1}{\mathrm {Q} }}{\frac {d\mathrm {Q^{2}Y} }{dy}}\int {\frac {\mathrm {P} dx}{\mathrm {T} ^{2}}}-2\mathrm {QY} \int \mathrm {\frac {P}{T^{2}}} {\frac {d\mathrm {T} }{dy}}dx\\=&{\frac {1}{\mathrm {Q} }}{\frac {d\mathrm {\left(Q^{2}Y\int {\cfrac {\mathrm {P} dx}{\mathrm {T} ^{2}}}\right)} }{dy}}\,;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c1a101f3b7f2ddfe11673f301276e89950a81668)
de sorte que l’équation précédente deviendra
![{\displaystyle \mathrm {\frac {P^{2}X}{T^{2}}} +{\frac {1}{\mathrm {Q} }}{\frac {d\mathrm {\left(Q^{2}Y\int {\cfrac {\mathrm {P} dx}{\mathrm {T} ^{2}}}\right)} }{dy}}=2\Phi (\mathrm {Z} )+\Pi (y),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/64815582bce9e0b9c4c6e5712b4f05651c7f9d61)
laquelle étant multipliée par
et intégrée derechef en regardant
comme constante, donnera, à cause de ![{\displaystyle d\mathrm {Z} =\mathrm {Q} dy,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b84dbbcdce7cb74930dcf1905e4c607957e2936f)
![{\displaystyle \mathrm {P^{2}X} \int {\frac {\mathrm {Q} dy}{\mathrm {T} ^{2}}}+\mathrm {Q^{2}Y} \int {\frac {\mathrm {P} dx}{\mathrm {T} ^{2}}}=2\int \Phi (\mathrm {Z} )d\mathrm {Z} +\Gamma (x)+\Delta (y),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de4c68795bd67fafa321f177b9e2f98ffbc0c278)
et
étant des fonctions quelconques de
et de ![{\displaystyle y.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83f72471aff7c6fbb27df0f971283a068efe091f)
Or, puisque
si l’on suppose en général
![{\displaystyle \Sigma (p)=-\int {\frac {dp}{\left[\Psi (p)\right]^{2}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60afd444b3c8831de612b5fa33ada86d05c0687f)
on aura
![{\displaystyle \int {\frac {\mathrm {P} dx}{\mathrm {T} ^{2}}}=-\Sigma \left(\int \mathrm {P} dx-\int \mathrm {Q} dy\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e06bd5dae35be2154ffeb46088e66cddc7cf2acc)
et
![{\displaystyle \int {\frac {\mathrm {Q} dy}{\mathrm {T} ^{2}}}=\Sigma \left(\int \mathrm {P} dx-\int \mathrm {Q} dy\right)\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/467e7abdb5af5f35fffa01cac07150cad9b7e264)