Substituant donc cette valeur de
dans l’équation différentielle
![{\displaystyle -{\frac {dy}{y}}={\frac {dx\log 10}{1+{\dfrac {t}{215}}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/959ef8b8418a217aa33a66283f8ae0388c3bd928)
du no 4, elle deviendra celle-ci
![{\displaystyle -{\frac {dy}{y}}={\frac {dx\log 10}{1+{\dfrac {p-qx}{215}}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fd5da2897aeff3af42fb6664c7440fd1316c5af)
dont l’intégrale est
![{\displaystyle \log {\frac {b}{y}}={\frac {215\log 10}{q}}\log {\frac {k}{1+{\dfrac {p-qx}{215}}}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2048bba61257f1752ad2ac84f84ee4829f166209)
étant une constante qu’il faut déterminer en sorte que lorsque
on ait
ce qui donnera
![{\displaystyle \log {\frac {b}{y}}={\frac {215\log 10}{q}}\log {\frac {1+{\dfrac {p-qa}{215}}}{1+{\dfrac {p-qx}{215}}}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/17037d9e04b38ab5e35f1348a3a7bb2202e473db)
et de là
![{\displaystyle {\frac {y}{b}}=\left({\frac {1+{\dfrac {p-qx}{215}}}{1+{\dfrac {p-qa}{215}}}}\right)^{\frac {215\log 10}{q}}=\left(1-{\frac {q}{215}}{\frac {x-a}{1+{\dfrac {c}{215}}}}\right)^{\frac {215\log 10}{q}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3193035b1be210c2a8e701b50f90b975365005c1)
d’où l’on tire
![{\displaystyle x-a=\left(1+{\dfrac {c}{215}}\right){\frac {1-\left({\dfrac {y}{b}}\right)^{\frac {q}{215\log 10}}}{\dfrac {q}{215}}}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b97b3224dcd9eeb2909c65688fdb38833b8055e)
Si la quantité
était infiniment petite, on aurait
![{\displaystyle 1-\left({\frac {y}{b}}\right)^{\frac {q}{215\log 10}}=-{\frac {q}{215\log 10}}\log {\frac {y}{b}}={\frac {q}{215}}\mathrm {L} {\frac {b}{y}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d9241a7aa8b8fdad1f20bbcd5854fd650899c1d)
donc
![{\displaystyle x-a=\left(1+{\frac {c}{215}}\right)\mathrm {L} {\frac {b}{y}},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5918d414fbaf83276a27e44da4a19934268eebbe)