![{\displaystyle {\frac {\operatorname {d} \omega }{\operatorname {d} t}}={\sqrt {4\varpi ^{2}\left({\frac {1}{\tau }}-{\frac {\operatorname {Sin} .\alpha }{T}}\right)^{2}+2m\operatorname {Cos} .\alpha \operatorname {Cos} .\beta -2m\operatorname {Cos} .\alpha \operatorname {Cos} .\omega }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/da3a7d05eac07fe001995fb4ee4ee87fc1bcbd1f)
(5)
Posons encore
![{\displaystyle x={\sqrt {4\varpi ^{2}\left({\frac {1}{\tau }}-{\frac {\operatorname {Sin} .\alpha }{T}}\right)^{2}+2m\operatorname {Cos} .\alpha \operatorname {Cos} .\beta -2m\operatorname {Cos} .\alpha \operatorname {Cos} .\omega }}.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/252bbca714b2636bad408861e769972746caa716)
(6)
d’où, en différentiant,
![{\displaystyle \operatorname {d} x={\frac {m\operatorname {Cos} .\alpha .\operatorname {d} \omega .\operatorname {Sin} .\omega }{\sqrt {4\varpi ^{2}\left({\frac {1}{\tau }}-{\frac {\operatorname {Sin} .\alpha }{T}}\right)^{2}+2m\operatorname {Cos} .\alpha \operatorname {Cos} .\beta -2m\operatorname {Cos} .\alpha \operatorname {Cos} .\omega }}}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7f296c6b8e413aaf27e92163ac8ec6d5d7b097b)
multipliant cette équation par l’équation (5), on en conclura
![{\displaystyle {\frac {\operatorname {d} x}{\operatorname {d} t}}=m\operatorname {Cos} .\alpha \operatorname {Sin} .\omega \,;\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/775a19e7a55790dd3387f1f7c73336ffabbaefb3)
(7)
mais de l’équation (6) on tire, en quarrant et transposant,
![{\displaystyle \operatorname {Cos} .\omega ={\frac {4\varpi ^{2}\left({\frac {1}{\tau }}-{\frac {\operatorname {Sin} .\alpha }{T}}\right)^{2}+2m\operatorname {Cos} .\alpha \operatorname {Cos} .\beta -x^{2}}{2m\operatorname {Cos} .\alpha }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec37481a14fc6e52dab5b45d4b5b9e310435f61f)
d’où
![{\displaystyle \operatorname {Sin} .\omega ={\frac {\sqrt {4m^{2}\operatorname {Cos} .^{2}\alpha -\left\{4\varpi ^{2}\left({\frac {1}{\tau }}-{\frac {\operatorname {Sin} .\alpha }{T}}\right)^{2}+2m\operatorname {Cos} .\alpha \operatorname {Cos} .\beta -x^{2}\right\}^{2}}}{2m\operatorname {Cos} .\alpha }},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/139c23fe3f836ffe72cfabf419ed420e8d4fb052)
remarquant alors que la quantité sous le radical se décompose en deux facteurs, et posant, pour abréger,
![{\displaystyle \left.{\begin{aligned}&G=2m(1+\operatorname {Cos} .\beta )\operatorname {Cos} .\alpha +4\varpi ^{2}\left({\frac {1}{\tau }}-{\frac {\operatorname {Sin} .\alpha }{T}}\right)^{2},\\\\&H=2m(1-\operatorname {Cos} .\beta )\operatorname {Cos} .\alpha -4\varpi ^{2}\left({\frac {1}{\tau }}-{\frac {\operatorname {Sin} .\alpha }{T}}\right)^{2},\end{aligned}}\right\}\quad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f8f80c26e702c59e8499773c2244efae99dfd84)
(8)
on aura