Page:Annales de mathématiques pures et appliquées, 1828-1829, Tome 19.djvu/371

${\frac {\operatorname {d} \omega }{\operatorname {d} t}}={\sqrt {4\varpi ^{2}\left({\frac {1}{\tau }}-{\frac {\operatorname {Sin} .\alpha }{T}}\right)^{2}+2m\operatorname {Cos} .\alpha \operatorname {Cos} .\beta -2m\operatorname {Cos} .\alpha \operatorname {Cos} .\omega }}.$ (5)

Posons encore

$x={\sqrt {4\varpi ^{2}\left({\frac {1}{\tau }}-{\frac {\operatorname {Sin} .\alpha }{T}}\right)^{2}+2m\operatorname {Cos} .\alpha \operatorname {Cos} .\beta -2m\operatorname {Cos} .\alpha \operatorname {Cos} .\omega }}.$ (6)

d’où, en différentiant,

$\operatorname {d} x={\frac {m\operatorname {Cos} .\alpha .\operatorname {d} \omega .\operatorname {Sin} .\omega }{\sqrt {4\varpi ^{2}\left({\frac {1}{\tau }}-{\frac {\operatorname {Sin} .\alpha }{T}}\right)^{2}+2m\operatorname {Cos} .\alpha \operatorname {Cos} .\beta -2m\operatorname {Cos} .\alpha \operatorname {Cos} .\omega }}}\,;$

multipliant cette équation par l’équation (5), on en conclura

${\frac {\operatorname {d} x}{\operatorname {d} t}}=m\operatorname {Cos} .\alpha \operatorname {Sin} .\omega \,;\qquad$(7)

mais de l’équation (6) on tire, en quarrant et transposant,

$\operatorname {Cos} .\omega ={\frac {4\varpi ^{2}\left({\frac {1}{\tau }}-{\frac {\operatorname {Sin} .\alpha }{T}}\right)^{2}+2m\operatorname {Cos} .\alpha \operatorname {Cos} .\beta -x^{2}}{2m\operatorname {Cos} .\alpha }},$

d’où

$\operatorname {Sin} .\omega ={\frac {\sqrt {4m^{2}\operatorname {Cos} .^{2}\alpha -\left\{4\varpi ^{2}\left({\frac {1}{\tau }}-{\frac {\operatorname {Sin} .\alpha }{T}}\right)^{2}+2m\operatorname {Cos} .\alpha \operatorname {Cos} .\beta -x^{2}\right\}^{2}}}{2m\operatorname {Cos} .\alpha }},$

remarquant alors que la quantité sous le radical se décompose en deux facteurs, et posant, pour abréger,

$\left.{\begin{aligned}&G=2m(1+\operatorname {Cos} .\beta )\operatorname {Cos} .\alpha +4\varpi ^{2}\left({\frac {1}{\tau }}-{\frac {\operatorname {Sin} .\alpha }{T}}\right)^{2},\\\\&H=2m(1-\operatorname {Cos} .\beta )\operatorname {Cos} .\alpha -4\varpi ^{2}\left({\frac {1}{\tau }}-{\frac {\operatorname {Sin} .\alpha }{T}}\right)^{2},\end{aligned}}\right\}\quad$(8)

on aura