Page:Annales de mathématiques pures et appliquées, 1828-1829, Tome 19.djvu/369

$g\operatorname {Cos} .\alpha \operatorname {Sin} \left(\theta -{\frac {2\varpi \operatorname {Sin} .\alpha }{T}}\right)\,;$

on aura donc pour l’équation différentielle du mouvement du centre de la sphère, dans le canal supposé immobile,

$r{\frac {\operatorname {d} ^{2}\theta }{\operatorname {d} t^{2}}}=g\operatorname {Cos} .\alpha \operatorname {Sin} \left(\theta -{\frac {2\varpi \operatorname {Sin} .\alpha }{T}}\right)\,;$

équation qui, en posant, pour abréger

$g=mr,\qquad \qquad \qquad$(1)

deviendra

${\frac {\operatorname {d} ^{2}\theta }{\operatorname {d} t^{2}}}=m\operatorname {Cos} .\alpha \operatorname {Sin} \left(\theta -{\frac {2\varpi \operatorname {Sin} .\alpha }{T}}\right).\qquad$(2)

En multipliant les deux membres de cette équation par

$2\left({\frac {\operatorname {d} \theta }{\operatorname {d} t}}-{\frac {2\varpi \operatorname {Sin} .\alpha }{T}}\right)\quad$ou$\quad 2.{\frac {\operatorname {d} .}{\operatorname {d} t}}\left(\theta -{\frac {2\varpi \operatorname {Sin} .\alpha }{T}}\right),$

elle deviendra

$2\left({\frac {\operatorname {d} \theta }{\operatorname {d} t}}-{\frac {2\varpi \operatorname {Sin} .\alpha }{T}}\right){\frac {\operatorname {d} ^{2}\theta }{\operatorname {d} t^{2}}}=$

$2m\operatorname {Cos} .\alpha .{\frac {\operatorname {d} .}{\operatorname {d} t}}\left(\theta -{\frac {2\varpi \operatorname {Sin} .\alpha }{T}}\right)\operatorname {Sin} .\left(\theta -{\frac {2\varpi \operatorname {Sin} .\alpha }{T}}\right)\,;$

ou bien

$\operatorname {d} .\left({\frac {\operatorname {d} \theta }{\operatorname {d} t}}-{\frac {2\varpi \operatorname {Sin} .\alpha }{T}}\right)^{2}=-2m\operatorname {Cos} .\alpha .\operatorname {d} .\operatorname {Cos} .\left(\theta -{\frac {2\varpi \operatorname {Sin} .\alpha }{T}}\right)\,;$

d’où, en intégrant

$\left({\frac {\operatorname {d} \theta }{\operatorname {d} t}}-{\frac {2\varpi \operatorname {Sin} .\alpha }{T}}\right)^{2}=A-2m\operatorname {Cos} .\alpha \operatorname {Cos} .\left(\theta -{\frac {2\varpi \operatorname {Sin} .\alpha }{T}}\right),$

$A$ étant la constante arbitraire.