111
RÉSOLUES.
![{\displaystyle \left({\frac {\operatorname {d} ^{2}y}{\operatorname {d} t^{2}}}-g'\operatorname {Cos} .^{2}\beta \right)\operatorname {Cos} .(\phi +\beta )-\left({\frac {\operatorname {d} ^{2}x}{\operatorname {d} t^{2}}}+g''\operatorname {Sin} .\beta \operatorname {Cos} .\beta \right)\operatorname {Sin} .(\phi +\beta )=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5f3fa2e97f0425bc5bc41f8da3b3c34de3e92664)
faisant, pour abréger,
![{\displaystyle g'\operatorname {Cos} .^{2}\beta =\xi ,\qquad g''\operatorname {Sin} .\beta \operatorname {Cos} .\beta =\xi ',}](https://wikimedia.org/api/rest_v1/media/math/render/svg/df21a9ddd759cd19306e28ad27d73f7667a3ddf1)
et conservant l’angle
on aura
![{\displaystyle \left({\frac {\operatorname {d} ^{2}y}{\operatorname {d} t^{2}}}-\xi \right)\operatorname {Cos} .\phi '-\left({\frac {\operatorname {d} ^{2}x}{\operatorname {d} t^{2}}}-\xi '\right)\operatorname {Sin} .\phi =0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/84d077f9fec70361e988574ab7a79af9d7f6157c)
Cette équation a la même forme que l’équation, (A) déjà traitée :
elle donnera
![{\displaystyle {\begin{aligned}&{\frac {\operatorname {d} ^{2}y}{\operatorname {d} t^{2}}}=\xi \operatorname {Cos} .^{2}\phi '-\xi '\operatorname {Sin} .\phi '\operatorname {Cos} .\phi ',\\\\&{\frac {\operatorname {d} ^{2}x}{\operatorname {d} t^{2}}}=\xi '\operatorname {Sin} .^{2}\phi '-\xi \operatorname {Sin} .\phi '\operatorname {Cos} .\phi '.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5d079922640e5b1abdb08782ee766f2e3bf71693)
On déduit de la première
![{\displaystyle {\tfrac {1}{2}}\left({\frac {\operatorname {d} y}{\operatorname {d} t}}\right)^{2}=k-{\frac {\xi }{3}}\left(2+\operatorname {Cos} .^{2}\phi '\right)\operatorname {Sin} .\phi '+{\frac {\xi '}{3}}\operatorname {Sin} .^{3}\phi ',}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf79dbeba7d0e926e1dc644d68faf6794cc77f9)
et de la seconde
![{\displaystyle {\tfrac {1}{2}}\left({\frac {\operatorname {d} x}{\operatorname {d} t}}\right)^{2}=k'+{\frac {\xi }{3}}\operatorname {Sin} .^{3}\phi '+{\frac {\xi '}{3}}\left(2-\operatorname {Sin} .^{2}\phi '\right)\operatorname {Cos} .\phi '.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db1a4ceed6aa59126fbdf7b809d53add9aefa559)
En faisant
ces deux dernières formules deviennent celles qu’on
a déjà trouvées ci-dessus ; lorsqu’on n’a pas égard au frottement.