110
QUESTIONS
![{\displaystyle \operatorname {d} x{\frac {\operatorname {d} ^{2}x}{\operatorname {d} t^{2}}}=-g'r\operatorname {d} \phi \operatorname {Cos} .\phi \operatorname {Sin} .^{2}\phi .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/067da39591572047bded8472092e01fadf4613a1)
La première donne
![{\displaystyle {\tfrac {1}{2}}\left({\frac {\operatorname {d} y}{\operatorname {d} t}}\right)^{2}=k-{\frac {g'r}{3}}\left(2+\operatorname {Cos} .^{2}\phi \right)\operatorname {Sin} .\phi \,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2b82886a272c5454c0a7a45e1e0f96d73c3f671)
et on tire de la seconde
![{\displaystyle {\tfrac {1}{2}}\left({\frac {\operatorname {d} x}{\operatorname {d} t}}\right)^{2}=k'-{\frac {g'r}{3}}\operatorname {Sin} .^{3}\phi \,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5bbb1f6f80c8e5130dd827176824c045d02c217a)
ce qui s’accorde avec le principe des forces vives
![{\displaystyle {\tfrac {1}{2}}\left({\frac {\operatorname {d} s}{\operatorname {d} t}}\right)^{2}=C-2g'y.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6e42cfbfda03df3ae95c6c882ecdfc6d41b4c7be)
Pour avoir égard au frottement, il suffit de rapporter le mouvement aux axes
Conservons les deux lettres
pour les deux axes
et prenons
pour
en posant, pour abréger,
![{\displaystyle gm.{\frac {b}{a+b}}=g'',}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a73491449ca7e0f8bf1311434c4a3b8df4ab8305)
nous aurons
![{\displaystyle \left({\frac {\operatorname {d} ^{2}u}{\operatorname {d} t^{2}}}-g'\operatorname {Cos} .\beta \right)\operatorname {Cos} .\phi '-\left({\frac {\operatorname {d} ^{2}z}{\operatorname {d} t^{2}}}+g''\operatorname {Sin} .\beta \right)\operatorname {Sin} .\phi '=0\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4220c034d96c4381665a1555cb531afeceaf2e4a)
mais
![{\displaystyle u={\frac {y}{\operatorname {Cos} .\beta }},\qquad z={\frac {x}{\operatorname {Cos} .\beta }}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/697a848742924356f1860ffb85d2c316495e5c9a)
donc, l’équation, ramenée aux premiers axes, deviendra