33
DE LA CYCLOÏDE.
au moyen de quoi nous aurons
![{\displaystyle \left.{\begin{aligned}x+x'&=\varpi r,\\y+y'&=2r\,;\end{aligned}}\right\}{\text{ d’où }}\left\{{\begin{aligned}\operatorname {d} x+\operatorname {d} x'&=0,\\\operatorname {dy} +\operatorname {d} y'&=0,\end{aligned}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f206448202016937ef5bf282163d57129c25d688)
Nous poserons en outre
![{\displaystyle \operatorname {Ang} .\mathrm {DCM} =2z,\qquad \operatorname {Ang} .\mathrm {D'CM} =2z',}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0cc24d97eb79461d43d1c89efd7b238756e4f4a7)
ce qui donnera
![{\displaystyle 2(z+z')=\varpi ,\qquad }](https://wikimedia.org/api/rest_v1/media/math/render/svg/e53cb877be10a5faaecfae8f9cdf11b3425d416f)
d’où
![{\displaystyle \qquad \operatorname {d} z+\operatorname {d} z'=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d893a11bd56ce4eee729bed06d5612a74fa10575)
Cela posé, nous aurons
![{\displaystyle {\begin{array}{lllll}Arc.\mathrm {MD} &=2rz,&Cord.\mathrm {MD} &=2r\operatorname {Sin} .z&=2r\operatorname {Cos} .z',\\Arc.\mathrm {MD'} &=2rz',&Cord.\mathrm {MD'} &=2r\operatorname {Sin} .z'&=2r\operatorname {Cos} .z.\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/02692b87ddea640cf66e3d8cb53703a65b6b4384)
Nous aurons encore
![{\displaystyle \mathrm {PD=P'D'=MN} =r\operatorname {Sin} .2z=2r\operatorname {Sin} .z\operatorname {Cos} .z=2r\operatorname {Sin} .z'\operatorname {Cos} .z',}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e3625e6ffe277c0f01e970f14e77ea7fb0426b2)
![{\displaystyle \mathrm {CN} =r\operatorname {Cos} .2z=r\left(\operatorname {Cos} .^{2}z-\operatorname {Sin} .^{2}z\right)=r\left(\operatorname {Sin} .^{2}z'-\operatorname {Cos} .^{2}z'\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/60735075bec609b8fa2314c2f5754141f71f4618)
mais, par la nature de la cycloïde,
![{\displaystyle \mathrm {OP=OD-DP} =Arc.\mathrm {MD-MN} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e5214b822445a89a6c28fd80a0e275fd4fcf04f)
![{\displaystyle \mathrm {MP=ND=CD-CN} \,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ccec7c3086f2e7a18bb44e12deae70fd8f9e2ff7)
donc, en substituant,
![{\displaystyle \left.{\begin{aligned}x&=2r(z-\operatorname {Sin} .z\operatorname {Cos} .z),\\\\y&=2r\operatorname {Sin} .^{2}z\end{aligned}}\right\}{\text{ d’où }}\left\{{\begin{aligned}x'&=2r(z'+\operatorname {Sin} .z'\operatorname {Cos} .z'),\\\\y'&=2r\operatorname {Sin} .^{2}z'\,;\end{aligned}}\right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/881445da59247b4d910a9f0a2ecab4c1ec0aca47)
donc encore
![{\displaystyle {\begin{array}{ll}\operatorname {d} x=4r\operatorname {d} z\operatorname {Sin} .^{2}z,&\operatorname {d} x'=4r\operatorname {d} z'\operatorname {Cos} .^{2}z',\\\operatorname {d} y=4r\operatorname {d} z\operatorname {Sin} .z\operatorname {Cos} .z,&\operatorname {d} y'=4r\operatorname {d} z'\operatorname {Sin} .z'\operatorname {Cos} .z'.\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b460ff46a128a0a0282dd4d9621a4313f70ea3fc)
De là on passerait facilement aux équations primitive et différentielle de la courbe, soit en
et
soit en
et
mais elles ne nous seront pas nécessaires.
Pour la commodité typographique, nous poserons encore