234
PUISSANCES
![{\displaystyle P_{{\frac {1}{4}}(k-2)}=\pm \left\{\operatorname {Sin} .mx\operatorname {Cos} .m\varpi \pm \operatorname {Cos} .mx\operatorname {Sin} .m\varpi \right.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6a3689d3db60c60596cac3b9a25dd204e638795)
![{\displaystyle +{\frac {m}{1}}\left[\operatorname {Sin} .(m-2)x\operatorname {Cos} .(m-2)\varpi \pm \operatorname {Cos} .(m-2)x\operatorname {Sin} .(m-2)\varpi \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3ea46abe26b54b8a79276e15b1387c19f73bdedb)
![{\displaystyle +{\frac {m}{1}}.{\frac {m-1}{2}}\left[\operatorname {Sin} .(m-4)x\operatorname {Cos} .(m-4)\varpi \pm \operatorname {Cos} .(m-4)x\operatorname {Sin} .(m-4)\varpi \right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e2a66dbeec87e71a9161fe87daed21f8f18432e)
![{\displaystyle \left.+\ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \ldots \right\}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cb2679115eecaebb606fd4a464f40a64cc472aa5)
or,
![{\displaystyle \operatorname {Cos} .m\varpi =\operatorname {Cos} .(m-2)\varpi =\operatorname {Cos} .(m-4)\varpi =\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/dc4f5f2025d0b0f2dda755193fe4dd38684908cc)
![{\displaystyle \operatorname {Sin} .m\varpi \ =\operatorname {Sin} .(m-2)\varpi \ =\operatorname {Sin} .(m-4)\varpi \ =\ldots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/33cc20e89f91e45ee8c74753a3005155a2ae37e5)
donc, en ayant égard à ces relations
![{\displaystyle P_{{\frac {1}{4}}(k-2)}=}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4a9c35cadef0344113426b06029f1cd7224f075)
![{\displaystyle {\begin{array}{l}\pm \operatorname {Cos} .m\varpi \left\{\operatorname {Sin} .mx+{\frac {m}{1}}\operatorname {Sin} .(m-2)x+{\frac {m}{1}}.{\frac {m-1}{2}}\operatorname {Sin} .(m-4)x+\ldots \right\}\\\\\mp \operatorname {Sin} .m\varpi \left\{\operatorname {Cos} .mx+{\frac {m}{1}}\operatorname {Cos} .(m-2)x+{\frac {m}{1}}.{\frac {m-1}{2}}\operatorname {Cos} .(m-4)x+\ldots \right\}\,;\end{array}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/82b85572badddc6b22446097516f4d789883c99f)
c’est-à-dire,
![{\displaystyle P_{{\frac {1}{4}}(k-2)}=\pm (P_{0}\operatorname {Sin} .m\varpi \pm Q_{0}\operatorname {Cos} .m\pi ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/76e93424fc2ce4303754ca5478b06ff747f33144)
Donc, si
est un multiple de
et que
soit négatif, on aura toujours la relation :
![{\displaystyle P_{0}\operatorname {Sin} .m\varpi \pm Q_{0}\operatorname {Cos} .m\pi =Q_{\frac {1}{2}}=0.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c11e106cb7737d4faa9c401fa2224644eb34845e)
2.o Les valeurs entièrement réelles de
qui existent (16) dans le cas où
est impair,
étant toujours négatif, se