132
CHAPITRE III.
et comme on a, d’autre part,
![{\displaystyle {\frac {d^{2}\eta _{1}}{dx^{2}}}=\varphi _{0},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/820886701216cc6113a00a29914a8fe3d574d09e)
on en conclura
![{\displaystyle \eta _{1}\ll \eta '_{1}\quad (\arg e^{\pm ix}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9d3c2721bfb8f1fc327a9a9f98a650893646d8b)
On trouve ensuite
![{\displaystyle z_{1}=\lambda f'_{1}\eta '_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1db17d64d22c7d9e4c213b1a60f3cbeb8048055b)
et, d’autre part,
![{\displaystyle [y_{1}]={\frac {-1}{[f_{1}]}}[f_{1}\eta _{1}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4eda907fed20f5e335ae0ac6358df07676c3c09e)
d’où
![{\displaystyle {\begin{aligned}\left[y_{1}\right]&\ll z_{1},\\y_{1}&\ll y'_{1}.\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e4cbb0295cac494444067828e4ff5a0937d13c18)
Il vient ensuite
![{\displaystyle \eta '_{2}=\varphi '_{1}(x,\,y'_{1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5518de2561aa1d20a0ffad54da6019e64379409b)
et, d’autre part,
![{\displaystyle {\frac {d^{2}\eta _{2}}{dx^{2}}}=\varphi _{1}(x,\,y_{1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7854a0557c008ba22cbe275e019eba9736e52a8f)
d’où
![{\displaystyle \eta _{2}\ll \eta '_{2}\,;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/70bd9e2b613d20dbc7cf07a4ee4e8e1a9370d56a)
puis
![{\displaystyle z_{2}=\lambda f_{1}\eta '_{2}+\lambda \theta '_{2}(x,\,y'_{1})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/efdd6ee243849ae0cf5b0c01dba2b8fff7852e4b)
et, d’autre part,
![{\displaystyle [y_{2}]={\frac {-1}{[f_{1}]}}[f_{2}\eta _{2}]-{\frac {1}{[f_{1}]}}[\theta _{2}],}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9f470d92db73f38a9c61dfd573b77e01f84f5e19)
d’où
![{\displaystyle [y_{2}]\ll z_{2},\quad y_{2}\ll y'_{2},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/61be8db3a71a64a040d4e54146e429011d23ad5e)
et ainsi de suite ; la loi est manifeste, on aura
![{\displaystyle y_{n}\ll y'_{n}\quad (\arg e^{\pm ix})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/87efec65db8d8f306299e7cb2f3f023c0fa393e3)
et
![{\displaystyle y\ll y'\quad (\arg \mu ,\,e^{\pm ix}).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e0e28849a8b316aa9e71126b28957bd27e4e021)
Si donc la série
![{\displaystyle y'=\mu y'_{1}+\mu ^{2}y'_{2}+\mu ^{3}y'_{3}+\dots }](https://wikimedia.org/api/rest_v1/media/math/render/svg/38e4a895a87c0e70d7ed2e93591b5c8382adf060)
converge, la série
(4)
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convergera a fortiori. Il me reste donc à établir que la série ![{\displaystyle y'}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3a535de94a2183d7130731eab8a83531d7c35c6b)