mettant la lettre
au lieu de
sous les signes
, et faisant
![{\displaystyle \int _{a}^{b}zf_{n}zdz=k_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8b1a127a866ade12ae41b92da36580dd96cdb8e1)
,
![{\displaystyle \int _{a}^{b}z^{2}f_{n}zdz=k'_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2eefb0429c4d42e782006c87e6f055d03096ff77)
,
![{\displaystyle \int _{a}^{b}z^{3}f_{n}zdz=k''_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/40f3122bfae424d1491a1d1e63af5cda5efe8765)
, etc.,
nous aurons, en séries convergentes,
![{\displaystyle {\begin{alignedat}{2}&\rho _{n}\cos {r_{n}}&{}={}&1-{\frac {x^{2}}{1\,{.}\,2}}k'_{n}+{\frac {x^{4}}{1\,{.}\,2\,{.}\,3\,{.}\,4}}k'''_{n}-{\text{etc.}},\\&\rho _{n}\sin {r_{n}}&{}={}&xk_{n}-{\frac {x^{3}}{1\,{.}\,2\,{.}\,3}}k''_{n}+{\text{etc.}}\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5932b9aaf4a143fed694e7074e78a67ffd88857d)
En faisant aussi
![{\displaystyle {\frac {1}{2}}(k'_{n}-k_{n}^{2})=h_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ab3eb112e37760cb845b758be8353a4814b38b4)
,
![{\displaystyle {\frac {1}{6}}(k''_{n}-3k_{n}k'_{n}+2k_{n}^{3})=g_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f8a25f714302780d01023fe9c215697295e83840)
, etc.,
on déduira de ces séries
![{\displaystyle {\begin{aligned}\rho _{n}&=1-x^{2}h_{n}+x^{4}l_{n}-{\text{etc.}},\\r_{n}&=xk_{n}-x^{3}g_{n}+{\text{etc.}}\;;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/73636df77221ce881bc152fd97d3df6d58440d2d)
et de cette valeur de
, on déduira ensuite
![{\displaystyle \log {\rho _{n}}=-x^{2}h_{n}+x^{4}(l_{n}-{\tfrac {1}{2}}h_{n}^{2})-{\text{etc.}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/326053d5ca67644f71a20aef63571145729722bb)
Faisons encore
![{\displaystyle \textstyle \sum k_{n}=\mu k}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b867a6cfc1d1ed030e7c22300d2ebb7aab50047a)
,
![{\displaystyle \textstyle \sum h_{n}=\mu h}](https://wikimedia.org/api/rest_v1/media/math/render/svg/664b4b622fbbf3b9b071c90c07cdf24562b8e270)
,
![{\displaystyle \textstyle \sum g_{n}=\mu g}](https://wikimedia.org/api/rest_v1/media/math/render/svg/426b299000e746b047518db2b4136689eb4072c6)
,
![{\displaystyle \textstyle \sum (l_{n}-{\frac {1}{2}}h_{n}^{2})=\mu l}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fbe861c8925dce847316aa3d58d6698597f7ffbf)
, etc. ;
les sommes
s’étendant, ici et dans tout ce qui va suivre, depuis
jusqu’à
; nous aurons
![{\displaystyle \log {\mathrm {Y} }=-\theta ^{2}=-x^{2}\mu h+x^{4}\mu l-{\text{etc.}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/46381f931991c6857cecaa80b86f18b2990065b2)
;
d’où l’on tire
![{\displaystyle {\begin{aligned}x&={\frac {\theta }{\sqrt {\mu h}}}+{\frac {l\theta ^{3}}{2\mu h^{2}{\sqrt {\mu h}}}}+{\text{etc.}},\\{\frac {dx}{x}}&={\frac {d\theta }{\theta }}+{\frac {l\theta d\theta }{\mu h^{2}}}+{\text{etc.}}\;;\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/975fd307391d9b68a14be045e6ae535a3e4c93a5)