![{\displaystyle =l^{(n+1)}L^{(n+1)}+l^{(n+2)}L^{(n+2)}+l^{(n+3)}L^{(n+3)}+{\text{etc. }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/62ef3106a51a922655f333430fd3e06a94f39744)
si generaliter per
exprimimus correctionem valoris approximati integralis
Hae correctiones
cum correctionibus
nexae erunt per aequationem
![{\displaystyle l^{(m)}=k^{(m)}-{\frac {1}{2}}mk^{(m-1)}+{\frac {1}{4}}\cdot {\frac {m.m-1}{1.2}}k^{(m-2)}-{\frac {1}{8}}\cdot {\frac {m.m-1.m-2}{1.2.3}}k^{(m-3)}+{\text{etc. }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/943c3ce69305627119f29228b25c23fe64191ce9)
Quo vero illas independenter eruere possimus, perpendamus, functionem
per substitutionem
transire in
![{\displaystyle {\begin{aligned}&2k(u^{-1}-u^{-2}+u^{-3}-u^{-4}+{\text{etc.}})\\+&4k^{\prime }(u^{-2}-2u^{-3}+3u^{-4}-4u^{-5}+{\text{etc.}})\\+&8k^{\prime \prime }(u^{-3}-3u^{-4}+6u^{-5}-10u^{-6}+{\text{etc.}})\\+&16k^{\prime \prime \prime }(u^{-4}-4u^{-5}+10u^{-6}-20u^{-7}+{\text{etc.}})\\+&{\text{ etc. }}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b060853c56282eb5041ad7d5f6f5ffbbbdb34a82)
sive in
![{\displaystyle {\begin{alignedat}{1}2ku^{-1}&+4(k^{\prime }-{\frac {1}{2}})u^{-2}+8(k^{\prime \prime }-{\frac {1}{2}}.2k^{\prime }+{\frac {1}{4}}k)u^{-3}\\&+16(k^{\prime \prime \prime }-{\frac {1}{2}}.3k^{\prime \prime }+{\frac {1}{4}}.3k^{\prime }-{\frac {1}{8}}k)u^{-4}+{\text{etc. }}\end{alignedat}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/918ebd517f9c6127b6b3d8dc4f43c46f13d985cb)
sive in
![{\displaystyle 2lu^{-1}+4l^{\prime }u^{-2}+8l^{\prime \prime }u^{-3}+16l^{\prime \prime \prime }u^{-4}+{\text{etc. }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0752d877e9c49a9eb8506b1c166e685305ebb896)
sive denique, quum a priori sciamus,
etc. usque ad
sponte evanescere, in
![{\displaystyle 2^{n+2}l^{(n+1)}u^{-(n+2)}+2^{n+3}l^{(n+2)}u^{-(n+3)}+2^{n+4}l^{(n+4)}u^{-(n+4)}+{\text{etc. }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4a2ece1f275e5f460e89082ffaf027f8b56697f)
At
quare quum
per substitutionem
transeant in
(art. 9), functio
per eandem substitutionem transibit in
Quodsi itaque seriem ex evolutione fractionis
oriundam per
designamus, erit
![{\displaystyle \Omega =2^{n+1}l^{(n+1)}u^{-(n+2)}+2^{n+2}l^{(n+2)}u^{-(n+3)}+2^{n+3}l^{(n+3)}u^{-(n+4)}+{\text{etc. }}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e302b5b369d3bead29b23e2133140194c28c1eee)
quo pacto coëfficientes
etc. quousque lubet erui poterunt.
Ita in exemplo art. 10 invenimus
![{\displaystyle {\begin{aligned}&U^{\prime \prime }=-{\frac {176}{13125}}u^{-1}-{\frac {304}{28125}}u^{-3}-{\frac {2576}{309375}}u^{-5}-{\text{etc. }}\\&\Omega =-{\frac {176}{13125}}u^{-7}-{\frac {832}{28125}}u^{-9}-{\frac {189856}{4296875}}u^{-11}-{\text{etc. }}\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db521d3649bb4220fd2b524bbeb54feb1b01631c)