Pagina:Werkecarlf03gausrich.djvu/196

$$=l^{(n+1)}L^{(n+1)}+l^{(n+2)}L^{(n+2)}+l^{(n+3)}L^{(n+3)}+{\text{etc. }}$$

si generaliter per ${\textstyle l^{(m)}}$ exprimimus correctionem valoris approximati integralis ${\textstyle \int (t-{\frac {1}{2}})^{m}\operatorname {d} t.}$ Hae correctiones ${\textstyle l^{(m)}}$ cum correctionibus ${\textstyle k^{(m)}}$ nexae erunt per aequationem

$$l^{(m)}=k^{(m)}-{\frac {1}{2}}mk^{(m-1)}+{\frac {1}{4}}\cdot {\frac {m.m-1}{1.2}}k^{(m-2)}-{\frac {1}{8}}\cdot {\frac {m.m-1.m-2}{1.2.3}}k^{(m-3)}+{\text{etc. }}$$

Quo vero illas independenter eruere possimus, perpendamus, functionem ${\textstyle \Theta }$ per substitutionem ${\textstyle t={\frac {1}{2}}u+{\frac {1}{2}}}$ transire in

$${\begin{aligned}&2k(u^{-1}-u^{-2}+u^{-3}-u^{-4}+{\text{etc.}})\\+&4k^{\prime }(u^{-2}-2u^{-3}+3u^{-4}-4u^{-5}+{\text{etc.}})\\+&8k^{\prime \prime }(u^{-3}-3u^{-4}+6u^{-5}-10u^{-6}+{\text{etc.}})\\+&16k^{\prime \prime \prime }(u^{-4}-4u^{-5}+10u^{-6}-20u^{-7}+{\text{etc.}})\\+&{\text{ etc. }}\end{aligned}}$$

sive in

$${\begin{alignedat}{1}2ku^{-1}&+4(k^{\prime }-{\frac {1}{2}})u^{-2}+8(k^{\prime \prime }-{\frac {1}{2}}.2k^{\prime }+{\frac {1}{4}}k)u^{-3}\\&+16(k^{\prime \prime \prime }-{\frac {1}{2}}.3k^{\prime \prime }+{\frac {1}{4}}.3k^{\prime }-{\frac {1}{8}}k)u^{-4}+{\text{etc. }}\end{alignedat}}$$

sive in

$$2lu^{-1}+4l^{\prime }u^{-2}+8l^{\prime \prime }u^{-3}+16l^{\prime \prime \prime }u^{-4}+{\text{etc. }}$$

sive denique, quum a priori sciamus, ${\textstyle l,}$ ${\textstyle l^{\prime },}$ ${\textstyle l^{\prime \prime },}$ ${\textstyle l^{\prime \prime \prime }}$ etc. usque ad ${\textstyle l^{(n)}}$ sponte evanescere, in

$$2^{n+2}l^{(n+1)}u^{-(n+2)}+2^{n+3}l^{(n+2)}u^{-(n+3)}+2^{n+4}l^{(n+4)}u^{-(n+4)}+{\text{etc. }}$$

At ${\textstyle \Theta ={\frac {T^{\prime \prime }}{T}};}$ quare quum ${\textstyle T,}$ ${\textstyle T^{\prime \prime }}$ per substitutionem ${\textstyle t={\frac {1}{2}}u+{\frac {1}{2}}}$ transeant in ${\textstyle {\frac {U}{2^{n+1}}},}$ ${\textstyle {\frac {U^{\prime \prime }}{2^{n}}},}$ (art. 9), functio ${\textstyle \Theta }$ per eandem substitutionem transibit in ${\textstyle {\frac {2U^{\prime \prime }}{U}}.}$ Quodsi itaque seriem ex evolutione fractionis ${\textstyle {\frac {U^{\prime \prime }}{U}}}$ oriundam per ${\textstyle \Omega }$ designamus, erit

$$\Omega =2^{n+1}l^{(n+1)}u^{-(n+2)}+2^{n+2}l^{(n+2)}u^{-(n+3)}+2^{n+3}l^{(n+3)}u^{-(n+4)}+{\text{etc. }}$$

quo pacto coëfficientes ${\textstyle l^{(n+1)},}$ ${\textstyle l^{(n+2)}}$ etc. quousque lubet erui poterunt.

Ita in exemplo art. 10 invenimus

$${\begin{aligned}&U^{\prime \prime }=-{\frac {176}{13125}}u^{-1}-{\frac {304}{28125}}u^{-3}-{\frac {2576}{309375}}u^{-5}-{\text{etc. }}\\&\Omega =-{\frac {176}{13125}}u^{-7}-{\frac {832}{28125}}u^{-9}-{\frac {189856}{4296875}}u^{-11}-{\text{etc. }}\end{aligned}}$$