Pagina:Werkecarlf03gausrich.djvu/194

$$z^{6}-{\frac {21}{13}}z^{4}+{\frac {105}{143}}zz-{\frac {35}{429}}=0$$

quo pacto facile praevidemus orturam esse functionem integram quarti ordinis. Iam ex applicatione praeceptorum praecedentium prodeunt sequentia:

$${\begin{aligned}&\zeta ^{\phantom {\prime \prime \prime \prime }}=7z^{6}-{\frac {105}{13}}z^{4}+{\frac {315}{143}}zz-{\frac {35}{429}}\\&\zeta ^{\prime {\phantom {\prime \prime \prime }}}=z^{6}-{\frac {21}{13}}z^{4}+{\frac {105}{143}}zz-{\frac {35}{429}}\\&\zeta ^{\prime \prime {\phantom {\prime \prime }}}=z^{4}-{\frac {10}{11}}zz+{\frac {5}{33}}\\&\zeta ^{\prime \prime \prime {\phantom {\prime }}}=zz-{\frac {3}{7}}\\&\zeta ^{\prime \prime \prime \prime }=1\\&{\begin{alignedat}{2}&\lambda ^{\phantom {\prime \prime \prime \prime }}={\frac {13}{42}}&&\qquad p={\frac {13}{6}}\\&\lambda ^{\prime {\phantom {\prime \prime \prime }}}=-{\frac {4719}{280}}&&\qquad p^{\prime }=-{\frac {4719}{280}}zz+{\frac {3333}{280}}\\&\lambda ^{\prime \prime {\phantom {\prime \prime }}}=-{\frac {147}{8}}&&\qquad p^{\prime \prime }=-{\frac {147}{8}}zz+{\frac {777}{88}}\end{alignedat}}\\&\eta ^{\phantom {\prime \prime \prime \prime }}=1\\&\eta ^{\prime {\phantom {\prime \prime \prime }}}=0\\&\eta ^{\prime \prime {\phantom {\prime \prime }}}={\frac {13}{42}}\\&\eta ^{\prime \prime \prime {\phantom {\prime }}}={\frac {20449}{3920}}zz-{\frac {14443}{3920}}\\&\eta ^{\prime \prime \prime \prime }={\frac {61347}{640}}z^{4}-{\frac {127413}{1120}}zz+{\frac {120263}{4480}}\\&{\begin{alignedat}{2}&Z^{\phantom {\prime \prime \prime \prime }}=z^{6}-{\frac {50}{39}}z^{4}+{\frac {283}{715}}zz-{\frac {256}{15015}};&&\qquad q^{\prime }=1\\&Z^{\prime {\phantom {\prime \prime \prime }}}={\frac {1}{3}}z^{4}-{\frac {22}{65}}zz+{\frac {323}{5005}}&&\qquad q^{\prime \prime }={\frac {1}{3}}\\&Z^{\prime \prime {\phantom {\prime \prime }}}=-{\frac {76}{2145}}zz+{\frac {632}{45045}}&&\qquad q^{\prime \prime \prime }=-{\frac {76}{2145}}\\&Z^{\prime \prime \prime {\phantom {\prime }}}=-{\frac {4}{3465}}&&\qquad q^{\prime \prime \prime \prime }=-{\frac {4}{3465}}\end{alignedat}}\end{aligned}}$$

Hinc tandem derivatur functio integra fractioni propositae aequivalens:

$$-{\frac {1859}{16800}}z^{4}-{\frac {1573}{29400}}zz+{\frac {7947}{39200}}$$

13.

Ad determinandum gradum praecisionis, qua formula nostra integralis ${\textstyle {R}A+{R}^{\prime }A^{\prime }+{R}^{\prime \prime }A^{\prime \prime }+{\text{etc.}}+{R}^{(n)}A^{(n)}}$ gaudet, statuamus generaliter

$${R}{a}^{m}+{R}^{\prime }{a}^{\prime m}+{R}^{\prime \prime }{a}^{\prime \prime m}+{\text{etc.}}+{R}^{(n)}a^{(n)m}={\frac {1}{m+1}}-k^{(m)}$$

ita ut ${\textstyle k^{(m)}}$ sit differentia inter integralis ${\textstyle \int t^{m}\operatorname {d} t}$ a ${\textstyle t=0}$ usque ad ${\textstyle t=1}$ sumti valorem verum atque approximatum. Habebimus itaque, singulis fractionibus in series evolutis,