Pagina:Werkecarlf03gausrich.djvu/207

I. Terminus unus, ${\textstyle n=0.}$

${\textstyle {\begin{aligned}U&=u,\;\;U^{\prime }=1,\;\;T=t-{\frac {1}{2}},\;\;T^{\prime }=1\\a&=0{,}5\\R&=1\end{aligned}}}$

Correctio formulae integralis proxime ${\textstyle ={\frac {1}{12}}L^{\prime \prime }.}$

II. Termini duo, ${\textstyle n=1.}$

${\textstyle {\begin{aligned}U{\phantom {{}^{\prime }}}&=uu-{\frac {1}{2}},\quad U^{\prime }=u\\T{\phantom {{}^{\prime }}}&=tt-t+{\frac {1}{6}},\quad T^{\prime }=t-{\frac {1}{2}}\\a{\phantom {{}^{\prime }}}&=0{,}2113248654\;\;051871\\a^{\prime }&=0{,}7886751345\;\;948129\\R{\phantom {{}^{\prime }}}&=R^{\prime }={\frac {1}{2}}\end{aligned}}}$

Correctio proxime ${\textstyle ={\frac {1}{180}}L^{\prime \prime \prime \prime }}$

III. Termini tres, ${\textstyle n=2.}$

${\textstyle {\begin{alignedat}{2}&U&&=u^{3}-{\frac {3}{5}}u,\;\;U^{\prime }=uu-{\frac {4}{15}}\\&T&&=t^{3}-{\frac {3}{2}}tt+{\frac {3}{5}}t-{\frac {1}{20}},\;\;T^{\prime }=tt-t+{\frac {11}{60}}\\&a&&=0{,}1127016653\;\;792583\\&a^{\prime }&&=0{,}5\\&a^{\prime \prime }&&=?{,}8872983346\;\;207417\\&R&&=R^{\prime \prime }={\frac {5}{18}}\\&R^{\prime }&&={\frac {4}{9}}\\\end{alignedat}}}$

Correctio proxime ${\textstyle ={\frac {1}{2800}}L^{\mathrm {vi} }.}$

IV. Termini quatuor, ${\textstyle n=3.}$

${\textstyle {\begin{array}{rl}U^{\phantom {\prime }}&=u^{4}-{\frac {6}{7}}uu+{\frac {3}{3^{3}}}\\U^{\prime }&=u^{3}-{\frac {11}{21}}u\\T^{\phantom {\prime }}&=t^{4}-2t^{3}+{\frac {9}{7}}tt-{\frac {2}{7}}t+{\frac {1}{70}}\\T^{\prime }&=t^{3}-{\frac {3}{2}}tt+{\frac {13}{2}}t-{\frac {5}{84}}\\&\;\;\quad a^{\phantom {\prime \prime \prime }}=0{,}0694318442\;\;029754\\&\;\;\quad a^{\prime {\phantom {\prime \prime }}}=0{,}3300094782\;\;075677\\&\;\;\quad a^{\prime \prime {\phantom {\prime }}}=0{,}6699905217\;\;924323\\&\;\;\quad a^{\prime \prime \prime }=0{,}9305681557\;\;970246\\R^{\phantom {\prime }}&=R^{\prime \prime \prime }=0{,}1739274225\;\;687284\;\;\;\log .=9{,}2403680612\\R^{\prime }&=R^{\prime \prime {\phantom {\prime }}}=0{,}3260725774\;\;312716\;\;\;\log .=9{,}5133142764\end{array}}}$

Horum coëfficientium expressio generalis ${\textstyle -{\frac {35}{144}}uu+{\frac {17}{48}}}$

Correctio proxime ${\textstyle ={\frac {1}{44100}}L^{\scriptscriptstyle VIII}}$