Pagina:Werkecarlf03gausrich.djvu/184

Pro ${\textstyle n={\phantom {0}}1}$ habemus ${\textstyle k^{\prime \prime }=-{\frac {1}{6}},}$ ${\textstyle k^{\prime \prime \prime }=-{\frac {1}{4}},}$ ${\textstyle k^{\prime \prime \prime \prime }=-{\frac {3}{10}}}$

Pro ${\textstyle n={\phantom {0}}2}$ invenimus ${\textstyle k^{\prime \prime \prime }=0,}$ ${\textstyle k^{\prime \prime \prime \prime }=-{\frac {1}{120}},}$ ${\textstyle k^{\scriptscriptstyle V}=-{\frac {1}{48}}}$

Pro ${\textstyle n={\phantom {0}}3}$ fit ${\textstyle k^{\prime \prime \prime \prime }=-{\frac {1}{270}},}$ ${\textstyle \quad k^{\scriptscriptstyle V}=-{\frac {1}{108}}}$

Pro ${\textstyle n={\phantom {0}}4\ldots k^{\scriptscriptstyle V}=0,}$ ${\textstyle k^{\scriptscriptstyle VI}=-{\frac {1}{2688}},}$ ${\textstyle k^{\scriptscriptstyle VII}=-{\frac {1}{768}}}$

Pro ${\textstyle n={\phantom {0}}5\ldots k^{\scriptscriptstyle VI}=-{\frac {1}{52?00}},}$ ${\textstyle k^{\scriptscriptstyle VII}=-{\frac {11}{15000}}}$

Pro ${\textstyle n={\phantom {0}}6\ldots k^{\scriptscriptstyle VII}=0,}$ ${\textstyle k^{\scriptscriptstyle VIII}=-{\frac {1}{38880}},}$ ${\textstyle k^{\scriptscriptstyle IX}=-{\frac {1}{8640}}}$

Pro ${\textstyle n={\phantom {0}}7\ldots k^{\scriptscriptstyle VIII}=-{\frac {167}{10588410}},}$ ${\textstyle k^{\scriptscriptstyle IX}=-{\frac {167}{2352980}}}$

Pro ${\textstyle n={\phantom {0}}8\ldots k^{\scriptscriptstyle IX}=0,}$ ${\textstyle k^{\scriptscriptstyle X}=-{\frac {37}{173?1504}},}$ ${\textstyle k^{\scriptscriptstyle XI}=-{\frac {37}{3145728}}}$

Pro ${\textstyle n={\phantom {0}}9\ldots k^{\scriptscriptstyle X}=-{\frac {865}{631351908}},}$ ${\textstyle k^{\scriptscriptstyle XI}=-{\frac {865}{114791256}}}$

Pro ${\textstyle n=10\ldots k^{\scriptscriptstyle XI}=0,}$ ${\textstyle k^{\scriptscriptstyle XII}=-{\frac {26927}{136500000000}},}$ ${\textstyle k^{\scriptscriptstyle XII}=-{\frac {26927}{21000000000}}}$

Pro valore pari ipsius ${\textstyle n}$ ubique hic fieri animadvertimus ${\textstyle k^{(n+1)}=0,}$ ac praeterea ${\textstyle k^{(n+3)}={\frac {n+3}{2}}k^{(n+2)};}$ pro valore impari ipsius ${\textstyle n}$ autem ubique prodit ${\textstyle k^{(n+2)}={\frac {n+2}{2}}k^{(n+1)}.}$ Ratio horum eventuum facile e considerationibus sequentibus depromitur.

Designemus generaliter per ${\textstyle l^{(m)}}$ differentiam inter valorem verum huius integralis ${\textstyle \int (t-{\frac {1}{2}})^{m}\operatorname {d} t}$ a ${\textstyle t=0}$ usque ad ${\textstyle t=1,}$ atque valorem eum, quem formula approximata profert, ita ut habeatur

$${\begin{array}{c}l^{(m)}=\int (t-{\frac {1}{2}})^{m}\operatorname {d} t-\left[(-{\frac {1}{2}})^{m}R+({\frac {1}{n}}-{\frac {1}{2}})^{m}R^{\prime }+({\frac {2}{n}}-{\frac {1}{2}})^{m}R^{\prime \prime }+({\frac {3}{n}}-{\frac {1}{2}})^{m}R^{\prime \prime \prime }+\right.{\text{etc.}}\\\left.+({\frac {1}{2}}-{\frac {1}{n}})^{m}R^{(n-1)}+({\frac {1}{2}})^{m}R^{(n)}\right]\end{array}}$$

integrali a ${\textstyle t=0}$ usque ad ${\textstyle t=1}$ accepto. Manifesto pro valore impari ipsius ${\textstyle m}$ evanescet tum valor verus integralis tum valor approximatus: erit itaque ${\textstyle l^{\prime }=0,}$ ${\textstyle l^{\prime \prime \prime }=0,l^{\scriptscriptstyle V}=0,l^{\scriptscriptstyle VII}=0}$ etc. sive generaliter ${\textstyle l^{(m)}=0}$ pro valore impari ipsius ${\textstyle m.}$ Pro valore pari autem ipsius ${\textstyle m,}$ formulae tribuimus formam hancce

$${\begin{array}{c}l^{(m)}={\frac {1}{2^{m}(m+1)}}-{\frac {2}{n^{m}}}(({\frac {1}{2}}n)^{m}R+({\frac {1}{2}}n-1)^{m}R^{\prime }+({\frac {1}{2}}n-2)^{m}R^{\prime \prime }+{\text{etc. }}\\+2^{m}R^{({\frac {1}{2}}n-2)}+R^{({\frac {1}{2}}n-1)})\end{array}}$$

si simul fuerit ${\textstyle n}$ par; vel hanc

$${\begin{aligned}l^{(m)}={\frac {1}{2^{n}}}({\frac {1}{m+1}}-{\frac {2}{n^{n}}}(n^{m}R&+(n-2)^{m}R^{\prime }+(n-4)^{m}R^{\prime \prime }+{\text{etc. }}\\&+3^{m}R^{({\frac {1}{2}}n-{\frac {3}{2}})}+R^{({\frac {1}{2}}n-{\frac {1}{2}})}))\end{aligned}}$$

si simul fuerit ${\textstyle n}$ impar.