Pagina:Werkecarlf03gausrich.djvu/182

pro ${\textstyle n}$ impari autem

$$={\frac {2\mu -n}{2^{n}}}({\frac {\alpha }{n}}+{\frac {\beta }{n-2}}+{\frac {\gamma }{n-4}}+{\frac {\delta }{n-6}}+{\text{etc.}})$$

In exemplo nostro habetur

${\textstyle U^{\prime \prime }=(25uu-25)(25uu-9)=625u^{4}-850uu+225,}$ adeoque

${\textstyle -12R^{\prime \prime }=-{\frac {1}{32}}(125-{\frac {850}{3}}+225)=-{\frac {25}{12}}}$ ut supra.

Observare convenit, fieri ${\textstyle U^{(n-\mu )}=U^{(\mu )},}$ adeoque ${\textstyle \int T^{(n-\mu )}\operatorname {d} t=\pm \int T^{(\mu )}\operatorname {d} t,}$ signo superiore valente pro ${\textstyle n}$ pari, inferiore pro impari. Quare quum facile perspiciatur, perinde haberi ${\textstyle M^{(n-\mu )}=\pm M^{(\mu )},}$ semper erit ${\textstyle R^{(n-\mu )}=R^{(\mu )},}$ sive ${\textstyle \mathrm {e} }$ coëfficientibus ${\textstyle R,R^{\prime },R^{\prime \prime }\ldots R^{(n)}}$ ultimus primo aequalis, penultimus secundo et sic porro.

4.

Valores numericos horum coëfficientium a Cotesio usque ad ${\textstyle n=10}$ computatos ex Harmonia Mensurarum huc adscribimus.

Pro ${\textstyle n=1}$ sive terminis duobus.
${\textstyle R=R^{\prime }={\frac {1}{2}}}$

Pro ${\textstyle n=2}$ sive terminis tribus.
${\textstyle R=R^{\prime \prime }={\frac {1}{6}},}$ ${\textstyle R^{\prime }={\frac {2}{3}}}$

Pro ${\textstyle n=3}$ sive terminis quatuor.
${\textstyle R=R^{\prime \prime \prime }={\frac {1}{8}},}$ ${\textstyle R^{\prime }=R^{\prime \prime }={\frac {3}{8}}}$

Pro ${\textstyle n=4}$ sive terminis quinque.
${\textstyle {R}={R}^{\prime \prime \prime }={\frac {7}{90}},}$ ${\textstyle R^{\prime }=R^{\prime \prime \prime }={\frac {16}{45}},}$ ${\textstyle R^{\prime \prime }={\frac {2}{15}}}$

Pro ${\textstyle n=5}$ sive terminis sex.
${\textstyle {R}={R}^{\scriptscriptstyle V}={\frac {19}{288}},}$ ${\textstyle {R}^{\prime }={R}^{\prime \prime \prime \prime }={\frac {25}{9}},}$ ${\textstyle {R}^{\prime \prime }={R}^{\prime \prime \prime }={\frac {25}{144}}}$

Pro ${\textstyle n=6}$ sive terminis septem.
${\textstyle R=R^{\scriptscriptstyle VI}={\frac {41}{846}},}$ ${\textstyle R^{\prime }=R^{\scriptscriptstyle V}={\frac {9}{35}},}$ ${\textstyle R^{\prime \prime }=R^{\scriptscriptstyle IV}={\frac {9}{280}},}$ ${\textstyle R^{\prime \prime \prime }={\frac {34}{105}}}$

Pro ${\textstyle n=7}$ sive terminis octo.
${\textstyle R=R^{\scriptscriptstyle VII}={\frac {751}{17280}},}$ ${\textstyle R^{\prime }=R^{\scriptscriptstyle VI}={\frac {3577}{17280}},}$ ${\textstyle R^{\prime \prime }=R^{\scriptscriptstyle V}={\frac {49}{640}},}$ ${\textstyle R^{\prime \prime \prime }=R^{\prime \prime \prime \prime }={\frac {2989}{17280}}}$

Pro ${\textstyle n=8}$ sive terminis novem.

${\textstyle R=R^{\scriptscriptstyle VIII}={\frac {989}{28350}},}$ ${\textstyle R^{\prime }=R^{\scriptscriptstyle VII}={\frac {2944}{14175}},}$ ${\textstyle R^{\prime \prime }=R^{\scriptscriptstyle VI}=-{\frac {464}{14175}},}$ ${\textstyle R^{\prime \prime \prime }=R^{\scriptscriptstyle V}={\frac {5248}{14175}},}$ ${\textstyle R^{\scriptscriptstyle IV}=-{\frac {454}{2835}}}$