Pagina:Werkecarlf03gausrich.djvu/182

pro $n$ 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

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

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

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

4.

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

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

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

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

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

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

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

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

Pro $n=8$ sive terminis novem.

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