Pagina:Werkecarlf03gausrich.djvu/183

Pro $n=9$ sive terminis decem.
$R=R^{IX}=\frac{2857}{89600},$ $R^{\prime }=R^{VIII}=\frac{15741}{89600},$ $R^{\prime \prime }=R^{VII}=\frac{27}{2240},$ $R^{\prime \prime \prime }=R^{VI}=\frac{1209}{5600},$ $R^{IV}=R^V=\frac{2889}{44800}$

Pro $n=10$ sive terminis undecim.
$R=R^X=\frac{16067}{598752},$ $R^{\prime }=R^{IX}=\frac{26575}{149688},$ $R^{\prime \prime }=R^{VIII}=-\frac{16175}{199584},$

$R^{\prime \prime \prime }=R^{VII}=\frac{5675}{12474},$ $R^{IV}=R^{VI}=-\frac{4825}{11088},$ $R^V=\frac{17807}{24948}.$

5.

Quum formula $\mathrm{\Delta }(AR+A^{\prime }{R}^{\prime }+A^{\prime }{R}^{\prime }+A^{\prime \prime }{R}^{\prime \prime }+\text{etc.}+A^{(n)}{R}^{(n)})$ integrale $\int y\mathrm{d}x$ ab $x=g$ usque ad $x=g+\Delta ,$ sive integrale $\mathrm{\Delta }\int y\mathrm{d}t$ a $t=0$ usque ad $t=1$ exacte quidem exhibeat, quoties $y$ in seriem evoluta potestatem $t^n$ non transscendit, sed approximate tantum, quoties $y$ ultra progreditur, superest, ut errorem, quem inducunt termini proxime sequentes, assignare doceamus. Designemus generaliter per $k^{(m)}$ differentiam inter valorem verum integralis $\int t^m\mathrm{d}\mathrm{t}$ a $t=0$ usque ad $t=1,$ atque valorem ex formula prodeuntem, ita ut sit $$\begin{array}{cc}& k=1-R-R^{\prime }-R^{\prime \prime }-R^{\prime \prime \prime }-\text{etc.}-R^{(n)}\\ & k^{\prime }=\frac{1}{2}-\frac{1}{n}(R^{\prime }+2R^{\prime \prime }+3R^{\prime \prime \prime }+\text{etc.}+n{R}^{(n)})\\ & k^{\prime \prime }=\frac{1}{3}-\frac{1}{nn}(R^{\prime }+4R^{\prime \prime }+9R^{\prime \prime \prime }+\text{etc.}+nn{R}^{(n)})\\ & k^{\prime \prime \prime }=\frac{1}{4}-\frac{1}{n^2}(R^{\prime }+8R^{\prime \prime }+27R^{\prime \prime \prime }+\text{etc.}+n^3{R}^{(n)})\end{array}$$ etc. Patet igitur, si $y$ evolvatur in seriem $$K+K^{\prime }t+K^{\prime \prime }tt+K^{\prime \prime \prime }{t}^3+\text{etc. }$$ differentiam inter valorem verum integralis $\int y\mathrm{d}t$ atque valorem approximatum formulae exprimi per $$Kk+K^{\prime }{k}^{\prime }+K^{\prime \prime }{k}^{\prime \prime }+K^{\prime \prime \prime }{k}^{\prime \prime \prime }+\text{etc.}$$ Sed manifesto $k,$ $k^{\prime },$ $k^{\prime \prime }$ etc. usque ad $k^{(n)}$ sponte fiunt $=0:$ correctio itaque formulae approximatae èrit

$$K^{(n+1)}{k}^{(n+1)}+K^{(n+2)}{k}^{(n+2)}+K^{(n+3)}{k}^{(n+3)}+\text{etc.}$$

Indolem quantitatum $k^{(n+1)},$ $k^{(n+2)}$ etc. infra accuratius perscrutabimur; hic sufficiat, valores numericos primae aut secundae, pro singulis valoribus ipsius $n,$ apposuisse, ut gradus praecisionis, quam formula approximata affert, inde aestimari possit.{{nop}