Pagina:Werkecarlf03gausrich.djvu/183

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

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

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

5.

Quum formula ${\textstyle \Delta (AR+A^{\prime }R^{\prime }+A^{\prime }R^{\prime }+A^{\prime \prime }R^{\prime \prime }+{\text{etc.}}+A^{(n)}R^{(n)})}$ integrale ${\textstyle \int y\operatorname {d} x}$ ab ${\textstyle x=g}$ usque ad ${\textstyle x=g+\Delta ,}$ sive integrale ${\textstyle \Delta \int y\operatorname {d} t}$ a ${\textstyle t=0}$ usque ad ${\textstyle t=1}$ exacte quidem exhibeat, quoties ${\textstyle y}$ in seriem evoluta potestatem ${\textstyle t^{n}}$ non transscendit, sed approximate tantum, quoties ${\textstyle y}$ ultra progreditur, superest, ut errorem, quem inducunt termini proxime sequentes, assignare doceamus. Designemus generaliter per ${\textstyle k^{(m)}}$ differentiam inter valorem verum integralis ${\textstyle \int t^{m}\mathrm {dt} }$ a ${\textstyle t=0}$ usque ad ${\textstyle t=1,}$ atque valorem ex formula prodeuntem, ita ut sit

$${\begin{aligned}&k=1-R-R^{\prime }-R^{\prime \prime }-R^{\prime \prime \prime }-{\text{etc.}}-R^{(n)}\\&k^{\prime }={\frac {1}{2}}-{\frac {1}{n}}(R^{\prime }+2R^{\prime \prime }+3R^{\prime \prime \prime }+{\text{etc.}}+nR^{(n)})\\&k^{\prime \prime }={\frac {1}{3}}-{\frac {1}{nn}}(R^{\prime }+4R^{\prime \prime }+9R^{\prime \prime \prime }+{\text{etc.}}+nnR^{(n)})\\&k^{\prime \prime \prime }={\frac {1}{4}}-{\frac {1}{n^{2}}}(R^{\prime }+8R^{\prime \prime }+27R^{\prime \prime \prime }+{\text{etc.}}+n^{3}R^{(n)})\end{aligned}}$$

etc. Patet igitur, si ${\textstyle y}$ evolvatur in seriem

$$K+K^{\prime }t+K^{\prime \prime }tt+K^{\prime \prime \prime }t^{3}+{\text{etc. }}$$

differentiam inter valorem verum integralis ${\textstyle \int y\operatorname {d} t}$ 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 ${\textstyle k,}$ ${\textstyle k^{\prime },}$ ${\textstyle k^{\prime \prime }}$ etc. usque ad ${\textstyle k^{(n)}}$ sponte fiunt ${\textstyle =0{:}}$ 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 ${\textstyle k^{(n+1)},}$ ${\textstyle k^{(n+2)}}$ etc. infra accuratius perscrutabimur; hic sufficiat, valores numericos primae aut secundae, pro singulis valoribus ipsius ${\textstyle n,}$ apposuisse, ut gradus praecisionis, quam formula approximata affert, inde aestimari possit.{{nop}