Pagina:Werkecarlf03gausrich.djvu/186

6.

Generaliter itaque loquendo praestabit, in applicanda methodo Cotesiana ipsi ${\textstyle n}$ tribuere valorem parem, seu terminorum multitudinem imparem in usum vocare. Perparum scilicet praecisio augebitur, si loco valoris paris ipsius ${\textstyle n}$ ad imparem proxime maiorem ascendamus, quum error maneat eiusdem ordinis, licet coëfficiente aliquantulum minori affectus. Contra ascendendo a valore impari ipsius ${\textstyle n}$ ad parem proxime sequentem, error duobus ordinibus promovebitur, insuperque coëfficiens notabilius imminutus praecisionem augebit. Ita si quinque termini adhibentur, sive pro ${\textstyle n=4,}$ error proxime exprimitur per ${\textstyle -{\frac {1}{2688}}K^{6}}$ vel per ${\textstyle -{\frac {1}{2688}}L^{6};}$ si statuitur ${\textstyle n=5,}$ error erit proxime ${\textstyle -{\frac {11}{52500}}K^{6}}$ vel ${\textstyle -{\frac {11}{52500}}L^{6},}$ adeoque ne ad semissem quidem prioris depressus: contra faciendo ${\textstyle n=6,}$ error fit proxime ${\textstyle =-{\frac {1}{38880}}K^{8}}$ vel ${\textstyle =-{\frac {1}{38880}}L^{8},}$ praecisioque tanto magis aucta, quo citius series, in quam functio evolvitur, iam per se convergit.

7.

Postquam haecce circa methodum Cotesii praemisimus, ad disquisitionem generalem progredimur, abiiciendo conditionem, ut valores ipsius ${\textstyle x}$ progressione arithmetica procedant. Problema itaque aggredimur, determinare valorem integralis ${\textstyle \int y\operatorname {d} x}$ inter limites datos ex aliquot valoribus datis ípsius ${\textstyle y,}$ vel exacte vel quam proxime. Supponamus, integrale sumendum esse ab ${\textstyle x=g}$ usque ad ${\textstyle x=g+\Delta ,}$ introducamusque loco ipsius ${\textstyle x}$ aliam variabilem ${\textstyle t={\frac {x-g}{\Delta }},}$ ita ut integrale ${\textstyle \Delta \int y\operatorname {d} t}$ a ${\textstyle t=0}$ usque ad ${\textstyle t=1}$ investigare oporteat. Respondeant ${\textstyle n+1}$ valores dati ipsius ${\textstyle y}$ hi ${\textstyle A,}$ ${\textstyle A^{\prime },}$ ${\textstyle A^{\prime \prime },}$ ${\textstyle A^{\prime \prime \prime }\ldots A^{(n)}}$ valoribus ipsius ${\textstyle t}$ inaequalibus his ${\textstyle a,}$ ${\textstyle a^{\prime },}$ ${\textstyle a^{\prime \prime },}$ ${\textstyle a^{\prime \prime \prime }\ldots a^{(n)},}$ designemusque per ${\textstyle {Y}}$ functionem algebraicam integram ordinis ${\textstyle n}$ hancce:

$${\begin{aligned}&A^{\phantom {\prime \prime }}{\frac {(t-a^{\prime })(t-a^{\prime \prime })(t-a^{\prime \prime \prime })\ldots (t-a^{(n)})}{(a-a^{\prime })(a-a^{\prime \prime })(a-a^{\prime \prime \prime })\ldots (a-a^{(n)})}}\\+&A^{\prime {\phantom {\prime }}}\,{\frac {(t-a)(t-a^{\prime \prime })(t-a^{\prime \prime \prime })\ldots (t-a^{(n)})}{(a^{\prime }-a)(a^{\prime }-a^{\prime \prime })(a^{\prime }-a^{\prime \prime \prime })\ldots (a^{\prime }-a^{(n)})}}\\+&A^{\prime \prime }{\frac {(t-a)(t-a^{\prime })(t-a^{\prime \prime \prime })\ldots (t-a^{(n)})}{(a^{\prime \prime }-a)(a^{\prime \prime }-a^{\prime })(a^{\prime \prime }-a^{\prime \prime })\ldots (a^{\prime \prime }-a^{(n)})}}\\+&{\text{ etc. }}\\+&A^{(n)}{\frac {(t-a)(t-a^{\prime })(t-a^{\prime \prime })\ldots (t-a^{(n-1)})}{(a^{(n)}-a)(a^{(n)}-a^{\prime })(a^{(n)}-a^{\prime \prime })\ldots (a^{(n)}-a^{(n-1)})}}\end{aligned}}$$

Manifesto valores huius functionis, si ${\textstyle t}$ alicui quantitatum ${\textstyle a,}$ ${\textstyle a^{\prime },}$ ${\textstyle a^{\prime \prime },}$ ${\textstyle a^{\prime \prime \prime }\ldots a^{(m)}}$ aequalis ponitur, coincidunt cum valoribus respondentibus functionis ${\textstyle y,}$ unde per-