Pagina:Werkecarlf03gausrich.djvu/197

adeoque correctio valoris approximati integralis

$$=-{\frac {11}{52500}}L^{\scriptscriptstyle VI}-{\frac {13}{112500}}L^{\scriptscriptstyle VIII}-{\frac {5933}{137500000}}L^{\scriptscriptstyle X}-{\text{etc. }}$$

15.

Coëfficiens ${\textstyle K^{(m)}}$ functionis ${\textstyle y}$ in seriem evolutae fit, per theorema Taylori, aequalis valori ipsius

${\textstyle {\frac {1}{1.2.3\ldots m}}\cdot {\frac {\operatorname {d} ^{m}y}{\operatorname {d} t^{m}}}}$ sive ${\textstyle {\frac {\Delta ^{m}}{1.2.3\ldots m}}\cdot {\frac {\operatorname {d} ^{m}y}{\operatorname {d} x^{m}}}}$

pro ${\textstyle t=0}$ sive ${\textstyle x=g;}$ perinde coëfficiens ${\textstyle L^{(m)}}$ est valor eiusdem expressionis pro ${\textstyle t={\frac {1}{2}}}$ sive ${\textstyle u=0}$ sive ${\textstyle x=g+{\frac {1}{2}}\Delta {:}}$ utrique coëfficienti ordinem ${\textstyle m}$ tribuemus. Generaliter itaque loquendo integratio nostra usque ad ordinem ${\textstyle n}$ inclus. exacta erit, quicunque valores pro ${\textstyle a,}$ ${\textstyle a^{\prime },}$ ${\textstyle a^{\prime \prime }\ldots a^{(n)}}$ accipiantur. Attamen hinc nihil obstat, quominus pro valoribus harum quantitatum scite electis praecisio ad altiorem gradum evehatur. Ita iam supra vidimus, in methodo Cotesii i.e. pro ${\textstyle a=0,}$ ${\textstyle a^{\prime }={\frac {1}{n}},}$ ${\textstyle a^{\prime \prime }={\frac {2}{n}},}$ ${\textstyle a^{\prime \prime \prime }={\frac {3}{n}}}$ etc. praecisionem sponte ad ordinem ${\textstyle n+1}$ inclus. extendi, quoties ${\textstyle n}$ sit numerus par. Generaliter patet, si valores ${\textstyle a,}$ ${\textstyle a^{\prime },}$ ${\textstyle a^{\prime \prime },}$ ${\textstyle a^{\prime \prime \prime }}$ etc. ita fuerint electi, ut in functione ${\textstyle T^{\prime \prime }}$ vel ${\textstyle U^{\prime \prime }}$ ab initio excidat terminus unus pluresve, praecisionem totidem gradibus ultra ordinem ${\textstyle n}$ promotum iri, quot termini exciderint. Hinc facile colligitur, quum multitudo quantitatum quas eligere conceditur sit ${\textstyle n+1,}$ per idoneam earum determinationem praecisionem semper ad ordinem ${\textstyle 2n+1}$ inclus. evehi posse, quo pacto adiumento ${\textstyle n+1}$ terminorum eundem praecisionis ordinem assequi licebit, ad quem attingendum ${\textstyle 2n+1}$ vel ${\textstyle 2n+2}$ terminos in usum vocare oporteret, si Cotesii methodum sequeremur.

16.

Totum hoc negotium in eo vertitur, ut pro quovis valore dato ipsius ${\textstyle n}$ functionem ${\textstyle T}$ eruamus formae ${\textstyle t^{n+1}+\alpha t^{n}+\alpha ^{\prime }t^{n-1}+\alpha ^{\prime \prime }t^{n-2}}$ etc. itaque comparatam, ut in producto

$$T(t^{-1}+{\frac {1}{2}}t^{-2}+{\frac {1}{3}}t^{-3}+{\frac {1}{4}}t^{-4}+{\text{etc.}})$$

evoluto potestates ${\textstyle t^{-1},}$ ${\textstyle t^{-2},}$ ${\textstyle t^{-3}\ldots t^{-(n+1)}}$ omnes nanciscantur coëfficientem ${\textstyle 0;}$ aut si magis placet, functionem ${\textstyle U}$ formae ${\textstyle u^{n+1}+\beta u^{n}+\beta ^{\prime }u^{n-1}+\beta ^{\prime \prime }u^{n-2}+}$ etc., cuius productum per ${\textstyle u^{-1}+{\frac {1}{3}}u^{-3}+{\frac {1}{3}}u^{-5}+{\frac {1}{7}}u^{-7}+}$ etc. liberum evadat a potesta-