Pagina:Werkecarlf03gausrich.djvu/201

Leviattentione adhibita elucet, singulas ${\textstyle V,}$ ${\textstyle {V}^{\prime },}$ ${\textstyle V^{\prime \prime },}$ ${\textstyle V^{\prime \prime \prime }}$ etc. ${\textstyle {W},}$ ${\textstyle {W}^{\prime },}$ ${\textstyle W^{\prime \prime },}$ ${\textstyle {W}^{\prime \prime \prime }}$ etc. fieri functiones integras indeterminatae ${\textstyle u;}$ terminum altissimum in ${\textstyle V^{(m)}}$ fieri ${\textstyle u^{m-1},}$ potestatesque ${\textstyle u^{m-2},}$ ${\textstyle u^{m-4},}$ ${\textstyle u^{m-6}}$ etc. abesse; terminum altissimam vero in ${\textstyle W^{(m)}}$ fieri ${\textstyle u^{m},}$ atque abesse potestates ${\textstyle u^{m-1},}$ ${\textstyle u^{m-3},}$ ${\textstyle u^{m-5}}$ etc. Per ea autem, quae supra demonstravimus, erit

$$\varphi ={\frac {1}{WW^{\prime }}}+{\frac {1}{3W^{\prime }W^{\prime \prime }}}+{\frac {2.2}{3.3.5W^{\prime \prime }W^{\prime \prime \prime }}}+{\frac {2.2.3.3}{3.3.5.5.7W^{\prime \prime \prime }W^{\prime \prime \prime \prime }}}+{\frac {2.2.3.3.4.4}{3.3.5.5.7.7.9W^{\prime \prime \prime \prime }W^{\prime \prime \prime \prime \prime }}}+{\text{etc.}}$$

ac proin generaliter

$${\begin{aligned}\varphi -{\frac {V^{(m)}}{W^{(m)}}}=&{\frac {2.2.3.3\ldots \ldots m.m}{3.3.5.5\ldots (2m-1)(2m+1)W^{(m)}W^{(m+1)}}}\\&+{\frac {2.2.3.3\ldots \ldots (m+1)(m+1)}{3.3.5.5\ldots \ldots (2m+1)(2m+3)W^{(m+1)}W^{(m+2)}}}\\&+{\text{etc. }}\end{aligned}}$$

Si igitur ${\textstyle \varphi -{\frac {V^{(m)}}{W^{(m)}}}}$ in seriem descendentem convertitur, eius terminus primus erit

$$={\frac {2.2.3.3\ldots mm\,u^{-(2m+1)}}{3.3.5.5\ldots (2m-1)(2m+1)}}$$

Productum vero ${\textstyle \varphi W^{(m)}}$ compositum erit e functione integra ${\textstyle V^{(m)}}$ atque serie infinita, cuius terminus primus

$$={\frac {2.2.3.3\ldots mm\,u^{-(m+1)}}{3.3.5.5\ldots (2m-1)(2m+1)}}$$

Hinc igitur sponte inventa est functio ${\textstyle U}$ ordinis ${\textstyle n+1;}$ quae conditioni in art. praec. stabilitae satisfacit, scilicet ut productum ${\textstyle \varphi U}$ liberum evadat a potestatibus ${\textstyle u^{-1},}$ ${\textstyle u^{-2},}$ ${\textstyle u^{-3}\ldots u^{-(n+1)}.}$ Scilicet non est alia quam ${\textstyle W^{(n+1)},}$ simulque patet, ${\textstyle U^{\prime }}$ aequalem fieri ipsi ${\textstyle V^{(m+1)},}$ nec non terminum primum ipsius ${\textstyle U^{\prime \prime }}$ esse

$$={\frac {2.2.3.3\ldots \ldots (n+1)(n+1)}{3.3.5.5\ldots \ldots (2n+1)(2n+3)}}.u^{-(n+2)}$$

Quodsi igitur pro ${\textstyle b,}$ ${\textstyle b^{\prime },}$ ${\textstyle b^{\prime \prime }\ldots b^{(n)}}$ accipiuntur radices aequationis ${\textstyle W^{(n+1)}=0,}$ valoresque coëfficientium ${\textstyle R,}$ ${\textstyle R^{\prime },}$ ${\textstyle R^{\prime \prime }\ldots R^{(n)}}$ per praecepta supra tradita eruuntur, formula nostra integralis praecisione gaudebit ad ordinem ${\textstyle 2n+1}$ ascendente, eiusque correctio exprimetur proxime per

$${\frac {1}{2^{2n+2}}}\cdot {\frac {2.2.3.3\ldots \ldots (n+1)(n+1)}{3.3.5.5\ldots \ldots (2n+1)(2n+3)}}L^{(2n+2)}={\frac {1.1.2.2.3.3\ldots \ldots (n+1)(n+1)}{2.6.6.10.10.14\ldots \ldots (4n+2)(4n+6)}}L^{(2n+2)}$$