Pagina:Werkecarlf03gausrich.djvu/192

singulas e binis praecedentibus per eandem legem derivamus, per quam functiones ${\textstyle \zeta ,}$ ${\textstyle \zeta ^{\prime },}$ ${\textstyle \zeta ^{\prime \prime },}$ ${\textstyle \zeta ^{\prime \prime \prime }}$ etc. inter se nexae sunt, scilicet per aequationes

$${\begin{aligned}&\eta ^{\prime \prime {\phantom {\prime \prime \prime }}}=\lambda \eta -p\eta ^{\prime }\\&\eta ^{\prime \prime \prime {\phantom {\prime \prime }}}=\lambda ^{\prime }\eta ^{\prime }-p^{\prime }\eta ^{\prime \prime }\\&\eta ^{\prime \prime \prime \prime {\phantom {\prime }}}=\lambda ^{\prime \prime }\eta ^{\prime \prime }-p^{\prime \prime }\eta ^{\prime \prime \prime }\\&\eta ^{\prime \prime \prime \prime \prime }=\lambda ^{\prime \prime \prime }\eta ^{\prime \prime \prime }-p^{\prime \prime \prime }\eta ^{\prime \prime \prime \prime }\end{aligned}}$$

etc. usque ad

$$\eta ^{(m)}=\lambda ^{(m-2)}\eta ^{(m-2)}-p^{(m-2)}\eta ^{(m-1)}$$

Manifesto ${\textstyle \eta ^{\prime \prime }=\lambda }$ hic est ordinis ${\textstyle 0;}$ ${\textstyle \eta ^{\prime \prime \prime }=-\lambda p^{\prime }}$ ordinis ${\textstyle k^{\prime }-k^{\prime \prime },}$ et perinde sequentes ${\textstyle \eta ^{\prime \prime \prime \prime },}$ ${\textstyle \eta ^{\prime \prime \prime \prime \prime }}$ etc. resp. ordinis ${\textstyle k^{\prime }-k^{\prime \prime \prime },}$ ${\textstyle k^{\prime }-k^{\prime \prime \prime \prime }}$ etc., ita ut ultima ${\textstyle \eta ^{(m)}}$ ascendat ad ordinem ${\textstyle k^{\prime }-k^{(m-1)}.}$

Porro consideremus tertiam functionum seriem, ${\textstyle \zeta -\zeta \eta ,}$ ${\textstyle \zeta ^{\prime }-\zeta \eta ^{\prime },}$ ${\textstyle \zeta ^{\prime \prime }-\zeta \eta ^{\prime \prime },}$ ${\textstyle \zeta ^{\prime \prime \prime }-\zeta \eta ^{\prime \prime \prime }}$ etc., inter cuius terminos quosvis ternos consequentes manifesto similis relatio intercedet, scilicet

$${\begin{aligned}\zeta ^{\prime \prime }-\zeta \eta ^{\prime \prime }&=\lambda (\zeta -\zeta \eta )-p(\zeta ^{\prime }-\zeta \eta ^{\prime })\\\zeta ^{\prime \prime \prime }-\zeta \eta ^{\prime \prime \prime }&=\lambda ^{\prime }(\zeta ^{\prime }-\zeta \eta ^{\prime })-p^{\prime }(\zeta ^{\prime \prime }-\zeta \eta ^{\prime \prime })\\\zeta ^{\prime \prime \prime \prime }-\zeta \eta ^{\prime \prime \prime \prime }&=\lambda ^{\prime \prime }(\zeta ^{\prime \prime }-\zeta \eta ^{\prime \prime })-p^{\prime \prime }(\zeta ^{\prime \prime \prime }-\zeta \eta ^{\prime \prime \prime })\end{aligned}}$$

Iam prima harum functionum fit ${\textstyle =0,}$ secunda ${\textstyle =\zeta ^{\prime }{:}}$ hinc facile colligitur, singulas per ${\textstyle \zeta ^{\prime }}$ divisibiles fore.

Hinc autem nullo negotio sequitur, loco fractionis ${\textstyle {\frac {Z}{\zeta }}}$ adoptari posse functionem integram ${\textstyle Z\eta ^{(m)},}$ quatenus quidem ipsi ${\textstyle {z}}$ non tribuantur alii valores nisi qui sint radices aequationis ${\textstyle \zeta ^{\prime }=0{:}}$ manifesto enim differentia ${\textstyle {\frac {Z(1-\zeta \eta ^{(m)})}{\zeta }}}$ pro tali valore ipsius ${\textstyle z}$ necessario evanescit, quum ${\textstyle 1-\zeta \eta ^{(m)}=\zeta ^{(m)}-\zeta \eta ^{(m)}}$ per ${\textstyle \zeta ^{\prime }}$ sit divisibilis.

Loco functionis ${\textstyle Z\eta ^{(m)}}$ etiam adoptari poterit eius residuum ex divisione per ${\textstyle \zeta ^{\prime }}$ ortum, cuius ordo erit inferior ordine functionis ${\textstyle \zeta ^{\prime }.}$

Ceterum hocce residuum commodius per algorithmum sequentem immediate eruere licet. Formentur aequationes sequentes

$${\begin{aligned}Z^{\phantom {\prime \prime \prime }}&=q^{\prime }\zeta ^{\prime }+Z^{\prime }\\Z^{\prime {\phantom {\prime \prime }}}&=q^{\prime \prime }\zeta ^{\prime \prime }+Z^{\prime \prime }\\Z^{\prime \prime {\phantom {\prime }}}&=q^{\prime \prime \prime }\zeta ^{\prime \prime \prime }+Z^{\prime \prime \prime }\\Z^{\prime \prime \prime }&=q^{\prime \prime \prime \prime }\zeta ^{\prime \prime \prime \prime }+Z^{\prime \prime \prime \prime }\end{aligned}}$$

etc. usque ad

$$Z^{(m-1)}=q^{(m)}\zeta ^{(m)}+Z^{(m)}$$