Pagina:Werkecarlf03gausrich.djvu/191

11.

Quoties ${\textstyle a,}$ ${\textstyle a^{\prime },}$ ${\textstyle a^{\prime \prime }}$ etc. vel ex parte vel omnes sunt irrationales, utilis erit transformatio functionis fractae, ex qua numeros ${\textstyle R,}$ ${\textstyle R^{\prime },}$ ${\textstyle R^{\prime \prime }}$ etc. derivamus, in functionem integram: principia talis transformationis, quum in libris algebraicis non inveniantur. hoc loco breviter explicabimus. Propositis scilicet tribus functionibus integris ${\textstyle Z,}$ ${\textstyle \zeta ,}$ ${\textstyle \zeta ^{\prime }}$ indeterminatae ${\textstyle z,}$ quaeritur functio integra, quae fractae ${\textstyle {\frac {Z}{\zeta }}}$ vice fungi possit, quatenus pro ${\textstyle z}$ accipitur radix quaecunque aequationis ${\textstyle \zeta ^{\prime }=0.}$ Supponemus autem, ${\textstyle \zeta }$ pro nullo horum valorum ipsius ${\textstyle z}$ evanescere, sive quod eodem redit, ${\textstyle \zeta }$ atque ${\textstyle \zeta ^{\prime }}$ nullum divisorem communem indeterminatum implicare. Exponentes potestatum altissimarum ipsius ${\textstyle z}$ in ${\textstyle \zeta }$ atque ${\textstyle \zeta ^{\prime }}$ per ${\textstyle k,}$ ${\textstyle k^{\prime }}$ denotabimus.

Dividatur sueto more ${\textstyle \zeta }$ per ${\textstyle \zeta ^{\prime },}$ donec residui ordo infra ${\textstyle k^{\prime }}$ depressus sit; statuatur residuum ${\textstyle ={\frac {\zeta ^{\prime \prime }}{\lambda }},}$ eiusque ordo ${\textstyle =k^{\prime \prime },}$ ita ut ${\textstyle {\frac {1}{\lambda }}z^{k^{\prime \prime }}}$ sit residui terminus altissimus; divisionis quotientem ponemus ${\textstyle ={\frac {p}{\lambda }}.}$ Perinde ex divisione functionis ${\textstyle \zeta ^{\prime }}$ per ${\textstyle \zeta ^{\prime \prime }}$ prodeat residuum ${\textstyle {\frac {\zeta ^{\prime \prime \prime }}{\lambda ^{\prime }}}}$ ordinis ${\textstyle k^{\prime \prime \prime },}$ quotiens ${\textstyle {\frac {p^{\prime }}{\lambda ^{\prime }}};}$ dein rursus e divisione functionis ${\textstyle \zeta ^{\prime \prime }}$ per ${\textstyle \zeta ^{\prime \prime \prime }}$ prodeat residuum ${\textstyle {\frac {\zeta ^{\prime \prime \prime \prime }}{\lambda ^{\prime \prime }}}}$ ordinis ${\textstyle k^{\prime \prime \prime \prime }}$ atque quotiens ${\textstyle {\frac {p^{\prime \prime }}{\lambda ^{\prime \prime }}}}$ et sic porro, donec in serie functionum ${\textstyle \zeta ^{\prime \prime },}$ ${\textstyle \zeta ^{\prime \prime \prime },}$ ${\textstyle \zeta ^{\prime \prime \prime \prime }}$ etc., quae singulae terminum suum altissimum coëfficiente ${\textstyle 1}$ affectum habebunt, perveniatur ad ${\textstyle \zeta ^{(m)}=1.}$ Hoc tandem evenire debere facile perspicitur, quum quaelibet functionum ${\textstyle \zeta ,}$ ${\textstyle \zeta ^{\prime },}$ ${\textstyle \zeta ^{\prime \prime },}$ ${\textstyle \zeta ^{\prime \prime \prime }}$ etc. cum praecedenti divisorem communem indeterminatum habere nequeat, adeoque certo divisio absque residuo fieri nequeat, quamdiu divisor fuerit ordinis maioris quam ${\textstyle 0.}$ Habebimus igitur seriem aequationum

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

etc. usque ad

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

ubi ${\textstyle \zeta ^{\prime \prime },}$ ${\textstyle \zeta ^{\prime \prime \prime },}$ ${\textstyle \zeta ^{\prime \prime \prime \prime }\ldots \zeta ^{(m)}}$ sunt functiones integrae ipsius ${\textstyle z}$ ordinis ${\textstyle k^{\prime \prime },}$ ${\textstyle k^{\prime \prime \prime },}$ ${\textstyle k^{\prime \prime \prime \prime }\ldots k^{(m)};}$ numeri ${\textstyle k^{\prime },}$ ${\textstyle k^{\prime \prime },}$ ${\textstyle k^{\prime \prime \prime }\ldots k^{(m)}}$ continuo decrescentes usque ad ultimum ${\textstyle k^{(m)}=0;}$ ${\textstyle p,}$ ${\textstyle p^{\prime },}$ ${\textstyle p^{\prime \prime },}$ ${\textstyle p^{\prime \prime \prime }}$ etc. quoque functiones integrae ipsius ${\textstyle z}$ ordinis ${\textstyle k-k^{\prime },}$ ${\textstyle k^{\prime }-k^{\prime \prime },}$ ${\textstyle k^{\prime \prime }-k^{\prime \prime \prime },}$ ${\textstyle k^{\prime \prime \prime }-k^{\prime \prime \prime \prime }}$ etc. (excepto casu, ubi ${\textstyle k<k^{\prime },}$ in quo manifesto statui debet ${\textstyle p=0}$).

His ita praeparatis formamus secundam seriem functionum integrarum ipsius ${\textstyle z,}$ puta ${\textstyle \eta ,}$ ${\textstyle \eta ^{\prime },}$ ${\textstyle \eta ^{\prime \prime },}$ ${\textstyle \eta ^{\prime \prime \prime }\ldots \eta ^{(m)}.}$ Et quidem statuemus ${\textstyle \eta =1,}$ ${\textstyle \eta ^{\prime }=0,}$ reliquas vero