Pagina:Werkecarlf03gausrich.djvu/193

scilicet deinceps dividendo ${\textstyle Z}$ per ${\textstyle \zeta ^{\prime },}$ dein residuum primae divisionis ${\textstyle Z^{\prime }}$ per ${\textstyle \zeta ^{\prime \prime },}$ tum residuum secundae divisionis per ${\textstyle \zeta ^{\prime \prime \prime }}$ ac sic porro. Quum residuum semper ad ordinem inferiorem pertineat quam divisor, ordo functionum ${\textstyle Z^{\prime },}$ ${\textstyle Z^{\prime \prime },}$ ${\textstyle Z^{\prime \prime \prime },}$ ${\textstyle Z^{\prime \prime \prime }}$ etc. erit resp. inferior quam ${\textstyle k^{\prime },}$ ${\textstyle k^{\prime \prime },}$ ${\textstyle k^{\prime \prime \prime },}$ ${\textstyle k^{\prime \prime \prime \prime }}$ etc.; ultima vero ${\textstyle Z^{(m)}}$ necessario fit ${\textstyle =0,}$ quum divisor ${\textstyle \zeta ^{(m)}}$ sit ${\textstyle =1.}$ Habemus itaque

$$Z=q^{\prime }\zeta ^{\prime }+q^{\prime \prime }\zeta ^{\prime \prime }+q^{\prime \prime \prime }\zeta ^{\prime \prime \prime }+q^{\prime \prime \prime \prime }\zeta ^{\prime \prime \prime \prime }+{\text{etc.}}+q^{(m)}\zeta ^{(m)}$$

Quatenus autem pro ${\textstyle z}$ solae radices aequationis ${\textstyle \zeta ^{\prime }=0}$ accipiuntur, fit ${\textstyle \zeta ^{\prime }=0,}$ ${\textstyle \zeta ^{\prime \prime }=\zeta \eta ^{\prime \prime },}$ ${\textstyle \zeta ^{\prime \prime \prime }=\zeta \eta ^{\prime \prime \prime },}$ ${\textstyle \zeta ^{\prime \prime \prime \prime }=\zeta \eta ^{\prime \prime \prime \prime }}$ etc., unde sub eadem restrictione erit

$${\frac {Z}{\zeta }}=q^{\prime \prime }\eta ^{\prime \prime }+q^{\prime \prime \prime }\eta ^{\prime \prime \prime }+q^{\prime \prime \prime \prime }\eta ^{\prime \prime \prime \prime }+{\text{etc.}}+q^{(m)}\eta ^{(m)}$$

Ordo vero huius expressionis necessario erit infra ${\textstyle k^{\prime }{:}}$ quum enim ordo quotientium ${\textstyle q^{\prime \prime },}$ ${\textstyle q^{\prime \prime \prime },}$ ${\textstyle q^{\prime \prime \prime }}$ etc. esse debeat infra ${\textstyle k^{\prime }-k^{\prime \prime },}$ ${\textstyle k^{\prime \prime }-k^{\prime \prime \prime },}$ ${\textstyle k^{\prime \prime \prime }-k^{\prime \prime \prime \prime }}$ etc., ordo singularum partium ${\textstyle q^{\prime \prime }\eta ^{\prime \prime },}$ ${\textstyle q^{\prime \prime \prime }\eta ^{\prime \prime \prime },}$ ${\textstyle q^{\prime \prime \prime \prime }\eta ^{\prime \prime \prime \prime }}$ etc. erit infra ${\textstyle k^{\prime }-k^{\prime \prime },}$ ${\textstyle k^{\prime }-k^{\prime \prime \prime },}$ ${\textstyle k^{\prime }-k^{\prime \prime \prime }}$ etc.

Denique adhuc observamus, si forte inter valores indeterminatae ${\textstyle z,}$ quos in fractione ${\textstyle {\frac {Z}{\zeta }}}$ substituere oporteat, rationales cum irrationalibus mixti reperiantur, magis e re fore, illos ab his separare atque hos solos in aequatione ${\textstyle \zeta ^{\prime }=0}$ comprehendere. Pro rationalibus enim valoribus calculi compendio opus non erit; pro irrationalibus autem calculus tanto simplicior erit, quo minor fuerit gradus functionis integrae, ad quam fractam reducere licet.

12.

Ecce nunc exemplum transformationis in art. praec explicatae. Proposita sit functio fracta

$${\frac {z^{6}-{\frac {50}{39}}z^{4}+{\frac {283}{715}}zz-{\frac {256}{15015}}}{7z^{6}-{\frac {105}{13}}z^{4}+{\frac {315}{143}}zz-{\frac {35}{429}}}}$$

in qua ${\textstyle z}$ indefinite repraesentat radices aequationis

$$z^{7}-{\frac {21}{13}}z^{5}+{\frac {105}{143}}z^{3}-{\frac {35}{429}}z=0$$

Si hic omnes septem radices complecti vellemus, ad functionem integram sexti ordinis delaberemur. Manifesto autem pro valore rationali ${\textstyle z=0}$ computus fractionis obvius est, datque valorem ${\textstyle {\frac {256}{1225}}{:}}$ quapropter seposita hac radice in aequatione sexti gradus subsistemus: