Pagina:Werkecarlf03gausrich.djvu/29

$Z$ rationaliter determinabilem fore asserit clar. de F., et proin realem. Hinc sequitur, radices aequationis $u'u'+u''u'+M''=0$ sub forma $p+q\surd {-1}$ contentas fore; eaedem vero manifesto aequationi $U'=0$ satisfacient: quare dabitur valor aliquis ipsius $u'$ sub forma $p+q\surd {-1}$ contentus. Iam coëfficiens $M'$ (eodem modo ut ante) rationaliter per $u'$ et coëfficientes in $Z$ determinari potest, adeoque etiam sub forma $p+q\surd {-1}$ contentus erit; quare aequationis $uu+u'u+M'$ radices sub eadem forma contentae erunt, simul vero aequationi $U=0$ satisfacient, i.e. aequatio haec habebit radicem sub forma $p+q\surd {-1}$ contentam. Denique hinc simili ratione sequitur, etiam $M$ sub eadem forma contineri, nec non radicem aequationis $zz+uz+M=0,$ quae manifesto etiam aequationi propositae $Z=0$ satisfaciet. Quamobrem quaevis aequatio ad minimum unam radicem formae $p+q\surd {-1}$ habebit.

11.

Obiectiones 1, 2, 3, quas contra Euleri demonstrationem primam feci (art. 8), eandem vim contra hanc methodum habent, ea tamen differentia, ut obiectio secunda, cui Euleri demonstratio tantummodo in quibusdam casibus specialibus obnoxia erat, praesentem in omnibus casibus attingere debeat. Scilicet a priori demonstrari potest, etiamsi formula detur, quae coëfficientem $M'$ rationaliter per $u'$ et coëfficientes in $Z$ exprimat, hanc pro pluribus valoribus ipsius $u'$ necessario indeterminatam fieri debere; similiterque formulam, quae coëfficientem $M''$ per $u''$ exhibeat, indeterminatam fieri pro quibusdam valoribus ipsius $u''$ etc. Hoc luculentissime perspicietur, si aequationem quarti gradus pro exemplo assumimus. Ponamus itaque $m=4,$ sintque radices aequationis $Z=0,$ hae $\alpha ,$ $\beta ,$ $\gamma ,$ $\delta .$ Tum patet, aequationem $U=0$ fore sexti gradus ipsiusque radices $-(\alpha +\beta ),$ $-(\alpha +\gamma ),$ $-(\alpha +\delta ),$ $-(\beta +\gamma ),$ $-(\beta +\delta ),$ $-(\gamma +\delta ).$ Aequatio $U'=0$ autem erit decimi quinti gradus, et valores ipsius $u'$ hi

$${\begin{array}{c}{\begin{array}{cccccc}2\alpha +\beta +\gamma ,&2\alpha +\beta +\delta ,&2\alpha +\gamma +\delta ,&2\beta +\alpha +\gamma ,&2\beta +\alpha +\delta ,&2\beta +\gamma +\delta ,\\2\gamma +\alpha +\beta ,&2\gamma +\alpha +\delta ,&2\gamma +\beta +\delta ,&2\delta +\alpha +\beta ,&2\delta +\alpha +\gamma ,&2\delta +\beta +\gamma ,\end{array}}\\{\begin{array}{ccc}\alpha +\beta +\gamma +\delta ,&\alpha +\beta +\gamma +\delta ,&\alpha +\beta +\gamma +\delta \end{array}}\end{array}}$$

Iam in hac aequatione, quippe cuius gradus est impar, subsistendum erit, habebitque ea revera radicem realem $\alpha +\beta +\gamma +\delta$ (quae primo coëfficienti in $Z$ mutato signo aequalis adeoque non modo realis sed etiam rationalis erit, si coëfficien-