Pagina:Werkecarlf03gausrich.djvu/358

tionis 13 radix positiva statuenda est $=G,$ atque vel negativa $=-G^{\prime },$ et $G^{\prime \prime }=0,$ vel radix negativa $=-G^{\prime \prime },$ et $G^{\prime }=0;$ coëfficientem $\gamma ^{\prime \prime }$ vel $\gamma ^{\prime }$ vero inveniemus per formulam

${\frac {1}{\sqrt {{\frac {AA}{aa}}+{\frac {BB}{bb}}-1}}}$

Ceterum in casu iam excluso, ubi punctum attractum in ipsa circumferentia ellipsis situm supponeretur, coëfficientes $\gamma$ et $\gamma ^{\prime },$ vel $\gamma$ et $\gamma ^{\prime \prime }$ evaderent infiniti, quod indicat, transformationem nostram ad hunc casum omnino non esse applicabilem.

8.

Quamquam formulae 15, 16 ad determinationem coëfficientium $\gamma ,$ $\gamma ^{\prime },$ $\gamma ^{\prime \prime }$ sufficere possent, tamen etiam elegantiores assignare licet. Ad hunc finem multiplicabimus aequationem [5] per $aabb-GG,$ unde prodit, levi reductione facta,

${\frac {aaAA(bb+G)}{aa+G}}-AAG+{\frac {bbBB(aa+G)}{bb+G}}-BBG+{\frac {aabbCC}{G}}-CCG=aabb-GG$

Sed e natura aequationis cubicae fit

summa radicum $\quad G-G^{\prime }-G^{\prime \prime }=AA+BB+CC-aa-bb$

productum radicum $\quad GG^{\prime }G^{\prime \prime }=aabbCC$

Hinc aequatio praecedens transit in sequentem:

${\frac {aaAA(bb+G)}{aa+G}}+{\frac {bbBB(aa+G)}{bb+G}}+G^{\prime }G^{\prime \prime }-G(G-G^{\prime }-G^{\prime \prime }+aa+bb)=aabb-GG$

quam etiam sic exhibere licet

${\frac {aaAA(bb+G)}{aa+G}}+{\frac {bbBB(aa+G)}{bb+G}}-(aa+G)(bb+G)+(G+G^{\prime })(G+G^{\prime \prime })=0$

Hinc valor coëfficientis $\gamma$ e formula prima in [15] transmutatur in sequentem:

$\gamma ={\sqrt {\frac {(aa+G)(bb+G)}{(G+G^{\prime })(G+G^{\prime \prime })}}}\ldots \ldots \ldots \ldots \;$[17]

Per analysin prorsus similem invenitur

$\gamma ^{\prime {\phantom {\prime }}}={\sqrt {\frac {(aa-G^{\prime })(bb-G^{\prime })}{(G+G^{\prime })(G^{\prime \prime }-G^{\prime })}}}\ldots \ldots \ldots \ldots \,$	[18]
$\gamma ^{\prime \prime }={\sqrt {\frac {(aa-G^{\prime \prime })(bb-G^{\prime \prime })}{(G+G^{\prime \prime })(G^{\prime }-G^{\prime \prime })}}}\ldots \ldots \ldots \ldots \,$	[19]