Pagina:Werkecarlf03gausrich.djvu/361

formula 17 suppeditat

$\gamma ={\sqrt {\frac {aabbAA}{(aa-bb)(aaCC+b^{4})}}}={\sqrt {\frac {aaCC+aabb}{aaCC+b^{4}}}}$

ac dein formulae 3, 4

${\begin{aligned}&\alpha ={\frac {\gamma (aa-bb)}{aA}}={\frac {\gamma bbA}{a(CC+bb)}}={\sqrt {\frac {bb(aa-bb)}{aaCC+b^{4}}}}={\sqrt {\frac {b^{4}AA}{(CC+bb)(aaCC+b^{4})}}}\\&\beta =0\end{aligned}}$

Valores coëfficientium $\gamma ^{\prime },$ $\gamma ^{\prime \prime }$ per formulas 18, 19 in hoc casu indeterminati manent, atque sic etiam valores coëfficientium reliquorum $\alpha ^{\prime },$ $\beta ^{\prime },$ $\alpha ^{\prime \prime },$ $\beta ^{\prime \prime }.$ Nihilominus per unum horum coëfficientium omnes quinque reliqui exprimi possunt, e.g. fit per formulam 6

$\alpha ^{\prime }={\frac {\gamma ^{\prime }aA}{aa-bb}}$

ac dein

$\beta ^{\prime }={\sqrt {(1-\alpha ^{\prime }\alpha ^{\prime }+\gamma ^{\prime }\gamma ^{\prime })}},\quad$$\gamma ^{\prime \prime }={\sqrt {(\gamma \gamma -1-\gamma ^{\prime }\gamma ^{\prime })}},\quad$$\alpha ^{\prime \prime }={\frac {\gamma ^{\prime \prime }aA}{aa-bb}},\quad$$\beta ^{\prime \prime }={\sqrt {(1-\alpha ^{\prime \prime }\alpha ^{\prime \prime }+\gamma ^{\prime \prime }\gamma ^{\prime \prime })}}$

Sed concinnius hoc ita perficitur. Ex

$\gamma \gamma =1+\alpha \alpha ,\quad \alpha \alpha ^{\prime }=\gamma \gamma ^{\prime },\quad 1=\alpha ^{\prime }\alpha ^{\prime }+\beta ^{\prime }\beta ^{\prime }-\gamma ^{\prime }\gamma ^{\prime }$

sequitur

$\beta ^{\prime }\beta ^{\prime }+{\frac {\gamma ^{\prime }\gamma ^{\prime }}{\alpha \alpha }}=1-\alpha ^{\prime }\alpha ^{\prime }+{\frac {\gamma \gamma \gamma ^{\prime }\gamma ^{\prime }}{\alpha \alpha }}=1$

Quapropter statuere possumus

$\beta ^{\prime }=\cos f,\quad \gamma ^{\prime }=\alpha \sin f,\quad \alpha ^{\prime }=\gamma \sin f$

Dein vero e formulis

$\varepsilon \alpha ^{\prime \prime }=\beta \gamma ^{\prime }-\gamma \beta ^{\prime },\quad$$\varepsilon \beta ^{\prime \prime }=\gamma \alpha ^{\prime }-\alpha \gamma ^{\prime },\quad$$\varepsilon \gamma ^{\prime \prime }=\beta \alpha ^{\prime }-\alpha \beta ^{\prime },\quad$$\varepsilon \varepsilon =1$

invenimus

$\alpha ^{\prime \prime }=-\varepsilon \gamma \cos f,\quad$$\beta ^{\prime \prime }=\varepsilon \sin f,\quad$$\gamma ^{\prime \prime }=-\varepsilon \alpha \cos f$

Valor anguli $f$ hic arbitrarius est, nec non pro lubitu statui poterit vel $\varepsilon =+1$ vel $\varepsilon =-1.$

11.

Si $G^{\prime },$ $G^{\prime \prime }$ sunt inaequales, valores coëfficientium $\gamma ,$ $\gamma ^{\prime },$ $\gamma ^{\prime \prime }$ per formulas 17, 18, 19 indeterminati esse nequeunt, sed quoties aliqua quantitatum