Pagina:Werkecarlf03gausrich.djvu/357

$\left.{\begin{aligned}\gamma &={\frac {1}{\sqrt {1-\left({\frac {aA}{aa+G}}\right)^{2}+\left({\frac {bB}{bb+G}}\right)^{2}}}}\\\gamma ^{\prime }&={\frac {1}{\sqrt {\left({\frac {aA}{aa-G^{\prime }}}\right)^{2}+\left({\frac {bB}{bb-G^{\prime }}}\right)^{2}-1}}}\\\gamma ^{\prime \prime }&={\frac {1}{\sqrt {\left({\frac {aA}{aa-G^{\prime \prime }}}\right)^{2}+\left({\frac {bB}{bb-G^{\prime \prime }}}\right)^{2}-1}}}\end{aligned}}\right\}$ [15]

Ex his aequationibus rite cum 5, 8, 11 combinatis etiam sequitur:

$\left.{\begin{aligned}\gamma &={\sqrt {\frac {G}{\left({\frac {AG}{aa+G}}\right)^{2}+\left({\frac {BG}{bb+G}}\right)^{2}+CC}}}\\\gamma ^{\prime }&={\sqrt {\frac {G^{\prime }}{\left({\frac {AG^{\prime }}{aa-G^{\prime }}}\right)^{2}+\left({\frac {BG^{\prime }}{bb-G^{\prime }}}\right)^{2}+CC}}}\\\gamma ^{\prime \prime }&={\sqrt {\frac {G^{\prime \prime }}{\left({\frac {AG^{\prime \prime }}{aa-G^{\prime \prime }}}\right)^{2}+\left({\frac {BG^{\prime \prime }}{bb-G^{\prime \prime }}}\right)^{2}+CC}}}\end{aligned}}\right\}$ [16]

Hae posteriores expressiones ostendunt, nullam quantitatum $G,$ $G^{\prime },$ $G^{\prime \prime }$ negativam esse posse, siquidem $\gamma ,$ $\gamma ^{\prime },$ $\gamma ^{\prime \prime }$ debent esse reales.

In casu itaque eo, ubi non est $C=0,$ necessario $G$ aequalis statui debet radici positivae aequationis $B,$ patetque adeo, $-G^{\prime }$ aequalem esse debere alteri radici negativae, atque $-G^{\prime \prime }$ aequalem alteri^[1]; utram vero radicem pro $-G^{\prime },$ utram pro $-G^{\prime \prime }$ adoptemus, prorsus arbitrarium erit.

Quoties $C=0,$ punctumque attractum intra curvam situm, duas radices negativas aequationis 13 necessario pro $-G^{\prime }$ et $-G^{\prime \prime }$ adoptare et proin $G=0$ statuere oportet. Quoniam vero in hoc casu formula prima in 16 fit indeterminata, formulam primam in 15 eius loco retinebimus, quae suppeditat

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

Quoties autem pro $C=0$ punctum attractum extra ellipsin iacet, aequa-

1. Proprie quidem ex analysi praecedenti tantummodo sequitur, $-G^{\prime }$ et $-G^{\prime \prime }$ satisfacere debere aequationi 13, unde dubium esse videtur, annon liceat, utramque $-G^{\prime }$ et $-G^{\prime \prime }$ eidem radici negativae aequalem ponere, prorsus neglecta radice tertia. Sed facile perspicietur, siquidem aequationis radix secunda et tertia sint inaequales, ex $-G^{\prime }=-G^{\prime \prime }$ sequi $\gamma ^{\prime }=\gamma ^{\prime \prime },$ $\alpha ^{\prime }=\alpha ^{\prime \prime },$ $\ell ^{\prime }=\ell ^{\prime \prime },$ et proin $-a^{\prime }a^{\prime \prime }-b^{\prime }b^{\prime \prime }+\gamma ^{\prime }\gamma ^{\prime \prime }=-a^{\prime }a^{\prime }-b^{\prime }b^{\prime }+\gamma ^{\prime }\gamma ^{\prime }=1,$ quod aequationi quartae in [1] est contrarium. Conf. quae infra de casu duarum radicum aequalium aequationis 13 dicentur.