Pagina:Werkecarlf03gausrich.djvu/162

[49]	$F(\alpha ,\beta ,\gamma ,1)=F(-\alpha ,-\beta ,\gamma -\alpha -\beta ,1)$
[50]	$F(\alpha ,\beta ,\gamma ,1).F(-\alpha ,\beta ,\gamma -\alpha ,1)=1$
[51]	$F(\alpha ,\beta ,\gamma ,1).F(\alpha ,-\beta ,\gamma -\beta ,1)=1$

in quarum prima $\gamma ,$ in secunda $\gamma -\beta ,$ in tertia $\gamma -\alpha$ debet esse quantitas positiva.

25.

Applicemus formulam 48 ad quasdam ex aequationibus art. 5. Formula XIII, statuendo $t=90^{0}={\tfrac {1}{2}}\pi ,$ fit ${\tfrac {1}{2}}\pi =F({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {3}{2}},1),$ sive aequivalet aequationi notae

${\tfrac {1}{2}}\pi =1+{\tfrac {1.1}{2.3}}+{\tfrac {1.1.3}{2.4.5}}+{\tfrac {1.1.3.5}{2.4.6.7}}+$ etc.

Quare quum per formulam 48 habeatur $F({\tfrac {1}{2}},{\tfrac {1}{2}},{\tfrac {3}{2}},1)={\frac {\Pi {\tfrac {1}{2}}.\Pi (-{\tfrac {1}{2}})}{\Pi {0}.\Pi {0}}},$ atque sit $\Pi {0}=1,$ $\Pi {\tfrac {1}{2}}={\tfrac {1}{2}}\Pi (-{\tfrac {1}{2}}),$ fit $\pi =(\Pi (-{\tfrac {1}{2}}))^{2}$ sive

[52]	$\Pi (-{\tfrac {1}{2}})={\sqrt {\pi }}$
[53]	$\Pi {\tfrac {1}{2}}={\tfrac {1}{2}}{\sqrt {\pi }}$

Formula XVI art. 5, quae aequivalet aequationi notae

$\sin nt=n\sin t-{\tfrac {n(nn-1)}{2\;.\;3}}{\sin t}^{3}+{\tfrac {n(nn-1)(nn-9)}{2\;.\;3\;.\;4\;.\;5}}{\sin t}^{5}-$ etc.

atque generaliter pro quovis valore ipsius $n$ locum habet, si modo $t$ limites $-90^{\circ }$ et $+90^{\circ }$ non transgrediatur, dat pro $t={\tfrac {1}{2}}\pi$

$\sin {\tfrac {n\pi }{2}}={\frac {n\Pi {\tfrac {1}{2}}.\Pi (-{\tfrac {1}{2}})}{\Pi (-{\tfrac {1}{2}}n).\Pi {\tfrac {1}{2}}n}}$

unde deducitur formula elegans

$\Pi {\tfrac {1}{2}}n.\Pi (-{\tfrac {1}{2}}n)={\frac {{\tfrac {1}{2}}n\pi }{\sin {\tfrac {1}{2}}n\pi }},$ sive statuendo $n=2z$

[54]	$\Pi (-z).\Pi (+z)={\frac {z\pi }{\sin z\pi }}$
[55]	$\Pi (-z).\Pi (z-1)={\frac {\pi }{\sin z\pi }}$

nec non scribendo $z+{\tfrac {1}{2}}$ pro $z$

[56]	$\Pi (-{\tfrac {1}{2}}+z).\Pi (-{\tfrac {1}{2}}-z)={\frac {\pi }{\cos z\pi }}$