Pagina:Werkecarlf03gausrich.djvu/367

adeoque valores integralium

$\int {\frac {\operatorname {d} T}{2\pi {\sqrt {mm{\cos T}^{2}+nn{\sin T}^{2}}}}},\quad \int {\frac {\operatorname {d} {T}^{\prime }}{2\pi {\sqrt {m^{\prime }m^{\prime }{\cos T^{\prime }}^{2}+n^{\prime }n^{\prime }{\sin T^{\prime }}^{2}}}}}$

si utriusque variabilis a valore $0$ usque ad valorem $360^{\circ }$ extenditur, inter se aequales. Et quum perinde ulterius continuare liceat, patet, his valoribus etiam aequalem esse valorem integralis

$\int {\frac {d\Theta }{2\pi {\sqrt {\mu \mu {\cos \Theta }^{2}+\mu \mu {\sin \Theta }^{2}}}}}$

a $\Theta =0$ usque ad $\Theta =360^{\circ },$ qui manifesto fit $={\tfrac {1}{\mu }}.$ Q. E. D.

17.

Ex aequatione, relationem inter $T$ et $T^{\prime }$ exhibente,

$(m-n)\sin T.{\sin T^{\prime }}^{2}=2m\sin T^{\prime }-(m+n)\sin T$

facile deducitur

${\begin{array}{l}{\sqrt {mm{\cos T^{\vphantom {\prime }}}^{2}+nn{\sin T^{\vphantom {\prime }}}^{2}}}=m-(m-n)\sin T.\sin T^{\prime }\\{\sqrt {m^{\prime }m^{\prime }{\cos T^{\prime }}^{2}+n^{\prime }n^{\prime }{\sin T^{\prime }}^{2}}}=m\operatorname {cotang} T.\operatorname {tang} T^{\prime }\end{array}}$

atque hinc, adiumento eiusdem aequationis,

${\begin{array}{l}{\phantom {=}}\sin T.\sin T^{\prime }.{\sqrt {mm{\cos T}^{2}+nn{\sin T}^{2}}}+m^{\prime }({\cos T}^{2}-{\sin T}^{2})\\=\cos T.\cos T^{\prime }.{\sqrt {m^{\prime }m^{\prime }{\cos T^{\prime }}^{2}+n^{\prime }n^{\prime }{\sin T^{\prime }}^{2}}}-{\frac {1}{2}}(m-n){\sin T}^{2}\end{array}}$

Multiplicata hac aequatione per

${\frac {\operatorname {d} T}{\sqrt {mm{\cos T}^{2}+nn{\sin T}^{2}}}}={\frac {\operatorname {d} T^{\prime }}{\sqrt {m^{\prime }m^{\prime }{\cos T^{\prime }}^{2}+n^{\prime }n^{\prime }{\sin T^{\prime }}^{2}}}}$

prodit

${\frac {m^{\prime }({\cos T}^{2}-{\sin T}^{2})\operatorname {d} T}{\sqrt {mm{\cos T}^{2}+nn{\sin T}^{2}}}}=-{\frac {{\frac {1}{2}}(m-n){\sin T^{\prime }}^{2}\operatorname {d} T^{\prime }}{\sqrt {m^{\prime }m^{\prime }{\cos T^{\prime }}^{2}+n^{\prime }n^{\prime }{\sin T}^{2}}}}+\operatorname {d} .\sin T^{\prime }\cos T$

Multiplicando hanc aequationem per ${\tfrac {m-n}{\pi }},$ substituendo $m^{\prime }(m-n)={\tfrac {1}{2}}(mm-nn),$ $(m-n)^{2}=4(m^{\prime }m^{\prime }-n^{\prime }n^{\prime }),$ ${\sin T}^{2}={\tfrac {1}{2}}-{\tfrac {1}{2}}({\cos T^{\prime }}^{2}-{\sin T^{\prime }}^{2}),$ et integrando, a valoribus $T$ et $T^{\prime }=0$ usque ad $360^{\circ },$ habemus:

${\begin{aligned}&(mm-nn)\int {\frac {({\cos T}^{2}-{\sin T}^{2}).\operatorname {d} T}{2\pi {\sqrt {mm{\cos T}^{2}+nn{\sin T}^{2}}}}}\\&=-{\frac {2(m^{\prime }m^{\prime }-n^{\prime }n^{\prime })}{\mu }}+2(m^{\prime }m^{\prime }-n^{\prime }n^{\prime })\int {\frac {({\cos T^{\prime }}^{2}-{\sin T^{\prime }}^{2})dT^{\prime }}{2\pi {\sqrt {m^{\prime }m^{\prime }{\cos T^{\prime }}^{2}+n^{\prime }n^{\prime }{\sin T^{\prime }}^{2}}}}}\end{aligned}}$