Pagina:Werkecarlf03gausrich.djvu/369

$\int {\frac {{\cos T}^{2}\operatorname {d} T}{2\pi (mm{\cos T}^{2}+nn{\sin T}^{2})^{\frac {3}{2}}}},\quad \int {\frac {{\sin T}^{2}\operatorname {d} T}{2\pi (mm{\cos T}^{2}+nn{\sin T}^{2})^{\frac {3}{2}}}}$

a $T=0$ usque ad $T=360^{\circ }$ extensorum, primo statim obvium est, haberi

$mmP+nnQ={\frac {1}{\mu }}\ldots \ldots \ldots \ldots \ldots \ldots$[24]

Porro fit

${\begin{alignedat}{1}{\frac {({\cos T}^{2}-{\sin T}^{2})\operatorname {d} T}{2\pi {\sqrt {mm{\cos T}^{2}+nn{\sin T}^{2}}}}}+&{\frac {(mm{\cos T}^{2}-nn{\sin T}^{2})\operatorname {d} T}{2\pi (mm{\cos T}^{2}+nn{\sin T}^{2})^{\frac {3}{2}}}}\\&={\frac {(mm{\cos T}^{4}-nn{\sin T}^{4})\operatorname {d} T}{\pi (mm{\cos T}^{2}+nn{\sin T}^{2})^{\frac {3}{2}}}}\\&=\operatorname {d} .{\frac {\cos T\sin T}{\pi {\sqrt {mm{\cos T}^{2}+nn{\sin T}^{2}}}}}\end{alignedat}}$

Integrando hanc aequationem a $T=0$ usque ad $T=360^{\circ },$ prodit

$-{\frac {\nu }{\mu }}+mmP-nnQ=0\ldots \ldots \ldots \ldots$[25]

E combinatione aequationum 24, 25 denique colligimus

$P={\frac {1+\nu }{2mm\mu }},\quad Q={\frac {1-\nu }{2nn\mu }}$

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