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The functions of Schläfli and Lobatschefsky. (English) JFM 61.0395.02
Es handelt sich um die gemeinsame Ableitung der Formeln, die {\it Schläfli} und {\it Lobatschefsky} für das Volumen eines doppelt rechtwinkligen Tetraeders im elliptischen bzw. hyperbolischen Raum gegeben haben. Dabei kommen Reihen von der Form $$ S(\alpha,\beta,\gamma)=\sum\limits_{n=1}^{\infty}\frac{(-X)^n}{n^2} (\cos(2n\alpha)-\cos(2n\beta)+\cos(2n\gamma)-1)-\alpha^2+\beta^2-\gamma^2 $$ zur Anwendung, wo ist: $$ \align X&=\frac{\sin\alpha\sin\gamma-D}{\sin\alpha\sin\gamma+D}, D=\sqrt{\cos^2\alpha\cdot\cos^2\gamma-\cos^2\beta},\\ 0&\leqq\alpha\leqq\frac\pi2,\quad0\leqq\beta\leqq\pi,\quad0\leqq\gamma\leqq\frac\pi2; \endalign $$ sie hängen mit dem {\it Schläfli}schen Resultat ${\ssize\frac18}\pi^2f(\alpha,\beta,\gamma)$ durch die Beziehung $$ S(\alpha,\beta,\gamma)={\ssize\frac12}\pi^2f({\ssize\frac12}\pi-\alpha,\beta, {\ssize\frac12}\pi-\gamma) $$ zusammen und werden als {\it Schläfli}sche Funktionen bezeichnet. Zum Schlüsse erfolgt Anwendung auf die regulären Polyeder im hyperbolischen Raum. (V 1.)
Reviewer: Volk, O.; Prof. (Würzburg)