Pagina:Werkecarlf03gausrich.djvu/366

16.

Quod attinet ad quantitates $P,$ $Q,$ manifesto quidem utraque fit

$={\frac {1}{2(G+G^{\prime })^{\frac {3}{2}}}}$

quoties $G^{\prime }=G^{\prime \prime },$ in omnibus vero reliquis casibus ad transscendentes sunt referendae. Quas quomodo per series exprimere liceat, abunde constat. Lectoribus autem gratum fore speramus, si hacce occasione determinationem harum aliarumque transscendentium per algorithmum peculiarem expeditissimum explicemus, quo per multos iam abhinc annos frequenter usi sumus, et de quo alio loco copiosius agere propositum est.

Sint $m,$ $n$ duae quantitates positivae, statuamusque

$m^{\prime }={\tfrac {1}{2}}(m+n),\quad n^{\prime }={\sqrt {mn}}$

ita ut $m^{\prime },$ $n^{\prime }$ resp. sit medium arithmeticum et geometricum inter $m$ et $n.$ Medium geometricum semper positive accipi supponemus. Perinde fiat

${\begin{array}{ll}m^{\prime \prime }={\frac {1}{2}}(m^{\prime }+n^{\prime }),&n^{\prime \prime }={\sqrt {m^{\prime }n^{\prime }}}\\m^{\prime \prime \prime }={\frac {1}{2}}(m^{\prime \prime }+n^{\prime \prime }),&n^{\prime \prime \prime }={\sqrt {m^{\prime \prime }n^{\prime \prime }}}\end{array}}$

et sic porro, quo pacto series $m,$ $m^{\prime },$ $m^{\prime \prime },$ $m^{\prime \prime \prime }$ etc., atque $n,$ $n^{\prime },$ $n^{\prime \prime },$ $n^{\prime \prime \prime }$ etc. versus limitem communem rapidissime convergent, quem per $\mu$ designabimus, atque simpliciter medium arithmetico-geometricum inter $m$ et $n$ vocabimus. Iam demonstrabimus, ${\tfrac {1}{\mu }}$ esse valorem integralis

$\int {\frac {\operatorname {d} T}{2\pi {\sqrt {mm{\cos T}^{2}+nn{\sin T}^{2}}}}}$

a $T=0$ usque ad $T=360^{\circ }$ extensi.

Demonstr. Supponamus, variabilem $T$ ita per aliam $T^{\prime }$ exprimi, ut fiat

$\sin T={\frac {2m\sin T^{\prime }}{(m+n){\cos T^{\prime }}^{2}+2m{\sin T^{\prime }}^{2}}}$

perspicieturque facile, dum $T^{\prime }$ a valore $0$ usque ad $90^{\circ },$ $180^{\circ },$ $270^{\circ },$ $360^{\circ }$ augeatur, etiam $T$ (etsi inaequalibus intervallis) a $0$ usque ad $90^{\circ },$ $180^{\circ },$ $270^{\circ },$ $360^{\circ }$ crescere. Evolutione autem rite facta, invenitur esse

${\frac {\operatorname {d} T}{\sqrt {mm{\cos T}^{2}+nn{\sin T}^{2}}}}={\frac {\operatorname {d} T^{\prime }}{\sqrt {m^{\prime }m^{\prime }{\cos T^{\prime }}^{2}+n'n'{\sin T^{\prime }}^{2}}}}$