Pagina:Werkecarlf03gausrich.djvu/72

$${\begin{array}{rl}mr^{m}\cos m\varphi +(m-1)Ar^{m-1}\cos(m-1)\varphi +(m-2)Br^{m-2}\cos(m-2)\varphi &\\+(m-3)Cr^{m-3}\cos(m-3)\varphi +{\text{ etc. }}+Lr\cos \varphi &=t'\\mr^{m}\sin m\varphi +(m-1)Ar^{m-1}\sin(m-1)\varphi +(m-2)Br^{m-2}\sin(m-2)\varphi &\\+(m-3)Cr^{m-3}\sin(m-3)\varphi +{\text{ etc. }}+Lr\sin \varphi &=u'\\mmr^{m}\cos m\varphi +(m-1)^{2}Ar^{m-1}\cos(m-1)\varphi +(m-2)^{2}Br^{m-2}\cos(m-2)\varphi &\\+(m-3)^{2}Cr^{m-3}\cos(m-3)\varphi +{\text{ etc. }}+Lr\cos \varphi &=t''\\mmr^{m}\sin m\varphi +(m-1)^{2}Ar^{m-1}\sin(m-1)\varphi +(m-2)^{2}Br^{m-2}\sin(m-2)\varphi &\\+(m-3)^{2}Cr^{m-3}\sin(m-3)\varphi +{\text{ etc. }}+Lr\sin \varphi &=u''\end{array}}$$

$${\frac {(tt+uu)(tt''+uu'')+(tu'-ut')^{2}-(tt'+uu')^{2}}{r(tt+uu)^{2}}}=y$$ Factorem $r$ manifesto e denominatore formulae ultimae tollere licet, quum $t',$ $u',$ $t'',$ $u''$ per illum sint divisibiles. Denique sit $R$ quantitas positiva determinata, arbitraria quidem, attamen maior maxima quantitatum $$mA\surd {2}{\text{, }}{\sqrt {mB\surd {2}}}{\text{, }}{\sqrt[{3}]{(mC\surd {2})}}{\text{, }}{\sqrt[{4}]{mD\surd {2}}}$$ abstrahendo a signis quantitatum $A,$ $B,$ $C$ etc., i.e. mutatis negativis, si quae adsint, in positivas. His ita praeparatis, dico, $tt'+uu'$ certo nancisci valorem positivum, si statuatur $r=R,$ quicunque valor (realis) ipsi $\varphi$ tribuatur.

Demonstratio. Statuamus $${\begin{array}{rl}R^{m}\cos 45^{o}+AR^{m-1}\cos(45^{o}+\varphi )+BR^{m-2}\cos(45^{o}+2\varphi )&\\+CR^{m-3}\cos(45^{o}+3\varphi )+{\text{ etc. }}+LR\cos(45^{o}+(m-1)\varphi )+M\cos(45^{o}+m\varphi )&=T\\R^{m}\sin 45^{o}+AR^{m-1}\sin(45^{o}+\varphi )+BR^{m-2}\sin(45^{o}+2\varphi )&\\+CR^{m-3}\sin(45^{o}+3\varphi )+{\text{etc.}}+LR\sin(45^{o}+m\varphi )&=U\\mR^{m}\cos 45^{o}+(m-1)AR^{m-1}\cos(45^{o}+\varphi )+(m-2)BR^{m-2}\cos(45^{o}+2\varphi )&\\+(m-3)CR^{m-3}\cos(45^{o}+3\varphi )+{\text{ etc. }}+LR\cos(45^{o}+(m-1)\varphi )&=T'\\mR^{m}\sin 45^{o}+(m-1)AR^{m-1}\sin(45^{o}+\varphi )+(m-2)BR^{m-2}\sin(45^{o}+2\varphi )&\\+(m-3)Cr^{m-3}\sin(45^{o}+3\varphi )+{\text{ etc. }}+Lr\sin(45^{o}+(m-1)\varphi )&=U'\end{array}}$$ patetque

I. $T$ compositam esse e partibus