Pagina:Werkecarlf03gausrich.djvu/73

$${\begin{array}{rl}&{\frac {R^{m-1}}{m\surd {2}}}\left[R+mA\surd {2}.\cos(45^{o}+\phi )\right]\\+&{\frac {R^{m-2}}{m\surd {2}}}\left[RR+mB\surd {2}.\cos(45^{o}+2\phi )\right]\\+&{\frac {R^{m-3}}{m\surd {2}}}\left[R^{3}+mC\surd {2}.\cos(45^{o}+3\phi )\right]\\+&{\frac {R^{m-4}}{m\surd {2}}}\left[R^{4}+mD\surd {2}.\cos(45^{o}+4\phi )\right]\\+&{\text{etc.}}\end{array}}$$

quas singulas, pro valore quolibet determinato reali ipsius $\varphi$, positivas evadere facile perspicitur: hinc $T$ necessario valorem positivum obtinet. Simili modo probatur, etiam $U,$ $T',$ $U'$ fieri positivas, unde etiam $TT'+UU'$ necessario fit quantitas positiva.

II. Pro $r=R$ functiones $t,$ $u,$ $t',$ $u'$ resp. transeunt in

$${\begin{array}{c}T\cos(45^{o}+m\varphi )+U\sin(45^{o}+m\varphi )\\T\sin(45^{o}+m\varphi )-U\cos(45^{o}+m\varphi )\\T'\cos(45^{o}+m\varphi )+U'\sin(45^{o}+m\varphi )\\T'\sin(45^{o}+m\varphi )-U'\cos(45^{o}+m\varphi )\end{array}}$$

uti evolutione facta facile probatur. Hinc vero valor functionis $tt'+uu'$, pro $r=R,$ derivatur $=TT'+UU',$ adeoque est quantitas positiva. Q.E.D.

Ceterum ex iisdem formulis colligimus, valorem functionis $tt+uu,$ pro $r=R,$ esse $TT+UU,$ adeoque positivum, unde concludimus, pro nullo valore ipsius $r,$ singulis $mA\surd {2},$ ${\sqrt {mB\surd {2}}},$ ${\sqrt[{3}]{mC\surd {2}}}$ etc. maiori, simul fieri posse $t=0,$ $u=0.$

2.

Theorema. Intra limites $r=0$ et $r=R,$ atque $\varphi =0$ et $\varphi =360^{o}$ certo exstant valores tales indeterminatarum $r,$ $\varphi ,$ pro quibus fiat simul $t=0$ et $u=0.$

Demonstratio. Supponamus theorema non esse verum, patetque, valorem ipsius $tt+uu$ pro cunctis valoribus indeterminatarum intra limites assignatos fieri debere quantitatem positivam, et proin valorem ipsius $y$ semper finitum. Consideremus integrale duplex

$$\iint ydrd\varphi$$

ab $r=0$ usque ad $r=R,$ atque a $\varphi =0$ usque ad $\varphi =360^{o}$ extensum, quod igitur valorem finitum plene determinatum nanciscitur. Hic valor, quem