Seite:NewtonPrincipien.djvu/644

	Isaac Newton: Mathematische Principien der Naturlehre

$\scriptstyle +{\frac {3}{256\alpha '}}$ $\scriptstyle +{\frac {3}{128\alpha '^{2}}}$ $\scriptstyle +{\frac {1}{16\alpha '^{3}}}$ $\scriptstyle +{\frac {3}{8\alpha '^{4}}}$ $\scriptstyle +{\frac {3}{2\alpha '^{5}}}$ $\scriptstyle -{\frac {3}{\alpha '^{6}}}$

$\scriptstyle {\frac {x^{10}}{r^{9}}}+{\frac {7}{1024\alpha '}}$ $\scriptstyle +{\frac {3}{256\alpha '^{2}}}$ $\scriptstyle +{\frac {3}{128\alpha '^{3}}}$ $\scriptstyle +{\frac {1}{16\alpha '^{4}}}$ $\scriptstyle +{\frac {3}{8\alpha '^{5}}}$ $\scriptstyle -{\frac {3}{2\alpha '^{6}}}$ $\scriptstyle +{\frac {1}{\alpha '^{7}}}$

$\scriptstyle {\frac {x^{12}}{r^{11}}}+{\frac {9}{2048\alpha '}}$ $\scriptstyle +{\frac {7}{1024\alpha '^{2}}}$ $\scriptstyle +{\frac {3}{256\alpha '^{3}}}$ $\scriptstyle +{\frac {3}{128\alpha '^{4}}}$ $\scriptstyle +{\frac {1}{16\alpha '^{5}}}$ $\scriptstyle -{\frac {3}{8\alpha '^{6}}}$ $\scriptstyle -{\frac {3}{2\alpha '^{7}}}$ $\scriptstyle +{\frac {1}{\alpha '^{8}}}$

$\scriptstyle {\frac {x^{14}}{r^{13}}}$

Hiernach wird

$\scriptstyle \int \limits _{-r}^{+r}y'dx=r^{2}\left\{{\frac {1}{\alpha '}}\left[2-1+{\frac {3}{20}}+{\frac {1}{56}}+{\frac {1}{192}}+{\frac {3}{1408}}+{\frac {7}{6656}}+{\frac {3}{5120}}\right]\right.$

$\scriptstyle +{\frac {1}{\alpha '^{2}}}\left[{\frac {2}{3}}-{\frac {3}{5}}+{\frac {3}{28}}+{\frac {1}{72}}+{\frac {3}{704}}\dots \right]+{\frac {1}{\alpha '^{3}}}\left[{\frac {2}{5}}-{\frac {3}{7}}+{\frac {1}{12}}\dots \right]$

$\scriptstyle \left.+{\frac {1}{\alpha '^{4}}}\left[{\frac {2}{7}}-{\frac {1}{3}}+{\frac {3}{44}}\dots \right]\right\}$ etc.

$\scriptstyle =r^{2}\left[{\frac {1,17711}{\alpha '}}+{\frac {0,1917}{\alpha '^{2}}}+{\frac {0,054}{\alpha '^{3}}}\right]$ = 0,11869r².

Der Halbkreis ist = ½r²π, mithin verhält sich NAn : NeFn = 1,57090 : 0,11869 = 794,1 : 60.

No. 262. S. 437. (Fig. 199.) Weil TH² = TS · TK ist, wird SK : ST = SK : $\scriptstyle {\frac {TH^{2}}{TK}}$ = TK · TK : TH² = (TK – TS)TK : TH² = TK² – TH² : TH² = (TK + TK)(TK – TH) : TH² endlich TK : ST = MH · HK : TH².

No. 263. S. 437. (Fig. 199.) Setzt man nämlich FG = Y, BG = y, GT = x, HT = h, KT = a; so ist y² = $\scriptstyle {\frac {b^{2}}{a^{2}}}$ (a² – x²), Y² = a² – x², mithin $\scriptstyle {\frac {FG^{2}-BG^{2}}{BG^{2}}}={\frac {FB\cdot Bf}{BG^{2}}}=\left(1-{\frac {b^{2}}{a^{2}}}\right)\left({\frac {a^{2}-x^{2}}{y^{2}}}\right)={\frac {a^{2}-b^{2}}{a^{2}}}\cdot {\frac {a^{2}}{b^{2}}}={\frac {KH\cdot HM}{HT^{2}}}$.

No. 264. S. 438. (Fig. 199.) Nach 1. wird nämlich TH : TK = $\scriptstyle {\sqrt {TS}}:{\sqrt {TK}}$.

No. 265. S. 438. (Fig. 199.) Setzet man, wie in Bem. 264. BG = y, FG = Y, TG = x, ferner $\scriptstyle \angle$ FTG = β und $\scriptstyle \angle$ FTB = α – β = γ; so ist allgemein tang α = $\scriptstyle {\frac {Y}{x}}$, tang β = $\scriptstyle {\frac {y}{x}}$, tang γ = $\scriptstyle {\frac {Yx-yz}{x^{2}+Yy}}$. Wenn aber

Empfohlene Zitierweise:
Isaac Newton: Mathematische Principien der Naturlehre. Robert Oppenheim, Berlin 1872, Seite 636. Digitale Volltext-Ausgabe bei Wikisource, URL: https://de.wikisource.org/w/index.php?title=Seite:NewtonPrincipien.djvu/644&oldid=- (Version vom 1.8.2018)