A Treatise On the Differential And Integral Calculus
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EXAMPLES. 165 Taking tlie differentials, we have — = dx loff a or -f- ^p =zy loo; a, y ^ dx ^ -^ "=" ■ and £ =y := J log 6j^ = ^ log a. Hence, representing log a by —, we shall have llh ^^l = t^^^^yl±. F. And y = i^, ; consequently, from r = ^^ — y ? ^^ shall have uc" ' 7/r 7ny ' for the radius of curvature : noticing, that the center of the circle must (see fig. P. 142) be taken on the concave side of the curve. 2. To find the radius of curvature at any point of the common cycloid. From p. 150... we have ~ = p = y, r and f=y- "^ - -'• dx -^ Hence we shall have i?^ + l = ?^ and {p^-i-lf=^]/^S. X -^ ' X X . 1 consequently, (^ + 1)'^ -^ ji?' = 2 \^r x (2r — a?) r= the ra- dius of curvature. Thus it is manifest that the radius of curvature equals twice the corresponding normal of the cycloid ; so that (see the fig. At p. 149) the radius of curva- 166 EADII OF CURVATURE, ETC. Ture at G equals twice the chord BF of the arc of the gener- ating circle, which corresponds to the cjcloidal arc GO.
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To quickly assess the difficulty of the text, read a short excerpt:
EXAMPLES. 165 Taking tlie differentials, we have — = dx loff a or -f- ^p =zy loo; a, y ^ dx ^ -^ "=" ■ and £ =y := J log 6j^ = ^ log a. Hence, representing log a by —, we shall have llh ^^l = t^^^^yl±. F. And y = i^, ; consequently, from r = ^^ — y ? ^^ shall have uc" ' 7/r 7ny ' for the radius of curvature : noticing, that the center of the circle must (see fig. P. 142) be taken on the concave side of the curve. 2. To find the radius of curvature at any point of the common cycloid. From p. 150... we have ~ = p = y, r and f=y- "^ - -'• dx -^ Hence we shall have i?^ + l = ?^ and {p^-i-lf=^]/^S. X -^ ' X X . 1 consequently, (^ + 1)'^ -^ ji?' = 2 \^r x (2r — a?) r= the ra- dius of curvature. Thus it is manifest that the radius of curvature equals twice the corresponding normal of the cycloid ; so that (see the fig. At p. 149) the radius of curva- 166 EADII OF CURVATURE, ETC. Ture at G equals twice the chord BF of the arc of the gener- ating circle, which corresponds to the cjcloidal arc GO.
What to read after A Treatise On the Differential And Integral Calculus?
You can find similar books in the "Read Also" column, or choose other free books by Theodore Strong to read onlineMoreLess
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