. An elementary treatise on the differential and integral calculus. Hence, Fig. 45. If s be measured from the lowest point V, to any point P(x, y), we have 178. The Involute of a Circle.—(See Art. 124) LetC be the centre of the circle,whose radius is r; APR is aportion of the involute, T andT are two consecutive pointsof the circle, P and Q twoconsecutive points of the in-volute, and <b the angle TCT = PTQ = dfaand PT = AT = r. .-. ds = PQ = r# ; FiS 46 /. s = rf(f)d(f) = |r2 + C. If the curve be estimated from A, (7 = 0, and we haves = | one circumference, 0 = 2n ; .\ s = \r
. An elementary treatise on the differential and integral calculus. Hence, Fig. 45. If s be measured from the lowest point V, to any point P(x, y), we have 178. The Involute of a Circle.—(See Art. 124) LetC be the centre of the circle,whose radius is r; APR is aportion of the involute, T andT are two consecutive pointsof the circle, P and Q twoconsecutive points of the in-volute, and <b the angle TCT = PTQ = dfaand PT = AT = r. .-. ds = PQ = r# ; FiS 46 /. s = rf(f)d(f) = |r2 + C. If the curve be estimated from A, (7 = 0, and we haves = | one circumference, 0 = 2n ; .\ s = \r (27r)2 = n circumferences, 0 = 2nn • .-. $ = \r (2nn)2 == THE CARDIOIDE. 353 179. Rectification in Polar Co-ordinates.—If the curve be referred to polar co-ordinates, we have (Art. 102), ds2 = rW f dr2;hence we get s = J (r2 + — j dd, s =J V + Hr*) dr 180. The Spiral of Archimedes.—From r = ad, wehave dO _ 1 dr ~~ a a J n v r (a2 + r2)l t a ,_ /r -f- Va2 + r22a a . /r 4- V «2 + ?,2\ (see Art. 172), from which it follows that the length of anyarc of the Spiral of Archimedes, measured from the pole, isequal to that of a parabola measured from its vertex, r anda having the same numerical values as y and p. 181. The Cardioide.—The equation of this curve isr = a (1 + cos 0). Here -=s = — « sin 0, «0 and hence 5 = J\_a2 (1 + cos 0)2 -f «2 sin2 0]*<Z0== a J(2 + 2 cos 0)*d0 cos -dd = 4:a sin ^ + tf. 354 LENGTHS OF CURVES IN SPACE. If we estimate the arc sfrom the point A, for which0 = 0, we have 5 = 0; .\ 67=0. Making 0 = n for thesuperior limit, we have s = 4:a sin - = Aa,
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