This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1892 Excerpt: ...The equiangular spiral, in which p oc r, is included as the case in which 334 Polar Curves. We shall next reduce the formula to a shape suited for application to curves given by their polar equations. We proved in Art. 205 335. This may easily be put in the r, 6 form thus: --Since 336. Tangential-Polar Form. In Art. ...
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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1892 Excerpt: ...The equiangular spiral, in which p oc r, is included as the case in which 334 Polar Curves. We shall next reduce the formula to a shape suited for application to curves given by their polar equations. We proved in Art. 205 335. This may easily be put in the r, 6 form thus: --Since 336. Tangential-Polar Form. In Art. 220 it was proved that giving us a formula for the radius of curvature suitable for p, equations. Ex. It is known that the general p, ifr equation of all epi-and hypocycloids can be written in the fomi p = A sin Bijr (p. 163, Ex. 6). Hence p = A sin Bifr-A Bhin Byr, and therefore P XP thus again proving the result of the Example in Art. 333. 337. Point of Inflexion. At a point of inflexion the radius of curvature is infinite. This is geometrically obvious from the fact that it is the radius of a circle which passes through three collinear points. We may hence deduce various forms of the condition for a point of inflexion; thus if /o =, we get = 0 from (A), =0 from (d), d$' Gty) '2d y' ' '(ac) = 0 from (E)' u]tffi=from () 2+2(S)2-=ofrom some of which have already been established otherwise. 338. List of Formulae. The formulae proved above are now collected for convenience. ds.. P = (o) f-' + -1-) (r2 + ri2)4 w Examples. 1. Apply formula (n) to the curves P2=ar, ap=73, 2 =--2. Apply formula (i) to the reciprocal spiral an =6. 3. Apply the polar formula for radius of curvature to show that the radius of the circle r--a cos 0 is-2 4. Show that for the cardioide r=a(l +cos 6) 4a 6, p = __cos-; i.e., a: Jr. Also deduce the same result from the pedal, equation of the curve, viz., 5. Show that at the points in which the Archimedean spiral r = aO intersects the reciprocal spiral r6 = a their curvatures are in the ratio 3:1 6. For ...
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