. An elementary treatise on the differential calculus founded on the method of rates or fluxions. - = • -, / n - a > a r sinJ0cos^0 sin <p therefore, by equation (5), r a 2a #a r p and, eliminating r by means of equation (1), we have 2a= p(e9 + £-ne) . (6) Passing to rectangular coordinates, we obtain / y y* 2a = V(x2 + /) [f**-^ + tr*»*-x7)m ... (7) * This curve is one of Cotes Spirals. For a discussion of these spirals, see Dy-namics of a Particle, by Tait and Steele, pp. 147-150, Fourth Edition, London, 1878. § XXXI.] THE PARABOLIC SPIRAL. 307 The Parabolic Spiral.285. If the axi
![. An elementary treatise on the differential calculus founded on the method of rates or fluxions. - = • -, / n ~ - a > a r sinJ0cos^0 sin <p therefore, by equation (5), r a _ 2a #a r p and, eliminating r by means of equation (1), we have 2a= p(e9 + £-ne) . (6) Passing to rectangular coordinates, we obtain / y y* 2a = V(x2 + /) [f**-^ + tr*»*-x7)m ... (7) * This curve is one of Cotes Spirals. For a discussion of these spirals, see Dy-namics of a Particle, by Tait and Steele, pp. 147-150, Fourth Edition, London, 1878. § XXXI.] THE PARABOLIC SPIRAL. 307 The Parabolic Spiral.285. If the axi Stock Photo](https://c8.alamy.com/comp/2AFRWEF/an-elementary-treatise-on-the-differential-calculus-founded-on-the-method-of-rates-or-fluxions-=-n-a-gt-a-r-sinj0cos0-sin-ltp-therefore-by-equation-5-r-a-2a-a-r-p-and-eliminating-r-by-means-of-equation-1-we-have-2a=-pe9-ne-6-passing-to-rectangular-coordinates-we-obtain-y-y-2a-=-vx2-f-tr-x7m-7-this-curve-is-one-of-cotes-spirals-for-a-discussion-of-these-spirals-see-dy-namics-of-a-particle-by-tait-and-steele-pp-147-150-fourth-edition-london-1878-xxxi-the-parabolic-spiral-307-the-parabolic-spiral285-if-the-axi-2AFRWEF.jpg)
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. An elementary treatise on the differential calculus founded on the method of rates or fluxions. - = • -, / n ~ - a > a r sinJ0cos^0 sin <p therefore, by equation (5), r a _ 2a #a r p and, eliminating r by means of equation (1), we have 2a= p(e9 + £-ne) . (6) Passing to rectangular coordinates, we obtain / y y* 2a = V(x2 + /) [f**-^ + tr*»*-x7)m ... (7) * This curve is one of Cotes Spirals. For a discussion of these spirals, see Dy-namics of a Particle, by Tait and Steele, pp. 147-150, Fourth Edition, London, 1878. § XXXI.] THE PARABOLIC SPIRAL. 307 The Parabolic Spiral.285. If the axis of the parabola y = 4cx . . (0 be conceived to be wrapped round a circle whose radius is a, and the ordinates corresponding to each point be laid off inthe direction of the radius of the circle, the curve thus de-termined will be the. parabolic spiral. Taking the pole at the cen-tre of the circle, and the radiuspassing through the vertex ofthe parabola as the initial line, we have x — a6 and y = r — a. Substituting these values of xand j in equation (1), we obtainthe polar equation. Fig. 54, (r — df = 4ca0. (2) The curve consists of two branches; the one determined bythe positive values of r — a is an infinite spiral without the circle ; the other branch passes through the centre when 6 = •—, 4C and emerges from the circle at the point at which and 6 = 3o8 CERTAIN HIGHER PLANE CURVES. [Art. 286. The Logarithmic or Exponential Curve, 286. The curve defined by the equation y = eJ ?x is called the exponential curve; and, since theequation can be written in the form Fig. 55. x = log y, it is also sometimes called the logarithmic curve. The axis ofx is an asymptote since £~°°= o.The curve defined by y = ax, or x = logay, is of the same general form. It passes, through the point (o, i)with an inclination whose tangent is log a. The Sinusoid.