An elementary course of infinitesimal calculus . Fig. 101. Fig. 101 shews the curve. The dotted branch correspondsto negative values of 6. Another mode of generation of this curve has been explainedin Art. 138. 368 INFINITESIMAL CALCtJLUS. [CH. IX 3°. The reciprocal spiral is defined by the equationr = aie (5). If y be the ordinate drawn to the initial line, we have sind f = r sin 6 = a 6 As 0 approaches the value zero, r becomes infinite, but yapproaches the finite limit a. Hence the line y = a is anasymptote. Y. Fig. 102. The dotted part of the curve in Fig. 102 corresponds tonegative values
An elementary course of infinitesimal calculus . Fig. 101. Fig. 101 shews the curve. The dotted branch correspondsto negative values of 6. Another mode of generation of this curve has been explainedin Art. 138. 368 INFINITESIMAL CALCtJLUS. [CH. IX 3°. The reciprocal spiral is defined by the equationr = aie (5). If y be the ordinate drawn to the initial line, we have sind f = r sin 6 = a 6 As 0 approaches the value zero, r becomes infinite, but yapproaches the finite limit a. Hence the line y = a is anasymptote. Y. Fig. 102. The dotted part of the curve in Fig. 102 corresponds tonegative values of 0. 141. The Llma^on, and Cardiold. If a point 0 on the circumference of a fixed circle ofradius |a be taken as pole, and the diameter through 0 asinitial line, the radius vector of any point Q on the cir-cumference is given by r = acos^ (1). If on this radius we take two points P, P at equal constantdistances c firom Q, the locus of these points is called a lima9on. Its equation is evidently r = acos ^ + c (2). This includes the paths both of P and of P, if 6 range from0 to 2ir. 140-141] SPECIAL CURVES. 369 If c a, r cannot vanish; see the curvetraced by Pj, Pj in the figure.
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