. An elementary treatise on the differential calculus founded on the method of rates or fluxions. Fig. 45. equation may also be written in the form g xxxi.] THE CARDIOID. >95 r — 4a siir \0, (2) In like manner, from equation (3), Art. 269, we obtain therectangular equation of the cardioid (x- + j2)2 — ^ax {x- + /-) — 4a*y (3) This curve occurs as a particular case of the epicycloid, seeArt. 296; and also of the hypocycloid, see Art. 299. Example. 1. Determine the minimum abscissa, and the maximum ordinate ofthe cardioid. Min. abscissa when 0 = 60° and r—a\Max. ordinate when 0 = 1200 and r =
. An elementary treatise on the differential calculus founded on the method of rates or fluxions. Fig. 45. equation may also be written in the form g xxxi.] THE CARDIOID. >95 r — 4a siir \0, (2) In like manner, from equation (3), Art. 269, we obtain therectangular equation of the cardioid (x- + j2)2 — ^ax {x- + /-) — 4a*y (3) This curve occurs as a particular case of the epicycloid, seeArt. 296; and also of the hypocycloid, see Art. 299. Example. 1. Determine the minimum abscissa, and the maximum ordinate ofthe cardioid. Min. abscissa when 0 = 60° and r—a\Max. ordinate when 0 = 1200 and r = 30. The Cartesian Ovals. 272. If a point move in such a way that fixed multiples ofits distances from two fixed points have a constant sum or differ-ence, the path described will be aCartesian oval. In other words,if we denote the distances of amoving point P from two fixedpoints A and B by r and r\ andthe distance ABby e, the locus ofP will be a Cartesian oval when rand r are connected by the linearrelation. Ir ± mr = ± ne, CO Fig. 46. in which /, m, and n denote numerical constants. The line AB is an axis of symmetry7, since points symmetri-cally situated with reference to this line have the same valuesof r and r . 296 CERTAIN HIGHER PLANE CURVES. [Art. 272. Inasmuch as any two of the four equations included in (1)may be regarded as differing in the sign of only one term, it isevident that the resulting curves cannot intersect, except in theparticular case when r or r admits of the value zero; that is,when A or B is a point of the locus. Moreover, an infinitely distant point cannot satisfy equation(1) if /and m have different values; hence in general the entirelocus of this equation consists of closed branches or ovals whichdo not intersect. 273. To derive the polar equation, A being the pole andPAB the vectorial angle 6, we have the relation r2 = r2 + c* — 2cr cos 6, and eliminating r between this equation and (1), we obtain (/2 - m*y + 2c(mt cos 6 ± In) r + (
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