An elementary course of infinitesimal calculus . Kg. 91. We proceed to notice in detail one or two of the caseswhich have specially important properties. Ex. 1. The cardioid. If in Art. 137 (3) we put 6 = a, we get x = 2a cos 6 + a cos 26, y = 2asm0+asin 29, whence a! + a = 2a(l+cos6)cos^, y = 2a (I + cos 6) sin (S). This shews that the radius vector drawn from the point (- a, 0)as pole is given by r = 2o(l + cos6i) (4).. Fig. 92. 138] SPECIAL CURVES. 357 This is otherwise evident from Fig, 92, where AP = iAN=i{OI + AM). The corresponding trochoids are given by x = 2a cos 0 + k cos 2$, i/=
An elementary course of infinitesimal calculus . Kg. 91. We proceed to notice in detail one or two of the caseswhich have specially important properties. Ex. 1. The cardioid. If in Art. 137 (3) we put 6 = a, we get x = 2a cos 6 + a cos 26, y = 2asm0+asin 29, whence a! + a = 2a(l+cos6)cos^, y = 2a (I + cos 6) sin (S). This shews that the radius vector drawn from the point (- a, 0)as pole is given by r = 2o(l + cos6i) (4).. Fig. 92. 138] SPECIAL CURVES. 357 This is otherwise evident from Fig, 92, where AP = iAN=i{OI + AM). The corresponding trochoids are given by x = 2a cos 0 + k cos 2$, i/=2a sin 6 + k sin 26. Referred to the point {-k, 0) as pole these formulae are equi-valent to r=2(a + kcosd) (5), which is the polar equation of the limagon (Art. 141). Thisequation, again, is easily obtained geometrically. Hx. 2. A circle rolls inside another of twice its radius. If in Art. 137 (6) we put b = ^a, we get x = acosd, y=0 (6); the tracing point on the circumference of the rolling circletraces out a diameter of the fixed circle. Again, the corresponding trochoidal curve is given by x = {b + k) cos 6, y={b — k)auiO (7), and is therefore an ellipse of semi-axes b±k. Moreover if therolling circle have a constant angular velocity, the motion of thetracing point is elliptic-harmonic.
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