An elementary course of infinitesimal calculus . 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); i.e. the tracing point on the circumfer

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An elementary course of infinitesimal calculus . 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); i.e. 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.. Fig. 93. 338 INFINITESIMAL, CALCULUS. [CH. IX These results also follow easily from geometrical considerations.The rolling circle passes always through the centre 0 of the fixedcircle; also, if P be the point of the rolling circle which initiallycoincides with A, the arc IP is equal to the arc I A. Hence, since the radii are as 1 : 2, the angle which the arc IP subtondsat the circumference of its circle must be equal to the anglewhich the arc I A subtends at the centre of its circle; that is, OPand OA coincide in direction, and P describes the fixed diameterOA. Again, since the angle POP is a right angle, the otherextremity of the diameter PP of the rolling circle describes thediameter of the fixed circle which is perpendicular to OA. HencePP is a line of constant length whose extremities move on twofixed straight lines at right angles to one another. It is knownthat under these circumstances any other point on PP describesan ellipse. Cf. Art. 163, Ex. 1. Hx. 3. A circle rolls on the o