An elementary course of infinitesimal calculus . The curve traced out by a point of the rolUng circle which isnot on the circumference is called an epitrochoid, or a hypo-trochoid, as the case may ba If k denote the distance of thetracing point from the centre of the rolling circle, the expressionsfor the coordinates x, y in the various cases are obtained bywriting k for h in the coeflScients (only) of the second terms. 138. Special Cases. 1°. If the radius of the fixed circle be infinitely great wefall back on the case of the cycloid. The correspondingequations are easily deduced from Art. 13
An elementary course of infinitesimal calculus . The curve traced out by a point of the rolUng circle which isnot on the circumference is called an epitrochoid, or a hypo-trochoid, as the case may ba If k denote the distance of thetracing point from the centre of the rolling circle, the expressionsfor the coordinates x, y in the various cases are obtained bywriting k for h in the coeflScients (only) of the second terms. 138. Special Cases. 1°. If the radius of the fixed circle be infinitely great wefall back on the case of the cycloid. The correspondingequations are easily deduced from Art. 137 (1), writing x + afor 00, aO = 60, and (finally) 0 = 0. 2°. Again, making the radius of the rolling circle infinite,we get the path described by a point of a straight line whichrolls on a fixed circle. The curve thus defined is called theinvolute of the circle; see Art. 161. The equations maybe obtained as limiting forms of Art. 137 (4), or they maybe written down at once from a figure. We find x = acoad + adem6\ y = asin 6- aOcos 6) ^ ?. Fig. 88. The corresponding trochoidal curve is x = (a + h)cos6 + ,\ ,. y = {a + h)sine - a6 cose] ^ where h = PQ in the figure, Q being the tracing point. Theparticular case of h=-a gives the spiral of Archimedes, seeArt. 140. 137-138] SPECIAL CURVES. 355 3°. If the radii a, b be commensurable, then after someexact number of revohitions the tracing point will havereturned to its original position, and its subsequent coursewill be a repetition of the previous path. In such cases thecurve is algebraic, since the trigonometrical functions can beeliminated between the expressions for x and y. Sometimesthe equation is more conveniently expressed in polar co-ordinates. Figs. 89, 90, 91 shew the epi- and hypo-cycloids in whichthe ratio of the radius of the rolling circle to that of the fixedcircle has the values 1, ^, ^, respectively.
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