. An elementary treatise on the differential calculus founded on the method of rates or fluxions. Fig. 64. Examples. 1. Show that if b > a in the case of the hypotrochoid, the curvemay also be generated as an epitrochoid. Put—:—ib = — tbr: then ib = ?//. The constants for the b T b — a curve as an epitrochoid are a* = —, b = -(b — a), and c = b — a. 2. Show that when b < a, in the case of the hypotrochoid, thecurve may be generated as an hypotrochoid with other values of theconstants. Put—:— rj) = ib; then ih = -?//. The new constants are b T T a — b a= —, b = T(a — b), and c = a — b. Si
. An elementary treatise on the differential calculus founded on the method of rates or fluxions. Fig. 64. Examples. 1. Show that if b > a in the case of the hypotrochoid, the curvemay also be generated as an epitrochoid. Put—:—ib = — tbr: then ib = ?//. The constants for the b T b — a curve as an epitrochoid are a* = —, b = -(b — a), and c = b — a. 2. Show that when b < a, in the case of the hypotrochoid, thecurve may be generated as an hypotrochoid with other values of theconstants. Put—:— rj) = ib; then ih = -?//. The new constants are b T T a — b a= —, b = T(a — b), and c = a — b. Since -r = , we have b b a a V b - + - = 1; a a hence, if, in one of the two modes of generation, the ratio of the radiusof the rolling circle to that of the fixed circle exceeds one-half, in theother it is less than one-half. 320 CERTAIN HIGHER PLANE CURVES. [Art. 3OO. The Four-Cusped Fig. 65. 300. In the case of thehypocycloid when b = \a, thecircumference of the rollingcircle is one-fourth the circum-ference of the fixed circle, andthe curve will have a cusp ateach of the four points wherethe coordinate axes cut thefixed circle, as represented inFig. 65. On substituting \a for bequations (2) Art. 297 become x=.\a cos tp + \a cos ypy — \a sin tp — \a sin yp (0 Substituting the values of cos $ip and sin 3^ from the for-mulas, cos yp = 4 cos9 tp — 3 cos ip, and sin 3^ = 3 sin tp — 4 sin3 ^, we have x — a cos3 tpy — a sin3 tp (2) whenceand X^ = #3 cos2 ^,y\ = #t sin2 ^. XXXI. THE FOUR-CUSPED HYPOCYCLOID. 321 Adding, we have 2 % (3) the rectangular equation of the curve. This equation, whenfreed from radicals, will be found to be of the sixth degree. Example. 1. In the case of the four-cusped hypocycloid, express in terms of?/ the tangent of the inclination of the chord BP, Fig. 65, and provethat this chord is perpendicular to the tangent to the curve at t
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