. Differential and integral calculus, an introductory course for colleges and engineering schools. rable), the ring includedbetween the fixed circle and the circle concentric with this andof radius a + 2 b in the case of the epicycloid, and a — 2b in thecase of the hypocycloid, is entirely covered by the curve. For it canbe proved, and is indeed geometrically evident, that through everypoint of this ring pass two branches of the curve. 95. Exercises. 1. What is the hypocycloid when b = \alWhat does each curve become when 6=0? 2. In the epicycloid show that Dxy= tan £(^r—J0 = tan(0 + £), and th
. Differential and integral calculus, an introductory course for colleges and engineering schools. rable), the ring includedbetween the fixed circle and the circle concentric with this andof radius a + 2 b in the case of the epicycloid, and a — 2b in thecase of the hypocycloid, is entirely covered by the curve. For it canbe proved, and is indeed geometrically evident, that through everypoint of this ring pass two branches of the curve. 95. Exercises. 1. What is the hypocycloid when b = \alWhat does each curve become when 6=0? 2. In the epicycloid show that Dxy= tan £(^r—J0 = tan(0 + £), and that consequently the tangent at P passes through T and the normalthrough N. 3. Prove a similar theorem concerning the hypocycloid. 4. Write the equations of the hypocycloid when b = \a. Determiney and the slopes of the cuspidal tan-gents. 5. The is the hypocycloid when b = \ a. It is also termed the hypocycloid of jourcusps. Write its equations and reducethem to the forms x = a cos30, y = a sin30, and thence obtain the equation z3 + y< al See Art. 91, exercise 132 DIFFERENTIAL CALCULUS §96
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