Rigge, William F., S.J.
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First Paragraph: One way of drawing a cardioid is to make a pen start in phase 90° from the center of a disk and move along a radius with simple harmonic motion, while the disk revolves with a uniform angular speed of the same period. If now, instead of a simple harmonic movement with the equation p = 1 — cos θ, the amplitude being unity, we give the pen a double harmonic motion, and write the equation p = (1 - cos θ) + (1 - cos mθ), in which m differs from unity by some small aliquot fraction; we shall then get a series of harmonic curves which have one variable parameter, and which must therefore have a common envelope. The problem before us is to find this envelope.