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The parameter can be removed to yield the equation
Expanding the Tschirnhausen cubic catacaustic shows that it is also a semicubical parabola:
The semicubical parabola was discovered in 1657 by William Neile who computed its arc length; it was the first algebraic curve (excluding the line) to be rectified. It is unique in that a particle following its path while being pulled down by gravity travels equal vertical intervals in equal time periods.