An elementary treatise on cubic and quartic curves, by A. B. Basset.

THE EVOLUTE OF AN ELLIPSE.

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and (2) and (3) become
h k
- sec + cosec = 1..................(4),
a b
cos3 b +sin (
_ _ _ + _ = I........................ (5).
cos sin
Let X = tan f, then (4) may be written in the form
h2X4 2hk 1'h2 k2 k2
+a2- + (X+x) + +- l x + -1) 2 +......(6).
a2 ab2 2 b2
This equation determines the value of b at the four points of
contact of the tangents from (h, k). Equation (5) determines the
value of X in terms of r, and is a sextic equation in X. The.sextic obviously contains (X - tan #)2 as a factor; and the quartic
factor will be found to be
X4+ 2 (X3 + ) tan + tan2 = 0............(7).
Since (6) and (7) are satisfied by the same value of X, it
follows that
h2 k2
- + = 1,
a2 b2
tan = ak/bh.
The first equation determines the locus of (h, k) which is an
ellipse; whilst the second gives the value of f in terms of (h, k).
When the evolute degrades into a four-cusped hypocycloid, the
locus becomes a circle.
The reciprocal theorem is as follows:If two tangents be drawn to the reciprocal curve from a point
on itself, the envelope of the chord of contact is an ellipse, which
becomes a circle when the curve is the reciprocal polar of a fourcusped hypocycloid.
340. To find the tangential equation of the evolute of the
evolute of an ellipse.
The equation of the normal to the evolute at 0 is
x cos r + y sin =p - p,
cos, sin.
whence = -- -p = p-P
p-p' p-p

B. C.

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