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

226

MISCELLANEOUS CURVES.

accordingly a2: + b22 =? P
( - p) '
(P -P)T2
Therefore a2$2 + b2n2 =p2 (~2 + p2),
also (p p)2 = p (p —2),
therefore 2 (p -_ p)2 (a- b2)2(b24 _- a 6 4)
(a2"2+br)" (~ 2 + b2)2
whence the required equation is
(a22 + b2n2)3 = (a2 - b2)2 (b214 - a2)4)2,
and is therefore a curve of the eighth class.
341. The evolute of an ellipse can also be generated in the
following manner, which can be proved directly or by orthogonally
projecting a four-cusped hypocycloid on a plane parallel to one of
the cuspidal tangents.
From any point P on an ellipse draw perpendiculars PM, PN
to the major and minor axes; draw ME parallel to the tangent at
P to meet CP in E; draw EL perpendicular to the major axis
cutting MN in R. Then MN is the tangent at R to the evolute of
an ellipse.
342. The evolute of an ellipse has two real single foci, which
are thefoci of the ellipse.
We have shown in ~ 65 that the tangential equation of the
curve
(xlA)R + (y/B) = 1
I 1
is + -= 1,
AsA22 + B2. =2
whence if (a, I) be the coordinates of any focus
(a+ ) -.)2-1 = 1..................
But A = (a2-b)/a, B= (2- b2)/b,
whence (1) becomes (a + t3)2 = a2 -b2,

and therefore

a= +(a2-b2)2, 3=.