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

THE FOUR-CUSPED HYPOCYCLOID.

221

This curve has a complex biflecnode at the origin, and from
~ 188 it has a pair of real biflecnodes at infinity which are situated
on the axes of x and y respectively. The quartic therefore belongs
to the seventh species; if therefore we employ the letters 8, /c, r, L
to denote simple singularities, and the symbols 8t2, T/c, to denote a
biflecnode and its reciprocal singularity, which by ~ 166 consists
of a pair of cusps having a common cuspidal tangent, Pliicker's
numbers for the reciprocal polar of a four-cusped hypocycloid are
n=4, =0, K=O, m= = 6, T=4, = 0, L2= 3,
and therefore the characteristics of the four-cusped hypocycloid
are
n=6, 8=4, c = O, m=4, 7=0, 1=0, 7-2, = 3.
All the four nodes are imaginary and are situated on the lines
x + y = 0; whilst two of the singularities T7c, are real and consist
of the two pairs of cusps on the axes of x and y respectively, and
their common cuspidal tangents. The third singularity 7/C2 consists of a pair of imaginary cusps at the circular points and their
common cuspidal tangent, which is the line at infinity. This may
be proved by writing (3) in the form
16 (a212 - r)3 + 27a212 (/2 - 72)2 = 0.
The reader will observe that when dealing with sextic and
other curves having compound singularities, the latter must be
considered as a whole, instead of the simple singularities of which
they are composed.
333. The portion of the tangent to a four-cusped hypocycloid,
which is intercepted by two real cuspidal tangents, is of constant
length.
WB
C O A
Let AB be a line of constant length a, which slides between
two lines at right angles. Through E the middle point of AB