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

;84

SPECIAL CUBICS.

accordingly
tan (09 + 0,)= lk/b.
Now
PON = POX- NOX
= 01+ 02- rr,
also
QOM= QOX-MOX
= 01 + 02 - r,
T
N Q
A M x
and
tan (01 + 02- 7r)=tan (0 + 02)
-= i/b = TX/AX
=tan TAX,
whence
PON= QOM= TAX.
140. The envelope of the chord of contact PQ is an ellipse.
Putting h = b =- a in (8) of ~ 130, the equation of the polar
conic is
U = 2bx2 kxy + b2x - bky = 0...............(6),
and by (5) of ~ 139, the equation of the two straight lines drawn
from the node to the points of contact is
V= 3by2-2kxy - bx2 = O..................(7),
whence U + XV = 0 is the equation of another conic which passes
through the points P, Q and also the origin. The easiest way of
determining the condition that this conic should represent two
straight lines is to observe that one of them must be of the form