An introduction to mathematics, by A. N. Whitehead.

CONIC SECTIONS 143
they are called, can be written as a perfect
square. For squaring out, we get
d2x2 +9dexy +e2y2 + 2gx +2fy +c;
so that by comparison a=d2, h=de, b=e2,
and therefore ab-h2 =d2e2 - (de)2 = 0. Hence
the necessary condition is automatically satisfied. The equation 2xy-4=0, where a=-b
=g=f=O, h=l, c= -4, represents a hyperbola. For the condition ab-h2 becomes
0-12, that is, -1, which is negative.
Some exceptional cases are included in the
general form of the equation which may not
be immediately recognized as conic sections.
By properly choosing the constants the equation can be made to represent two straight
lines. Now two intersecting straight lines
may fairly be said to come under the Greek
idea of a conic section. For, by referring to
the picture of the double cone above, it will
be seen that some planes through the vertex,
V, will cut the cone in a pair of straight lines
intersecting at V. The case of two parallel
straight lines can be included by considering
a circular cylinder as a particular case of a
cone. Then a plane, which cuts it and is
parallel to its axis, will cut it in two parallel
straight lines. Anyhow, whether or no the
ancient Greek would have allowed these
special cases to be called conic sections, they