X

Now a b − h 2 becomes 1 × 1 − 0 2 , that is, 1 , and is therefore positive. Hence the circle is a particular case of an ellipse, as it ought to be. Generalising, the equation of any circle can be put into the form a ( x 2 + y 2 ) + 2 g x + 2 f y + c = 0 . Hence a b − h 2 becomes a 2 − 0 , that is, a 2 , which is necessarily positive. Accordingly all circles satisfy the condition for ellipses. The general form of the equation of a parabola is ( d x + e y ) 2 + 2 g x + 2 f y + c = 0 , so that the terms of the second degree, as

they are called, can be written as a perfect square. For squaring out, we get $d^2{x}^2+2dexy+e^2{y}^2+2gx+2fy+c;$ so that by comparison $a=d^2$, $h=de$, $b=e^2$, and therefore $ab-h^2=d^2{e}^2-(de{)}^2=0$. Hence the necessary condition is automatically satisfied. The equation $2xy-4=0$, where $a=b=g=f=0$, $h=1$, $c=-4$, represents a hyperbola. For the condition $ab-h^2$ becomes $0-1^2$, that is, $-1$, which is negative.

plus 0.75em minus 0.25em The limitation, introduced by saying that,

when the general equation represents any locus, it represents a conic section, is necessary, because some particular cases of the general equation represent no real locus. For example $x^2+y^2+1=0$ can be satisfied by no real values of $x$ and $y$. It is usual to say that the locus is now one composed of imaginary points. But this idea of imaginary points in geometry is really one of great complexity, which we will not now enter into.

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 are certainly included among the curves represented by the general algebraic form of the second degree. This fact is worth noting; for it is characteristic of modern mathematics to include among general forms all sorts of particular cases which would formerly have received special treatment. This is due to its pursuit of generality.