I 11 THEORY OF EQUATIONS 17

If is the cross-ratio (a — fi) (y — 8)-r- (a — 8) (y— /3) of the
roots a, j8, y, 8 of the quartic,

{(0 + 1) (0-2) (0-4)} 2 / 3 = 27 (f-4> + l)>J*.
We note the cases = 0, —1, J(l + -/— 3).
If = 0, the quartic has two equal roots and I 3 = 27 J 2.
If = -l, J"=0; and if = ■§ (1 + V^8), 1=0*

* The range of points (a, 0), (3, 0), (7, 0), (8, 0) and the pencil of lines
y = ax, y = fix, y = yx, y = dx are said to be respectively 'harmonic' or
• equianharmonic ' in these two cases.
CHAPTEE II

SINGULAR POINTS

§ 1. Inflexions, &c.
Suppose that the Cartesian equation of a curve of degree n
arranged in ascending powers of x and y is

= a + b^ + b^y + c x 2 + 2c 1 xy + c 2 y 2 + d x 3 + ....
If the origin lies on the curve, the equation must be satisfied
by x = and y = ; and therefore a = 0.
