Invariant Theory of Conies. 465

The line pairs are the three pairs of opposite sides of the
quadrangle whose summits are the points of intersection of 8^
Sz. Their equation is formed by eliminating "k between (958)
and $! - kSz = 0. Thus we get

A^3 = 0. (959)

If S2 = 0 denote a line pair, A2 vanishes, being the discrimi-
nant, and equation (958) reduces to the quadratic

0, . (960)

showing that through the points of intersection of a conic Si
and a line pair S2 can be drawn two other line pairs, their
equation is found, by eliminating k between (960) and Si - hS*,
to be
Ai$22 - ®i$2£i + ®2$i2 = 0. (961)
If $o = 0 be the square of a line, say (A,,.)3, then not only does
A2 vanish identically, but also ®a, and ®l becomes A>? or Sij
then the equation (958) reduces to A! - #Si = 0, and only one
line pair can be drawn, viz.,
A! (Ax)2 - Si Si = 0, (962)
which will evidently be the tangent pair to Si at the points
where it meets Xz. This will give the equation of the asymp-
totes if Xx = 0 be the line at infinity.
The equation (958) is the fundamental one in the invariant
theory of conies. It was first given by Lame, in his Examen
des Differ entes Methodes. See FIEDLEB'S Translation of SALMON'S
Conic Sections. I shall call it LAMP'S EQJJATIOIT.
EXERCISES.
1. Find the equation of the bisectors of the angles of the line pair
ax2 + Zhxy 4- %2 = 0,
the axes being oblique.