Thus variables have been found such that

A = 2yiy2 + ay^^ cA -B=b^^ + b^yi (4),

and the reduction is now complete. Geometrically, the distinction in
(3) depends on whether ^2=0 is or is not one of the pair of lines
cA-B = 0\ when it is not, we take ^s = as the harmonic conjugate
of ya = with respect to this pair of lines. Then y/ = is the tangent
to -4 = at its second intersection with y^, as in Fig. 4 1.
Fig. 4.
If^ however y x^ is not present in {cA - B) in (3), we may write
cA-B = 2y2 (bxs + ky^) = ^y^ys,

* Here the method is not really dififerent from method (i) : with more variahles
the two methods are quite distinct (see note f on p. 41).
t It will be seen that we are dealing with the same case as the second on p. 82 ;
but from a different point of view.
B. 3



84 A FAMILY OF QUADRATIC FORMS [CH. HI

SO that A = 2yi'ya + ay^^lb^,

where y/ =!/i + k (ak'/b^) y^ - (ak/b) y^.
Thus the reduced forms are

A=2y^y2 + ay^^y cA-B=2y^s (5).
Here, from the geometrical point of view, yj = is one of the two
lines cA -B = and ys = is taken to be the other ; yi =0 is deter-
mined as before. The conies will be seen to osculate* at P, as in Fig. 5.
Osculating
Contact [3]




Fig. 5.
We note here that if a = in equation (5), XA-B = always
represents a pair of straight lines (see § 4 below).
