[The line y-\ = t(x-\) through the cusp (1, 1) meets the curve
where a; = 1 + i(<-3) 2 , y = 1 + %t (<-3) 8 .]



X 4 UNICURSAL CURVES 149

Ex. 2. Express rationally in terms of a parameter the coordinates of
any point on the curves
(i) x 3 + y 3 = Saxy.
(ii) (2x li + 2y* + l)(y-x) = x*--6xy + y\
v (iii) x s + y s + 4xz(x + y) = 0.
. (iv) (x + y) (x - yf = ixyz.
Ex. 3. Express the coordinates of any point on the curve
(x'*-y 2 y i -(x' i -y*)(x + 3y)-Sx 2 + 6xy + y* =
rationally in terms of a parameter.
[A variable conic through the double points

(0,0), x = —y = co , x = y = oo
and one other fixed point E on the curve meets this quartic in one
more point. Choose B as the point on the curve consecutive to the
cusp x = y = oo . The conic is then

(x + y-t)(x-y-l) = *
meeting the quartic where

_ ( f l + 2t-2){t i + t-l) _ (tf + 2t-2)(? + t-3)

X ~ 2{t* + t-2) ' y ~ 2(t* + t-2)

See § 6, Ex. 1 and Ch. Ill, Fig. 3.]

Ex. 4. Express rationally in terms of a parameter the coordinates of
any point on the curves

(i) a:V + 2x*y + xy 2 -3x' i + 2xy-2y* = 0.
(ii) 2y*(2x , *-5x + 4:)-xy(x-2)(x-g) + 2x' i (x-2y = 0.
(iii) »y + 12 a 3 (Sx + 2y) + 108a 4 = 0.
(iv) xy(2xy-hx-10y) + 50(x-yf = 0.
