(xiii) The curve has acnodes at (0, oo ), (oo , 0) and a triple point at
(0, 0). It has a branch asymptotic to x + y + 6 = and an oval lying
between the asymptote and the axes of reference. Consider its inter-
sections with y = tx or txy + x + y = 0.
(xiv) (x + l)(x + 3)(y+l)(y-S) + lSxy + 12x + 6y + 9 = 0,
(xv) Asymptotes x 2 (25y 2 -36) = 0. Solve for x and «. See
Ch. XVIII, Fig. 6. '

(xvi) Asymptote's x = ±2, y = ±\.
(xvii) Asymptotes xhf = 0. Solve for x and y.
(xviii) Asymptotes x = ±l,y = ±B. Biflecnodes at (0, oo ) and (oo , 0).
Bitangents x = y, Vx = y, 3x + y = 4, &c.
(xix) Asymptotes x*= 0. Solve for y. See Ch. XVIII, Fig. 2.
§6.
Ex. III. Trace

Newton's diagram (Fig. 4) gives us as
the approximation at the origin

2/ 4 = 2x\
which is the pair of parabolas
y 2 = ± </2x.
Hence there are two branches of the
curve touching each other and touching
the axis of y at the origin. The origin is F ..
called a tacnode in such a case. It is
sometimes called also a ' double cusp ' ; but this nomen-
clature is open to objection, as we shall see later.*
As in § 5 we find the parallel asymptotes
y+1 = and y— 2 = 0.
— -V 1



* A tacnode is the point of contact of two ordinary branches of a curve.
