172 
AN EIGHTH MEMOIR ON QUANTICS. 
[405 
298. We 
see 
that 
0 = oo , gives 
X = CO , 
y = — oo, point at infinity, the direction of 
parallel to axis of x. 
the curve 
'0 = 9, 
УУ 
x = 0, 
у = 0, the origin. 
0 — 8, 
УУ 
ж = — 1, 
¿/ = + 1, the node, tangent parallel to axis of y. 
УУ 
_ 43 2 5 
Л ~ 2187» 
У ~ 2 Sr> f 1 angent parallel to the axis of y. 
0 = 4, 
УУ 
r — — 14 5 
x — ТГГ’ 
у = Ц-, tangent parallel to axis of x. 
0 = 0, 
УУ 
x = — 1, 
у — + 1, the node. 
0 = -l, 
УУ 
^ = - -#> 
y=0- 
0 = — 16, 
У У 
x = —l 6ff, 
у = — 41§, the cusp of the bicorn. 
0= -GO , 
УУ 
x = — oo , 
у = — oo, point at infinity, direction of curve 
axis of x. 
parallel to 
299. The 
Nodal Cubic 
is shown along with the Bicorn, Plate, fig. 2; 
it consists 
of one continuous line, passing from a point at infinity, through the cusp of the 
bicorn, on to the node-cusp, then forming a loop so as to return to the node-cusp, 
again meeting the bicorn at the origin, and finally passing off to a point at infinity, 
the initial and ultimate directions of the curve being parallel to the axis of x. 
300. It may be remarked that, inasmuch as one of the branches of the cubic 
touches the bicorn at the node-cusp, the node-cusp counts as (4 + 2 =) 6 intersections; 
the intersections of the cubic with the bicorn are therefore the cusp, the node-cusp, 
and the origin, counting together as (2+6 + 1=) 9 intersections, and besides these the 
point at infinity on the axis of x, counting as 3 intersections. This may be verified 
by substituting in the equation of the cubic the bicorn 0-values of x and y. But 
to include all the proper factors, we must first write the equation of the cubic in the 
homogeneous form 
(9x + 8y + 5 zf z — (y — zf (25 z — 9 y) = 0, 
and herein substitute the values 
x : y : z = — (0 + 2) (0 3 — 0 2 + 20 — 4) : (0 + 2) 2 (0 — 3) 0 : (0 + 1) (f> 3 ; 
the result is found to be 
0 3 {((f) + 1) (40 2 + 60 - 9) 2 - (20 + 3) 2 (40 3 + 40 2 + 180 + 27)} = 0, 
that is 
— 90 3 (0 + 2) (40 + 3) 2 = 0; 
and considering this as an equation of the order 12, the roots are 0 = 0, 3 times, 
0 = — 2, 1 time; 0= — f, 2 times, and 0 = oo, 6 times. 
301. The cubic curve divides the plane into 3 regions, which may be called 
respectively the loop, the antiloop, and the extra cubic; for a point within the loop 
or antiloop, 0(.'c, y) is =—, for a point in the extra cubic yfr(x, y) is =+. If in 
conjunction with the cubic we consider the discriminatrix, or line y = 0, then we have 
in all six regions, viz. y being = +, three which may be called the loop, the triangle,