536 PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QIT ANTICS 
we have 
A = J, 
B =T¥8-J 2 (y— 1) 5 
I) = J% 
^i == T2'8 J 3 (^’ ~\-y), 
N=^J e {9(l+.tf-8(l+^)(l-y)-(l-y) s }, 
=25rJ 6 - {/— %/' + + 9* s + 1 ly + 10*} ■ 
It thus becomes necessary to consider the curve 
+(#, y) —y 3 — 3 y 1 + 8 xy + 9x 2 + 1 ly + 1 Ox = 0 , 
the equation whereof may also be written 
9tf+%+5=(3/— l)x/25— 9y. 
297. This is a cubic curve, viz. it is a divergent parabola having for axis the line 
9#+4y + 5 = 0, and its ordinates parallel to the axis of x; and having moreover a node 
at the point x=—l, y=-\- 1, that is, at the node-cusp of the bicorn ; the curve is thus 
a nodal cubic ; we may trace it directly from the equation, but it is to be noticed that 
qua nodal cubic it is a unicursal curve ; the coordinates x, y are therefore rationally 
expressible in terms of a parameter 4 ; and it is easy to see that we in fact have 
81(*+1)= W-8), 
9(y-l)=-+(+-8), 
whence also 
dy — 1 8 (+ — 4) 
dx 4 /( 34 / — 16 ) 
298. We see that 
+=co , gives x=co , y= — co , point at infinity, the direction of the curve parallel 
to axis of x. 
4 = 9, 
4—8, 
J 16 
4— Tt 
+=4, 
4=0, 
4=-i, 
+= — 16, 
4=—cc , 
x=0, y=0, the origin. 
x= — 1, y=-\-l, the node, tangent parallel to axis of y. 
§4, y=~\ ri ~ , tangent parallel to the axis of y. 
T — ‘t a * 
— 2 18 
X= — 1 
x= — 
y—^Q-, tangent parallel to axis of x. 
y—4- 1, the node. 
y= o. 
x= — 76ff, y= — 41-f, the cusp of the bicorn. 
x — — co, y= — co, point at infinity, direction of curve parallel to axis 
of x. 
299. The Nodal Cubic is shown along with the Bicorn, Plate, fig. 2 ; it consists of one 
