528 
PROFESSOR CAYLEY’S EIGHTH MEMOIR ON QUANTICS. 
curve <p(x, y ) = 0, which is Professor Sylvester’s Bicorn : this curve divides the plane 
into certain regions, and if we attend to the solid figure and instead of the curve consider 
the cylinder, then to each region of the plane there correspond in solido two regions, 
one in front of, the other behind the plane region, and of these regions in solido , one is 
facultative, the other is non-facultative (viz. for given values of (x, y), whatever be the 
sign of <p(x, y), then for a certain sign of z, z<p(x , y) will be positive or the solid region 
will be facultative, and for the opposite sign of z , z<p(x, y) will be negative or the 
region will be facultative). It hence appears that we may attend only to the plane 
regions, and that (the proper sign being attributed to z , that is to J) each of these may 
be regarded as facultative. It is to be added that the discriminatrix is in the present 
case the plane ?/=0, or, if we attend only to the plane figure, it is the line y = 0 ; so that 
in the plane figure the separation into regions is effected by means of the Bicorn and 
the line y— 0. 
277. Reverting to the equation of the Bicorn, and considering first the form at infi- 
nity, the intersections of the curve by the line infinity are given by the equation 
y 3 (Zy — £’)=0, viz. there is a threefold intersection y 3 = 0, and a simple intersection 
2y—x=0; the equation y 3 = 0 indicates that the intersection in question is a point of 
inflexion, the tangent at the inflexion (or stationary tangent) being of course the line 
infinity ; the visible effect is, however, only that the direction of the branch is ultimately 
parallel to the axis of x. The equation 2 y — a’=0 indicates an asymptote parallel to 
this line, and the equation of the asymptote is easily found to be 2 y — #+5 — 0. 
278. The discussion of the equation would show that the curve has an ordinary cusp ; 
and a cusp of the second kind, or node-cusp, equivalent to a cusp and node ; the curve 
is therefore a unicursal curve, or the coordinates are expressible rationally in terms of a 
parameter <p ; we in fact have 
r — -(? + 2)(f 3 - ^ + 2y-4) (? + 2) 2 (f-3) 
whence also 
i=4W+ 2 )- 
279. The curve may be traced from these equations (see Plate, fig 2, where the bicorn 
is delineated along with a cubic curve afterwards referred to) : as <p extends from an 
indefinitely small positive value s through infinity to — 1 — s, we have the upper branch 
of the curve, viz. 
<p = s, gives x=oo , y= — go , point at infinity, the tangent being horizontal. 
<p = oo , gives x= — 1, y=+, the node-cusp, tangent parallel to axis of y. 
<p = — 2, gives #=0, y= 0, the tangent at this point being the axis of x. 
<p = — 1 — g, gives x=cc , y~- j-, point at infinity along the asymptote. 
And as <p extends from x= — 1+e to x—— s, we have the lower branch, viz. 
<& = — ! + £, gives x=—cc , y=— oo , point at infinity along the asymptote 
