PROFESSOR CAYLEY’S EIGHTH MEMOIR OH QHANTICS. 
537 
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 (4-1-2 = ) 6 intersections ; the inter- 
sections of the cubic with the bicorn are therefore the cusp, the node-cusp, and the 
origin, counting together as (2-f6-f 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 <p-values of x and y. We must, however, to 
include all the proper factors, first write the equation of the cubic in the homogeneous form 
(9x+8y+5zyz-(y-zy(25z-9y)=0, 
and herein substitute the values 
x:y: *=-(<p+2)(<P 3 -<P 2 + 2<p-4) : (<p + 2) 2 (<p-3)<p : (<p + l>p 3 ; 
the result is found to be 
<p 3 {(<pH-l)(4<p 2 + 6<p— 9) 2 — (2<p-|-3) 2 (4<p 3 + 4<p 2 -l-18<p+27)} = 0, 
that is 
-9<p 3 (<p+2X4<p + 3) 2 =0; 
and considering this as an equation of the order 12, the roots are <p = 0, 3 times, 
<p= — 2, 1 time; <p = — f, 2 times, and <p = co , 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, 
\p(x, y) is = — , for a point in the extra cubic \p(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, and the upper 
region; and y being = — , three which maybe called the right, left, and under regions 
respectively ; the triangle and the under region form together the antiloop. 
302. It is now easy to discuss Hermite’s two sets of criteria; the first set becomes 
y=+> y— 1=-, 
y=+, y- 1= -b 
J(x+y) =+, 
%-i X*+y)=+> 
-y) 
4>(x, y)== — 5 character 5 r 
^(x, y)= — 
•#*7 y) = +i 
\p(x, y)— — , cannot exist. 
character r+4?, 
Referring to the Plate, fig. 4, which shows a portion of the cubic and the bicorn, 
then 1° the conditions y=-\~, -^(x, y)=— imply that the point (#, y) is within the loop 
or within the triangle of the cubic; the condition y— 1=— brings it to be within the 
triangle, and for any point within the triangle we have x +?/= — , whence also the con- 
dition J(x-\-y)= + becomes J = — ; hence the conditions amount to J= — , (x, y) within 
the triangle ; but by the general theory (#, y), being within the triangle, that is, in the 
region P or T, if J = — , will of necessity be within the region P ; so that the condi- 
mdccclxvii. 4 D 
