405] 
AN EIGHTH MEMOIR ON QUANTICS. 
171 
Article Nos. 296 to 303.—Comparison with the Criteria No. 283: the Nodal Cubic. 
296. ioi the discussion of Hermites results, it is to be observed that in the 
notation of the present Memoir we have 
A = J, 
B =-K = - T 
D = L, 
A= 161- JA = T ig (2 11 X - J 3 + JD), 
N = 18 L 2 — JKL — K 3 
= ^ {3 2 .2 22 L 2 - UJL (J 2 - D) - (J 2 - DY], 
or, putting as above, 
2 U L — J 3 D 2 11 L J 2 — D 
x = /3 » y = J*’ and therefore 1 + X =~JY > 1 - y = —jr- > 
we have 
A = J, 
B = its (V ~ 1)» 
D = J 2 y, 
Bi — J 3 + y), 
{»(1+ «>■-8(1+*)(!-?)-(!-?)■}. 
= . [if — 3,f + 8xy 4- Oa- + 11 y + lOj.j. 
It thus becomes necessary to consider the curve 
■'/r ( x > y) = y 3 — 3y 2 + & x y + 9# 2 + 11 y + 10# = 0, 
the equation whereof may also be written 
9# + 4y + 5 = (y — 1) V25 — 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 — — 1, 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 qud nodal cubic it is a unicursal curve; the coordinates x, y are 
therefore rationally expressible in terms of a parameter -yjr; and it is easy to see that 
we in fact have 
81(#+1)= ^(yjr — 8), 
9 (y — 1) = — (ty — 8), 
whence also 
dy _ — 18 (^ - 4) 
dx ^(3^ — 16) 
22—2