[494 
EXAMPLE OF A SPECIAL DISCRIMINANT. 
47 
u. (1871), 
are such that 
is any number 
ninant of the 
efficients their 
the particular 
the coefficients 
i (besides the 
termination of 
ression of the 
blem. I have, 
8 nodes and 
he coefficients 
Ey this in the 
and 2 = 0 the 
494] 
Assume x = kz, y = gz, the first two equations give 
3 (nkg + r) g + g = 0, 
3 (nkg + r) k + i = 0, 
whence also 
6nk 2 g 2 + 6rkg + gk + ig = 0, 
and the equation of the curve gives 
6nk 2 g 2 + 12 rkg + 4gk + 4 ig + c = 0, 
whence eliminating gk + ig we find 
18 nk 2 g 2 + 12 rkg — c = 0. 
Moreover the first two equations give 
9 (nkg — r) 2 kg — ig = 0, 
or putting kg = 6 we have 
18n0 2 + 12r# - c = 0, 
9 (7i0 + r) 2 0 — ig = 0, 
from which 0 is to be eliminated. 
The equations are 
18w0 2 + 12r0 - c = 0, 
9n 2 0 3 + 18 nr0 2 + 9r 2 # — ig = 0, 
and thence 
\8n 2 0 3 + 36?ir0 2 + 18r 2 0 - 2ig = 0, 
18n 2 0 3 + 12nr0 2 - end =0, 
24?ir# 2 + (18r 2 + cn) 6 — 2ig = 0, 
18?ir# 2 + 12r 2 0 — cr =0, 
(dr 2 + Sen) 0 — big + 4cr = 0, 
g - % ~ 4cr = 2 % ~ 2cr . 
- 6r 2 + Sen 3 2r 2 + cn 5 
or substituting in 18?i0 2 + \2r0 — c = 0, this is 
8n (3ig - 2cr) 2 + 8r (3ig - 2cr) (2r 2 + cn) - c (2r 2 + cn) 2 = 0. 
Hence, developing, the special discriminant is 
□ = - 1 chi 2 
+ 12 c 2 nr 2 
— 7 2 cginr 
— 36 cr 4 
+ 72 g 2 i 2 n 
+ 48 gir 3 , 
which is as it should be of the degree 5, =3.3 2 — 11.2.