254 
PROFESSOR M. J. M. HILL OH THE LOCUS OF SINGULAR POINTS 
(A.) The Discriminant. 
It reduces to 
0 
cr 
a/3 
gz 
a/3 
hz 
gz 
hz 
hz 2 
The Conic Node 
Locus 
is z 
= 0 . 
= — (^ a — gfi) 
v. 
In this case equations (143), (161), (162), (163) are equivalent to the three 
equations 
a 2 (x — a) + a/3 (y — b) + gz = 0, 
a/3 (x — a) + /3 2 (y — b) + hz = 0, 
g (x — a) + h (y — b) + kz — 0, 
the only solutions of which (unless g/3 — ha — 0) are 
x — a, y — b, z = 0. 
Hence there is now only one system of values of the parameters satisfying (143), 
(161), (162), (163). 
The same value of the parameter b would be obtained from the equation (181) 
which becomes in this case, after changing x into £, 
2b^(uJJ - VW) + ~ (uw - V 3 ) = 0. 
Now 
uU -VW = (g/3 - ha) az, 
uw — V 3 = 2 (ha — g/3 ) ayz + (ha 2 — g 3 ) 2 3 ; 
therefore, 
^ (uXJ — VW) = (g/3 — ha) a, 
0 
^ (uw — V 3 ) =2 (ha — g/3) ay + 2 (ha 2 — g 2 ) z. 
On the conic node locus z = 0. 
Therefore the equation for b is 
2 b (g/3 — ha) a + 2 (ha — g/3) ay = 0. 
Therefore 
b = y. 
There is only one conic node, since uv — W 3 = 0, and, therefore, equation (181) 
reduces to a simple equation for b. 
