236 
PROFESSOR M. J. M. HILL OR THE LOCOS OF SINGULAR POINTS 
Hence there are in general two conic nodes at every point of the locus of ultimate 
intersections. 
(B.) It follows from (146) and (147) by means of (158) that A, dA/dx both vanish 
at points on the locus of ultimate intersections. 
By symmetry 0A /dy, dA/dz also vanish. 
Hence A contains C 3 as a factor. 
Example 11 .—Locus of two Conic Nodes. 
Let the surfaces be 
( a C + S 3 -£ 2 ) (* — «) 3 + 2 (/3£ + Sexy) (x — a) (y — b) + (y£ + e z y 2 ) (y — bf 
+ 2 gC (x — a) + 2 hi (y — b) + Tct? = 0, 
where £ = z — cx — dy ; and a, /3, y, S, e, c, d, g, h, k are fixed constants; a, b the 
arbitrary parameters ; n = 1 or 2. 
(A.) The Discriminant. 
This can be formed by solving the equations 
(a£ + § 3 ^ 3 ) (a; — a) -f (/3£ + Sexy) (y — b) + gl= 0, 
+ Sexy) (x — a) + (y£ + Cy 2 ) (y — b) + h'C, = 0, 
for a, b ; and substituting in 
gl (x — a) + hi (y — b) + K u . 
The values of a, b are (after removing the factor £ which makes them indetermi¬ 
nate) given by 
_ __ (h/3 — gy) £ + ey (hdx — gey) 
X a ~ («7 - /3 2 ) £ + (uehf - 2/3Sexy + 7 SW) ’ 
, _ (r//3 — 7a) £ — 3.i' (hSx — gey) 
J (a 7 — /3 2 ) £ + (o>.e~if — 2j3hexy + ySh; 2 ) 
Substituting these values, and multiplying by the rationalising factor 
uv — w 2 = £ [(ay — /3 2 ) £ + (aeh/ 2 — 2 /3Sexy -f- ySar 3 )] 
the result is 
« (ay - y8 3 ) £" + 2 + k (aeV “ 2/dSe.tt/ 4. y3V) £’ + 1 
— (a/r — 2 ,8gh + yf) £ 3 — {hSx — geyf £ 2 . 
This might also have been obtained from the form (146). 
