246 
PROFESSOR M. J. M. HILL ON THE LOCUS OP SINGULAR POINTS 
There is not a double set of equal values as in Art. 15 (see especially Example 10 
of that article), where the degree of the equation of the system of surfaces in the 
parameters is higher than the second. 
If the values given above for a, b be substituted in the left-hand side of (8), and 
the result multiplied b}^ the rationalising factor uv — W 3 , which in this case is 
— (2 xymz fi- m 2 z 2 ), the result is 
r oY) l 
— (2 xymz + m¥) j —--—- — g 2 z 2 l = m 2 3 {2 g 2 xy — ( c 2 — g 2 ) mz}, 
I *4” oibz 
which is the same value for the discriminant as before. 
It will be noticed that the factor 2 enters once through the rationalising factor, and 
twice from the remaining part. 
(B.) The Node Locus is 2 = 0 . 
Substituting a? = a + X, y = b Y, 2 = Zin the equation, it becomes 
(bX - al + cZ) 3 - g 2 Z 2 - 2 mXYZ = 0. 
Hence the new origin is a binode. There are no other singular points on the 
surface. 
The biplanes are bX. — aY cZ E gZ = 0. 
They intersect in the straight line 6X — aY = 0, Z = 0. 
Hence the binode locus is 2 = 0, and the edge of the binode, which lies in the 
binode locus, satisfies the condition for contact with the binode locus. 
(C.) The Locus (c 3 — g 2 ) mz — 2 g 2 xy = 0 is an Ordinary Envelope. 
To prove this it is necessary to satisfy at the same time 
( bx — ay + cz ) 2 — g 2 z 3 — 2 mz {x — a) (y — b) = 0 . . . . (a), 
(c 3 — g 2 ) mz — 2 g 2 xy =0.(e), 
2 b {bx - ay + cz) — 2m {y — b)z — 2a (bx — ay + cz) — 2 vi (x — a) z 
- 2 ghy ~ - 2(fx 
2c (bx — ay + cz) — 2g l z — 2m ( x — a) (y — b) / n 
m (c 2 - g 2 ) ’ ' ' 
Multiplying numerator and denominator of the first ratio in (£) by x, of the second by 
y, and of the third by 2 ; adding the numerators to form a new numerator, and the 
denominators to form a new denominator, and reducing by (a) and (e), each of the 
above ratios 
2 (xy — ab) 
