AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 
251 
the result is 
m2 3 o 
- — z 6 . 
2 xy + mz 
If this be expanded in ascending powers of z, the lowest is the third power. 
But the rationalising factor applied to form the discriminant, viz. — mz (2 xy -j- mz) 
contains the factor z. Hence, the factor z 4 is accounted for. 
The discriminant is as before 
mz 4 (2 xy -j- mz — m). 
(B.) The Uniplanar Node Locus is z — 0 . 
Put x = a + X, y = b + Y, 2 = Z in the equation. It becomes 
(bX - aY + Z ) 2 - Z 3 - 2 mXYZ = 0 . 
Hence the new origin is a uniplanar node. 
Hence z = 0 is the uniplanar node locus. 
(C.) The Envelope Locus is 2 xy + mz — m — 0 . 
The equation can be written 
ary 9, — 2 ab (xy + mz) -j- Irx 9 + 2 a(m — 1) yz + 26 (m + 1) xz + p 
= p — z 3 + z 3 + 2 mxyz. 
Let p be determined as a function of x, y, z, so that the left-hand side of the 
equation may break up into factors linear with regard to a, h. 
Then 
p = 2 3 — 2 mxyz 
2 xy + 
It may then be verified that the equation can be written 
a + - 2 [ — b (xy + mz) + (to — 1 ) yz\ 
J' 
- ^z(2xy + mz) | b - y + — 
y- 
2 xy + mz 
£ (2 xy + mz — m) 
2 xy + mz 
Hence it may be concluded that 2 xy + mz — m — 0 will touch the surface where 
both the factors of the left-hand side vanish, i.e., where 
a + — [— b (xy + mz) + (m — 1) yz] = 0, 
b - V + : 
yz 
2 xy + mz 
= 0 , 
2 k 2 
