AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 
221 
Putting X 3 — ea/c/3, and multiplying by /3 3 /a, this becomes 
a~/3 : (3p 2 — A 2 ) 2 z 4 + 4pa/3 (3 Jr — p 2 ) (ae 3 fi- ./3c 3 ) z 3 — 4A 2 z 2 (ae 3 — /3c 3 ) 2 
— 32ae 3 /3c 3 A 3 z 2 + 8c 3 e/3X [pa/3z 3 (3A " 3 — p 2 ) — 2 A 3 z 2 (ae 3 -f- /Sc 3 )]. 
Substituting for X its two values dz x/f a ffi), multiplying the results together, and 
reducing, it becomes 
{a 2 /3 3 (3p 2 - A 2 ) 3 z 2 + 4a/3p (ae 3 + /Sc 3 ) (3A 2 - p 3 ) 2 - 4 A 3 (ae 3 -/Sc 3 ) 2 } 2 
- 64a 3 /SW(p 3 + A 2 ) 3 2 2 
This is the same value as before for the discriminant. 
(B.) The Surface z = 0 is a Binode Locus such that the Edge of the Binode touches 
the Binode Locus. 
Transforming the equation by means of x = a + X, y = h + Y, 2 = Z, it becomes 
aX 3 -f- |SY 3 + 3(cX + eY -f gZ) 2 - h 2 Z 2 = 0 . 
Hence the new origin is a binode. 
Hence the binode locus is 2 = 0 . 
The biplanes are 
3*(cX + eY + gZ) - hZ = 0 . 
3“ (cX + eY + gZ) + liZ = 0 . 
The equations of the edge are therefore 
cX + eY= 0 , Z= 0 . 
It lies therefore in the plane Z = 0 , i.e., in the plane 2 = 0 . 
Hence it may be considered to touch the binode locus. 
The condition (76) is satisfied at every point on the binode locus. 
Hence the factor z 4, is accounted for. 
(C.) The Surface 
{a 3 /3 3 (3p 2 — A 2 ) 2 z 2 -j- 4pa/3 (ae 3 + /3c 3 ) (3 h 2 — p 3 ) z — ill 2 (ae 3 — /Sc 3 ) 3 } 2 
— 64 a 3 /3 3 c 3 e 3 (p 3 -fi A 2 ) 3 z 2 = 0 
is an ordinary Envelope. 
This may be proved by finding the tangent planes parallel to the plane 2 = 0 . 
Hence it is necessary to satisfy at the same time 
