AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 263 
Hence, if <f>(x , y, z) = 0, then, in order that the above equations may be satisfied, 
cix -j- b -J- y = 0, 
2 — 0, 
The locus of these points is the curve 
2 = 0, y, z) = 0. 
This belongs to one of the exceptional cases enumerated in the Section VI. of this 
paper. 
Example 17.— This example, shows the difference between the cases when the equation 
is of the Second Degree in the parameters and those in which it is of a Higher 
Degree, so jar as regards Binode and Unode Loci, 
Let the surfaces be 
cl(x — a) 3 + 3/3 (x — a) 2 z + cz 3 -fi ‘3d (y — b) 2 + ez 2 = 0. 
(A.) The Discriminant, 
It is the same as that of the equation 
«X 3 + 3/LX-Z + 3dY 2 Z + (c‘ 2 3 + ez 2 ) Z 3 = 0. 
Therefore 
S = - 
T = 4 d?z 2 [a 3 (e + cz) -(- 2/3%}. 
Therefore 
A = l6c/% 4 (a 4 (e + czf + 4a 3 /3% (e + cz)}. 
(B.) The Locus of Biplanar Nodes is 2 — 0. 
For putting x = a -j- X, y = b + Y, z — Z, the equation becomes 
aX 3 + 3/3K 2 Z + cZ 3 + 3 dY 2 + eZ 2 = 0. 
The edge of the biplanes is given by Y = 0, Z = 0. 
Hence the edge of the biplanes lies in the biplanar node locus 2—0, and, therefore, 
satisfies the condition for contact with the biplanar node locus. 
Hence the factor 2 4 is accounted for (Art. 15). 
(C.) If e — 0, the Locus of Uniplanar Nodes is 2 = 0, 
In this case, 
A = 1 6d 6 («V 2 + 4a 3 /3 3 c) 2 9 . 
Hence the factor 2 fi is accounted for (Art. 12), 
