AND LINES IN THE INTERSECTIONS OF A SYSTEM OF SURFACES. 
Hence, when x = £, y = rj, z = £, 
[«i, Oq] [<2 1} &i] 
[ a n ^ 1 ] [^n ^ 1 ] 
is of order e 1 '*. 
Hence 
[a x , oq] [«!, ^i] 
[ a i> ^1] [^n -fi] 
is of order e, and therefore vanishes at points on the conic node locus. 
Similarly it can be shown that the term 
j [cc 2 , a 2 ] [%, /a] 
/j / 
vanishes at points on the conic node locus. 
Therefore A, dA/dx, 3 2 A/3x 2 all vanish on the conic node locus. 
Therefore A contains C 3 as a factor. 
Example 9.— Conic Node Locus which is also an Envelope. 
Let the surfaces be 
a (x — a) 3 + S/3 (y — bf 3y(x ~ a) z ~j~ Sz' 2 = 0, 
where a, /3, y, S are fixed constants ; a, b the arbitrary parameters. 
(A.) TheD iscriminant. 
It is the same as that of the equation 
a X 3 + 3/3Y 2 Z + SyzXZ 2 + SYZ 3 = 0 . 
Therefore 
S = afLyZ, 
T = 4« 2 /3 3 Sz 2 . 
Therefore 
A = 164/3 8 z 3 (aS 2 Z fi- 4y 3 ). 
(B.) The Conic Node Locus , which is cdso an Envelope , is z — 0 . 
To prove this, transform the equation by means of x = a 4 X, y = b 4 Y) £ = Z. 
It becomes 
aX 3 4 3/3Y 3 4 3yXZ 4 8 Z 3 = 0 . 
Hence the new origin is a conic node, and one of the tangent planes of the conic 
node is Z = 0 . 
Hence the conic node locus is % — 0 , and it is also an envelope. 
