AND LINES IN THE INTERSECTIONS OP A SYSTEM OF SURFACES. 
195 
Hence it is necessary to find x, y, z, so that 
f=rx( x — af + /3 (y — 6) 3 -f 3 [c (,x - a) + gzf = 0 . . 
Therefore 
therefore 
3 7T. = a ( x ~ a Y + 2c [ c ( x ~ «) + 9*] 
N fjf = 2 9 [o(x~a) + gz] 
y — b, 
a(x — of -f 3 [c (x — a) + gzf — 0, 
a (x — a) 2 + 2c [c {x — a) + gz] = 0 ; 
[c (x — ci) + gz] [c (x — a) + 3 gz] = 0. 
The solution c (x — a) + gz = 0 is inconsistent with (85). 
Hence it is necessary to take 
c (x — a) + 3gz = 0. 
Substituting in (83), 
(9gaz — 4c 3 ) = 0. 
= 0 . 
= 0 . 
#0 . 
■ ( 82 ), 
• (83), 
• (84), 
, (85). 
The solution z— 0, gives x = a, y — b, and therefore belongs to the unode locus. 
The solution 9 gaz — 4c 3 = 0 
gives 
ct. {x — ct)' -f- 2c~ {x — ct) -fi 8cY9a —— 0, 
and therefore 
x — a — — 4c~/3a, 
or 
x — a — — 2o 3 /3a. 
Hence at every point y, 4c 3 /9ya on this locus, the locus is touched by two non- 
consecutive surfaces of the system, viz., those whose parameters are given by 
ci = £ + 4c 3 /3a, b — y; 
and 
a = ^ + 2c 2 /3a, b = y. 
This accounts for the factor 
(9 gaz — 4c 3 ) 3 
in the discriminant. 
2 c 2 
