503] THE VERTICES OF CONES WHICH SATISFY SIX CONDITIONS. 137 
whence 
A=PP', B = -(PQ' + P'Q), G = QQ'; 
then 
4 AC-B 2 = -(PQ'-P'Q) 2 ; 
and we have 
P, Q = z a x-zx a , W a x-y a z, 
P', Q' = z b x-zx b , w b x-y b z, 
whence 
PQ’ - P'Q = (z a w b - z b w a ) x 2 
+ [VaZb “ yb z a ~ (X a W b ~ X b W a )\ XZ 
+ - °c b y a ) P; 
viz. if (a, b, c, f, g, h) are the coordinates of the line ab, this is 
= hx~ + (a — f) xz + cz 2 . 
Hence, omitting the constant factor (y — v) 4 (v—\) 4 (\—y) 4 {that is (Xp —X y ) 4 (X y — \ a ) 4 (\ a —X^) 4 }> 
the foregoing equation norm 2 = 0 becomes 
[ha; 2 + (a — f) xz + c^ 2 ] 4 = 0, 
and the intersections of the quadric with the surface are obtained by combining the 
equation xw — yz = 0 with the several equations 
{ha? 2 + (a — f) zx + c z 2 } 8 = 0, 
{{x - \ a y) (x - Xpy) {x - X y y)}* = 0, 
{h« 2 + (a — f) zx-j- cz 2 } 4 = 0 ; 
viz. these are 
lines (a, ¡3, 7, 8) 
each 
8 
times 
16 
line (x = 0, z = 0) 
16 
35 
16 
lines a, /3, 7 
each 
8 
33 
24 
line (x = 0, y= 0) 
24 
33 
24 
lines [ab, a, /3, 7] each 
4 
>3 
8 
line (x = 0, z — 0) 
8 
33 
8 
(16 + 24 + 8) x 2 = 48 + 48 
But it is clear that the lines (x = 0, y = 0) and (x = 0, z=0) are introduced by the 
process of elimination, and are no part of the intersection. The complete intersection 
consists of the lines (a, ¡3, 7, 8) each 8 times, the lines (a, /3, 7) each 8 times, and 
the lines [ab, a, /3, 7] each 4 times. But the last-mentioned lines being only double 
lines on the surface, this means that the two sheets each touch the quadric surface, 
or that the lines are tacnodal. 
C. VIII. 
18