If, instead, we substitute for 6 the onefold value = , we find 
a/3y 
— (— x + y + z) (x — y + z) (x + y — z) + xyz = 0, 
or, what is the same thing, 
a? + y 3 + z s — (yz 2 + y'-z + zx 2 + z 2 x + xy- + x-y) + 3 xyz — 0, 
which is the one-with-twofold centre cubic. 
32. Recollecting that 
-i* 
(0 + A,) (6 + /-l) (0 + v) 
we deduce for the twofold value of k 
ia 3 /3y 
k, = k. 2 = 
(¡3y - a! 1 ) (ya - 0 2 ) (a/3 - r) ’ 
■ a :i /3y 
(0y + y a + a/3) 3 ’ 
and for the one-with-twofold value, 
{(x 2 + /3“ + y 2 )" 
— ^ 0L S /3 Sr f 
(2a 2 -4- 0y) (20 s + y a) (2y 2 + a/3) 
jor/3y 
(0 - 7 ) 2 (7 - a) 2 (a - 0) s '