322 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
[349 
27. Assume that 6 has its twofold value = — the equations 
x ■ V ■ QZyi : : 6+~v ’ 
substituting also therein for X, /a, v the values a -3 , /3~ 3 , 7 -3 , give for the coordinates of 
the twofold centre 
r .... Z=z a3 @V . #*7« . a @T 
^ ¡3y — a 2 ' yu — /3 2 ' a/3 — 7 2 ’ 
but in virtue of the relation a + /3 + 7 = 0 we have 
/3y — «2 = /37 + 7a + a/3 = 7a — /3 2 = a/3 — y 2 ; 
or the values are x : y : z = a 2 : /3 2 : y 2 . Hence also we have as the equation of the 
locus of the twofold centre, 
\/x + \/y + ^/z=0, 
or, what is the same thing, 
x 2 + y 2 + z 2 — 2yz — 2zx — 2 xy = 0, 
which is a conic touching the lines x = 0, y = 0, 2 = 0 at their intersections with the 
lines y — z — 0, s — ¿c = 0, x — y = 0 respectively, or, what is the same thing, in the points 
(0, 1, 1), (1, 0, 1), (1, 1, 0) respectively. 
28. 
Similarly, if 6 has its one-with-twofold value 
the equations 
x ' y ' z 0 + x : 0 + fi : 0 + v > 
substituting also therein for X, y, v the values a 3 , /3~ 3 , 7 3 , give for the coordinates of 
the one-with-twofold centre 
a? By /3 s yen y’’a(3 
X : V '' Z ~ 2a 2 + (3y : 2/3 2 + yu : 2 7 2 + a/3’ 
but in virtue of a + /3 + y = 0 we have 
2a 2 + /37 = a 2 — a (/3 + 7) + /3y — (a — /3) (a — 7) = - (7 — a) (a — £), 
and similarly 
2/3 2 + ya = — (a — $) (£ — 7), 2y 2 + a/3 = - (/3 - 7) (7 - a); 
and thence these values are 
x : y : s = a 2 (/3 - 7) : /3 2 (7 - a) : y 2 (a - /3), 
for the coordinates of the one-with-twofold centre. 
We thence deduce 
y +z = /3°- 7 - /3 2 a + 7 2 a - y 2 /3 = (¡3y -a/3- ay) (/3 - 7) = (/3y + a 2 ) (/3 - 7),