349] ON A CASE OF THE INVOLUTION OF CUBIC CURVES, 
349 
or, as this may be written, 
(y + z) | x (x + y + x) — 4y^| = 0, 
so that the cubic locus breaks up into the line y+z = 0 and into the conic 
x (x + y + z) — tyz = 0. 
74. I say that the critic centres lie, one of them on the line, and the other two 
on the conic. 
In fact, putting \ = | (ya + v) the equation in 0 is 
0 3 — 0 (/¿V + J (ya + vf) — % /jlv([a + v) = 0, 
that is 
and we have 
jtf + \ (ya + *>) j j# 2 - i (fi + v)0- /iv| = 0, 
^ 0+%(/J, + v)'0 + /jl'0 + v' 
75. Hence if 0+^(/j, + v) = 0, we obtain 
x : y : * = —T 
whence also 
I (/* + v ) ' i O “ v) ' | (fi - v) 
:^ V : -1 : 1; 
ya + v 
(/j, + v) x + (fi - v) y = 0, 
(ya + v) x — (ya — v) z =0, 
y + z =0, 
so that the corresponding critic centre lies on the line y-Vz — 0; the last-mentioned 
equations, restoring the value 4 A in place of ya + v, may also be written 
4<\x -f (ya — v) y = 0, 
4Xx — (fi — v) z = 0, 
y + z =0. 
76. If on the other hand 
0 2 ~ 2 + v) 0 — /jlv = 0, 
or, as this equation may be written, 
o0' 2 — (0 + 2ya) (0 + 2v) = 0, 
then observing that in general, in virtue of the equation 
1 1 1__2 
0 + X + 0 + ya + 0 + v 0