349] ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 353 
86. If a critic centre lie on the line y — z= 0, then of the other two critic 
centres, one lies on this same line and the other is the point x = 0, x + y + z = 0, or 
say the point (0, 1, —1). And in this case the line \x + yy 4- vz = 0 passes through 
the last-mentioned point; that is, we have y = v. Conversely, starting from the equation 
y = v, so that the line \x 4- yy 4- vz = 0 is \x 4- y {y + z) = 0, a line through the inter 
section of the lines x = 0, x + y + z= 0, the equation in 0 is 
(0 + y) (6 2 — y0 — 2\y) = 0, 
where the factor 0 + y = 0 corresponds to the critic centre # = 0, x J t-y + z = Q, or 
(0, 1, — 1), (it will presently be shown that this is so), and the quadric equation 
0- — yd — 2\y = 0 corresponds to two critic centres on the line y — z = 0. We have 
x ■ V ■ z = : ’ 
and thence y — z = 0; and 0 (x — y) = — \x 4- yy, which substituted in the equation 
0 2 — y0 — 2Xy = 0 gives 
(\x - yy) {(X + y)x- 2yy] - 2\y (x - yy = 0, 
and the two critic centres are given as the intersections of this conic by the line 
y~z = 0. 
87. Consider for a moment the case v = y-f e, where e is ultimately 
equation in 0 is 
1 - 1 1 = 
0 + X 6 + y 6 + y + e 0 ’ 
then if a root is 6 = — y + Ae, we have 
1 1 1 2 
Ae 4- X — y Ae (A + 1) e Ae —y 
so that, e being indefinitely small, we have 
\ + = 0, that is, 2A + 1 = 0 or A = — 4, 
A A + l 2 
and then 
6 — — y — ^6, ^ + X = X — y — 0 4* y = — 2" e ; 0v =e, 
which gives 
i 1 1 _ 1 _2 2 
^ 0 + \ 0 + y 0 + v \ — y—\e e e 
or, e being indefinitely small, x \ y : z = 0:1:—1, so that the factor 0 + y 
sponds, as mentioned above, to the critic centre (0, 1, — 1). 
= 0, the 
= 0 corre 
ct V. 
45