352 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
[349 
these give {y = 0, 2 = 0) or else (— x + y + 2 = 0, vz 2 — /xy 2 — 0), that is, one of the three 
critic centres is the angle {y = 0, z— 0) of the triangle; and the other two are the 
intersections of the line —x+y+z= 0 with the pair of lines vz 2 — ¡xy 2 = 0. 
It should be remarked that, given the critic centre y — y 1 = 0, z = z x = 0, the re 
maining two centres cannot be determined as the intersection of the polar — x + y + z = 0 
with the conic 
+ 3/i (*i~0 + Oi - yi) = 0 
X y z 
inasmuch as the equation of this conic becomes the identity 0 = 0. 
83. The critic centres for the case in question, X = 0, may also be determined by 
means of the equation of the cubic through the three centres; in fact, since A = 0, 
the equation Xa + /x/3 + vy = 0 becomes ¡xfi + v<y = 0, that is ¡3 : 7 = v : — /x ; and the 
equation of the cubic therefore is 
and since the ratio a : ¡3 is arbitrary we have the two equations 
! = 0, 
X x+y+z ’ y z x + y + z 
which resolve themselves into the above-mentioned two equations, —x+y + z — 0, 
vz 2 — ¡xy 2 = 0. 
84. Consider a critic centre the coordinates of which are (0, y lt z x ), that is, which 
is an arbitrary point on the side x = 0 of the triangle: it is to be remarked that 
there is not any position of the line Xx + /xy + vz = 0, which properly gives rise to such 
a critic centre. 
For writing ¿r x = 0 the equations 
-x 1 + y 1 +z 1 _ x 1 -y 1 + z! = x 1 + y 1 - z 1 
Xx! fxy, vz! 
№ 
give fx = 0, v = 0, that is, the line Xx+ /xy + vz = 0 is found to be x = 0; but in this 
[yz + k{x + y + z) 2 ^j = 0, which irrespectively of the value of k has 
case the cubic is x 
nodes at the points x = 0, yz + k{y + zf = 0, and which only for the value k = 0 acquires 
a third node at the point y = 0, 2 = 0: the case is a singular and exceptional one. 
85. If notwithstanding we assume a critic centre at the point (0, y i} 2^, then 
the other two critic centres are by the general theorem given as the intersection of the 
line 
(3/1 + Z X ) X - (y, - 2i) (y - z) = 0 
with the conic (pair of lines) x (y — z) = 0, that is, we have a twofold centre x = 0, y — z = 0, 
or what is the same thing a twofold centre (0, 1, 1).