349] 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
327 
36. The equation of the other tangent is 
13yx + yay + af3z — 0, 
or, what is the same thing, 
This is in fact equivalent to 
X , y z = 0 ; 
a 2 , ß- 7 2 
a 2 (£ - 7)> /3 2 (7 - a X 7 2 (« ~ /3) 
for, developing the determinant, we find 
x. /3 2 7 2 (2a - /3 — 7) + 2/. 7 2 a 2 (2/3 — 7 — a) + z. a 2 /3 2 (27 — a — ¡3) — 0, 
or, what is the same thing, 
hence the line 
which is one of the tangents at the one-with-twofold centre, is also the line joining 
this point with the twofold centre. 
37. The equation of the tangents at a critic centre or node may be obtained in 
a different form, involving, instead of the parameter 0, the coordinates (x, y, z) of the 
node. We have 
(0 + \)x=q( x + y+z), 
2 
or, what is the same thing, 
and similarly 
thence also 
(6 + 4\) x = 0 (x — 2x + 2y + 2z) = 0 (— x + 2y + 2z), 
= 4 0 [2yz + x* - (y - z) 2 }, 
— \0 (x 2 — y 2 — z 2 + 4>yz) 
and