318 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
[349 
18. In particular, taking in succession for the point (a, b, c) the point of inter 
section of the line Xx + yy + vz = 0, with the lines x = 0, y — 0, z — 0, x + y + z = 0, the 
critic centres lie on the conics 
v fi 2 (v - fi) _ Q 
y Z X + y + z 
X v 2 (X — v) _ q 
Z X X+y+z 
/f _ x _ 2(y — X) _ 0 
x y x + y + z 
y — v v — X X — y 
~T~ "T" V' : 
X y z 
which are useful for the construction of the critic centres for a given line Xx + yy + vz = 0. 
The last of the four conics passes through the point (1, 1, 1) which is the harmonic of 
the line x + y+z = 0 in regard to the triangle formed by the lines x = 0, y = 0, 0 = 0; 
and I call it the harmonic conic. 
Article Nos. 19 to 21. General Formulce for the Critic Centres. 
19. I consider the involution 
xyz + k (x + y + z'f (Xx + +y + vz) = 0. 
Writing the equation in the form 
and differentiating with regard to x, y, z respectively, we obtain 
— kX (x + y + zf = yz (- x + y + z), 
— ky (x + y + z) 3 = zx ( x — y + z), 
— kv(x+y+z) 3 = xy( x + y — z), 
which determine the coordinate ratios x : y : z of the node or critic centre ; and the 
corresponding value of k. 
20. Writing the equations under the form 
— k(x+ y+zf _ —x + y + z _x — y + z _x + y — z 2 
xyz Xx yy vz 6’ 
where 6 is an auxiliary parameter to be determined, we find