344 
ON A CASE OF THE INVOLUTION OF CUBIC CURVES. 
[349 
61. The fourth conic through the critic centres is the conic 
y — v v — X X — a 
-— + + —- = o, 
x y z 
which it will be observed passes through the vertices of the triangle x = 0, y = 0, z — 0, 
and also through the point (1, 1, 1) which is the harmonic of the line x + y + z = 0 
in regard to the triangle: I call it the ‘ harmonic conic.’ Representing the equation by 
f <7 h A 
- + - + - =0, 
X y z 
or, what is the same thing, 
2fyz + 2 gzx + 2 hxy = 0, 
we have f=y — v, g = v — X, li = X —y, and therefore f+g+h= 0. 
62. It is easy to show that the coordinates of the pole of the line x + y + z = 0 
in regard to the harmonic conic are x : y : z =f 2 : g 2 : h~; these values satisfy the 
condition fx + fy+fz — 0, that is, the pole in question lies on the twofold centre 
conic. 
63. The equation of the tangents to the harmonic conic at its intersection with 
the line x + y + z = 0 (which tangents meet of course in the last-mentioned pole, that 
is in a point of the twofold centre conic) is found to be 
2fgh (x + y + z'y + □ (2fyz + 2gzx + 2hxy) = 0 ; 
if for shortness 
□ = P + p 2 + h 2 - 2gh - 2 hf - 2fg, 
or what is the same thing 
□ = - 4 (gh + hf+fg), =2 (f 2 + g- + h 2 ). 
64. We have identically 
— 6fgh (x 2 + y 2 + z 2 — 2yz — 2 zx — 2 xy) 
= 2fgh (x + y + zf + □ (2fyz + 2gzx + 2hxy) -8 (fx + gy + hz) (ghx + hfy +fgz) r 
so that the tangents in question meet the twofold centre conic 
X 2 + y 2 + z 2 — 2 yz — 2 zx — 2 xy = 0, 
at its intersections with the lines fx+gy + hz = 0, and ghx + hfy + fgz = 0: the latter 
of these is in fact the tangent of the conic at the point (/ 2 , g 2 , h 2 ) of intersection 
of the two tangents. Hence the two tangents meet at the point (f 2 , g 2 , h 2 ) of the 
twofold centre conic and they besides meet the conic at its points of intersection with 
the line fx + gy + hz = 0.