351] 
ON CUBIC CONES AND CURVES. 
405 
10. A non-singular cubic cone (simplex or complex) may be represented by an 
equation of the canonical form 
x 3 4- y 3 + z 3 + Olxyz = 0, 
where the coordinates x, y, z and the parameter l are all real; the invariants of the 
form are S = — l + l i , T =1 — 20I s — 81 s . It is to be noticed that the form in question 
cannot represent a singular cone; we find as the condition that it may do so 
R = 64S’ 3 -T 2 = -(l + 81 3 ) 3 = 0, 
but when this condition is satisfied, the cone breaks up into a system of three planes; 
thus for the real root 1 = —^, we have 
a? + y 3 + z 3 — ’3xyz = (x + y + z) (x + coy + arz) (x + co 2 y + coz), 
where co is an imaginary cube root of unity; and by merely writing cox, co 2 x successively 
in place of x, we see that the like decomposition occurs from the imaginary roots 
l = — \ co, 1 = co 2 . 
11. The equation a? + y 3 + z 3 + Olxyz = 0 is in general transformable into the form 
(X+Y+Z) 3 +0kXYZ = 0, 
where X, Y, Z are linear functions of the original coordinates, such that X — 0, F = 0, Z = 0 
are the equations of the tangent planes at three inflexions in the plane X + F+ Z = 0; 
if however the three tangent planes meet in a line, then X, Y, Z will satisfy identi 
cally a certain linear equation, and it is clear & priori that the transformation must 
fail. The condition for the three tangent planes meeting in a line is S = — l +1 4, = 0, 
that is, we have 
1 = 0, 1, co, or co 2 ; 
and attending only to the real roots / — 0, 1 = 1, it will be presently seen that for 
1 = 0 the tangent planes at the three real inflexions do not, for 1 = 1, they do meet 
in a line. Hence the simplex neutral cone corresponds to the value 1 = 1, that is, the 
equation is 
x 3 + y 3 + z 3 + Oxyz = 0, 
and this equation is not transformable into the form (X + Y + Zf + OhXYZ = 0, which 
is that employed in the sequel for the general discussion of the simplex and complex 
cones. The theory on which the foregoing conclusion depends is as follows. 
On the condition S= 0. Nos. 12 to 17. 
12. A cubic has in general nine inflexions, which lie by threes on twelve planes, 
viz. denoting the inflexions by 1, 2, 3, 4, 5, 6, 7, 8, 9, the planes may be taken to be 
123, 
456, 
789, 
147, 
258, 
369, 
159, 
267, 
348, 
168, 
249, 
357, 
that is, we have four systems, each of three planes passing through the nine inflexions.