414 
ON CUBIC CONES AND CURVES. 
I 
[351 
And there is corresponding to l = ±— 1) or k = — 3 + V3, a special form which 
might be called the simplex harmonic, viz. the tangents from any line of the cone 
form a harmonic system. It is to be observed that, qua simplex cone, the four 
tangents are two of them real, two imaginary. 
I = 1; k = 0 (form fails). The cone is simplex neutral. 
I between 1 and go , or k between 0 and go ; the cone is simplex quadrilateral. 
32. It will be observed that the crunodal and cuspidal kinds of cones do not 
present themselves in the foregoing investigations; the reason is that the crunodal 
kind admits of no representation in the form x? + y 3 + z 3 + Qlxyz = 0, and (inasmuch as 
there is only one real inflexion) it admits of no real representation in the other form 
(X + Y + Zf+ 6kXYZ = 0; the cuspidal kind admits of no representation in either of 
the two forms. 
I conclude with a discussion not absolutely required for the purpose of the memoir, 
but which is of interest in regard to the form (X + Y + Zf + Q/cXYZ =0. 
Reduction of the Hessian to the form (X' + F + Z'f + Qk'X'Y'Z' — 0. 
33. The cubic cone 
a? + y s + z 3 + Qlxyz = 0, 
has for its Hessian 
or say 
if 
Hence writing 
we have 
— P (x? + y 3 + z 3 ) + ( 1 + 2l 3 ) xyz = 0, 
a? + f + z 3 + 61'xyz = 0, 
1 + 21 3 
l’ = - 
6P • 
X' = — 2 l'x + y + z, 
Y = x — 21' y + z, 
Z' = x + y — 21' z, 
(X' + Y + ZJ + 6k'X' Y Z = 8( ' 2 ff)f 4p 1)5 (a* + f + *• + 61'xyz) 
Hence the equation of the Hessian is 
{X' + F + Z'f + Qk'X'Y'Z' = 0, 
4(f-iy 
l-2i'+4Z' ! ' 
where the value of k' is