408 
ON CUBIC CONES AND CURVES. 
[351 
where it may be noted that the equator X + Y+ Z = 0 does not cut the two trilateral 
regions (+ + + +) and ( ); and further that the line X = Y = Z which is the 
harmonic of the equator X + Y + Z = 0 in regard to the system of the three tangents 
JF^=0, lies wholly in the two trilateral regions (+ + + +) and ( ). 
19. The equation in question, 
(X+ Y+Zy + QkXYZ= 0, 
shows that, as above stated, the cone lies wholly in the 8 trilateral regions, or in the 
6 quadrilateral regions, viz. if k be negative, it lies wholly in the 8 trilateral regions, 
and if k be positive, it lies wholly in the 6 quadrilateral regions. Let k be negative, 
then the positive quantity — , which is 
XYZ 
~ (X +Y+Z) 3 ’ 
if we attend only to the values of X, Y, Z which have the same sign (that is, to 
points in one of the two trilateral regions), has a maximum value = corresponding 
to X = F= Z. And if — exceeds this value, that is, if — k < f, or, what is the 
same thing, if k lie between the values 0 and — then the equation — jr- = ^ ^ 
& 2 1 Qk (X+Y + Z) 3 
cannot be satisfied in the assumed manner, that is, by values of X, Y, Z having the 
same sign; and thus no portion of the cone lies in the two trilateral regions: in the 
contrary case, that is, if k lie between the values — oo, — |, the equation can be so 
satisfied, and a portion of the cone lies in the two trilateral regions. 
Hence k being negative, we have as follows : 
k between — go and — f, the cone is complex, 
k = — f, the cone is acnodal, 
k between — f and 0, the cone is simplex trilateral ; 
and k being positive, or say 
k between 0, oo, the cone is simplex quadrilateral. 
20. It is to be remarked that for k = 0, the cone as represented by the foregoing 
equation degenerates into the threefold plane (X + Y + Zf = 0. The value k = 0 
corresponds however to the value 1 = 1 of the parameter l in the equation 
a? + y 3 + £ 3 + 6lxyz = 0, that is, it corresponds to the simplex neutral cone, represented 
by the equation 
a? + y 3 + z 3 + §xyz = 0, 
which, as already remarked, is not transformable into the form (X + Y + Z) 3 + QkXYZ = 0: 
this leads to the consideration of the transformation in question.