ON THE CUBICAL DIVERGENT PARABOLAS. 
103 
399] 
for the equation in X; the quadrinvariant I is =0, and hence the discriminant, 
= — 27»/-, is negative; that is, the roots are two real, two imaginary, as already 
mentioned. 
Considering the simplex forms, first, if c = 0, then for the curve 
y 2 = a? + 2d, 
it appears that R lies at infinity, I within the curve; and for the curve 
y 2 = ofi — 2 d, 
that R lies without the curve, I at infinity. 
It further appears that when d = 0, or for the curve, 
y 2 = a? + 3 ex, 
R, I lie equidistant from the vertex, R without, / within the curve. 
Hence in the curve 
y- = x 3 + Sex + 2d, 
since, when d = 0, the points R, I are equidistant from the vertex, and for c = 0, the 
point R is at infinity, it is easy to infer by continuity that the points R, I lie R 
without, I within the curve, / being nearer to the vertex. 
And similarly in the curve 
y n - = a? + 3 ex — 2d, 
that the points R, 1 lie R without, I within the curve, R being nearer to the vertex. 
Again, in the curve 
y 2 = a? — 3cx + 2d, 
since, in the curve y 1 = a? + Sex + 2d, R is without, I within the curve, and as c 
becomes = 0, R passes off to infinity, it appears that c having changed its sign, or 
for the curve now in question, R having passed through infinity, will be situate within 
the curve; that is, R, I lie each of them within the curve. 
And similarly for the curve 
y- — a? — Sex — 2d, 
it appears that R, I lie each without the curve. 
Hence, finally, for the simplex forms, we have the 7 species of Plucker, viz. 
y 2 — x? — Sex + 2d, c 3 < d 2 , 
simplex ampullate, R, I within the curve; 
y 2 = a?— Sex - 2d, c 3 < d 2 , 
simplex campaniform, R, I without the curve; 
y 2 = sc 3 + 2d, 
simplex neutral, / within the curve, R at infinity;