ON THE CUBICAL DIVERGENT PARABOLAS.
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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;