ON THE CUBICAL DIVERGENT PARABOLAS. 
102 
[399 
but of the simplex species, there are five which are to the eye campaniform, and the 
three complex species have with each other a close resemblance in form. 
I remark as regards the simplex forms, that the tangents at the two inflexions 
meet in a point R on the axis, and that the ampullate, the neutral, and campaniform 
forms are distinguished from each other according to the position of R, viz. for the 
ampullate form, R lies within the curve, for the campaniform form R lies without the 
curve, and for the neutral form, R is at infinity. It is to be observed, as regards the 
complex forms, that here R always lies without the curve, between the infinite branch 
and the oval. 
The further division of the simplex and complex forms so as to obtain the 7 + 3 
species of Plticker, may be effected by considering in conjunction with the point R 
a certain other point I on the axis; it is to be remarked that excluding the 
inflexion at infinity the cubical divergent parabola has in all eight inflexions, two real 
and six imaginary, viz. the inflexions lie by pairs on four ordinates, or if x be the 
abscissa corresponding to an inflexion, x is determined by a quartic equation; this 
equation has always two real and two imaginary roots, each of the imaginary roots 
gives a pair of imaginary inflexions; one of the real roots gives a positive value for 
y 2 and therefore two real inflexions, the tangents at these meet in the above-mentioned 
point R on the axis; the other real root gives a negative value for y 2 and therefore 
two imaginary inflexions, but the tangents at these meet in a real point on the axis, 
and this I call the point I. It is clear that for each of the four pairs of inflexions 
the tangents at the two inflexions meet at a point on the axis, so that if X be the 
abscissa of such point, then X is determined by a quartic equation; two of the roots 
of this equation are imaginary, the other two roots are real, and correspond to the 
points R and I respectively. 
The equation of the curve being as above 
y 2 = a? — Sex + 2d, 
then the coordinate x belonging to a pair of inflexions is found by the equation 
ж 4 — 6 cx- + 8 dx — 3c 2 = 0, 
or what is the same thing, 
(1, 0, -c, 2d, -3с 2 $я, 1) 4 =0, 
(the invariant / is =0, and hence the discriminant, = — 27J-, is negative, or the roots 
are two real, two imaginary, as already mentioned): the corresponding value of X is 
easily found to be 
У __ x 3 + Sex — 4/ 
Л = ’ 
and we thence obtain 
3 cX 4 — 4dX 3 — 6c 2 X 2 + 12 cdX — (c 3 + 4 dr) — 0, 
or what is the same thing, 
(3c, —d, — c 2 , Scd, — c 3 — 4/ 2 }£X, 1) 4 = 0,