104 
ON THE CUBICAL DIVERGENT PARABOLAS. 
[399 
y n - — a?— 2 d, 
simplex campaniform quasi-neutral, R without the curve, I at infinity; 
y 2 = a? + Sex + 2 d, 
simplex campaniform, R without and further from, I within and nearer to the curve ; 
y 2 = sc 3 + Sex, 
simplex campaniform equidistant, viz. R and I are equidistant from the curve, 
R without and I within; 
y 2 = a? + Sex — 2 d, 
simplex campaniform, R without and nearer to, I within and further from the curve. 
Passing to the complex forms, suppose for a moment that a is the diameter of 
the oval and /3 the distance of the oval from the vertex of the infinite branch ; the 
equation of the curve then is y 2 = x (x — a) (x — a — ¡3), or changing the origin so as to 
make the term in x 2 to vanish, this is 
y 2 = (® +1« + \$) + ££) 0- -§£), 
or, what is the same thing, 
y 2 = a?-1 (a 2 + a/3+ fi 2 )x - (a -¡3)(2a +/3) (a+ 2$), 
or comparing this with y 2 = a? — Sex + 2d, d is = +, 0 or —, as a < /3, a = ¡3, a. > ¡3, or say 
as the oval is smaller, mean, or larger; viz. the magnitude of the oval is estimated 
by the relation which the diameter thereof bears to the distance of the oval from 
the infinite branch. In the case d = 0, or for the curve y 2 = x a — Sex it appears (as 
for the corresponding simplex form y 2 = af+ Sex) that the points R, I are equidistant 
from the point x = 0, which is in the present case the middle vertex, or vertex of 
the oval which vertex is nearest to the infinite branch. As the oval diminishes, so 
that the curve becomes ultimately acnodal, I remaining within the oval ultimately 
coincides with the acnode; and as the oval increases so that the curve becomes 
ultimately crunodal, R remaining between the oval and the infinite branch, ultimately 
coincides with the crunode; and it hence easily appears by continuity that for 
a smaller oval I is nearer to, R further from the middle vertex; while for a larger 
oval, I is further from, R nearer to the middle vertex. Hence for the complex forms 
the species are 
y 2 — X s — Sex + 2d, 
smaller oval, I nearer to, R further from the middle vertex; 
y 2 = x 3 — Sex, 
mean oval, R and I equidistant from the middle vertex ; 
y 2 = ct? — Sex — 2d, 
larger oval, I further from, R nearer to the middle vertex: and the division into 
species is thus completed. 
Cambridge, June 16, 1865.