399] 
101 
399. 
ON THE CUBICAL DIVERGENT PARABOLAS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. ix. (1868), 
pp. 185—189.] 
Newton reckons five forms, viz. these are the simplex, the complex, the crunodal, 
the acnodal, and the cuspidal, but as noticed by Murdoch, the simplex has three 
different forms, the ampullate, the neutral, and the campaniform. We have thus the 
8 forms at once distinguishable by the eye. 
Pliicker has in all 13 species, the division into species being established or completed 
geometrically by reference to the asymptotic cuspidal curve (or asymptotic semi-cubical 
parabola), and analytically as follows, viz. writing the equation in the form 
y 2 = x? — Sex + 2d, 
the different species are 
simplex, y 2 = x? — Sex + 2d^ ^ ampullate, 
„ y 2 = x? — Sex — 2dj ’ campaniform, 
„ y 2 = x? + 2d, neutral, 
f = a? — 2d, campaniform, 
„ y 2 = x? + Sex + 2d, „ 
„ y 2 = a? + Sex, >> 
„ y 2 = x 3 + Sex — 2d, j, 
complex, y n - = x? — Sex + 2d 
„ y n - = x? — Sex — 2d 
„ y 2 — x? — Sex, 
acnodal, y 2 = x? — Sex — 2c V (c), 
crunodal, y 2 — x? Sex + 2c V (c), 
cuspidal, y 2 = x?\