284 
[345 
345. 
ON THE INFLEXIONS OF THE CUBICAL DIVERGENT 
PARABOLAS. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. vi. (1864), 
pp. 199—203.] 
The five divergent parabolas, species 67 to 71, of Newton’s Enumeratio Linearum 
tertii Ordinis (1704), are included under the general equation y 2 = ax?+ 3bx 2 + 3cx +d : 
there are two general forms, or forms without singularities, viz. the parabola cum ovali, 
sp. 67, and the parabola pura, sp. 71; two forms having a double point, viz. the nodata, 
sp. 68, and the punctata, sp. 69, according as the double point is one with real 
branches, or is a conjugate or isolated point; and finally the cuspidata or semicubical 
parabola, sp. 70, which has a cusp. In the nomenclature of my short note “ On Curves 
of the Third Order,” British Assoc. Report for the Year 1861, Notices &c. p. 2, the five 
parabolas are the complex, the simplex, the crunodal, the acnodal, and the cuspidal; 
the distinction there made of the simplex kind of cones of the third order into three 
subspecies, applies to the simplex parabola, and for this particular case was, as I have 
since ascertained, noticed in Murdoch’s Newtoni Genesis Gurvarum per Umbras, 8vo. 
Lond. 1746, pp. 1—126. It may be remarked that in this very interesting and valuable 
work the number of species is given as 78, viz. the author includes the four species 
added by Stirling, and the other two usually considered to have been added by Cramer 
(one of them the author himself attributes to Cramer), and that the demonstration of 
Newton’s theorem is effected in the most complete way by showing in what manner 
the five cones are each of them to be cut so as to obtain the 78 species of cubic curves. 
The analytical investigation of the points of inflexion of the above-mentioned 
divergent parabolas, that is, the curves defined by the equation 
y 2 = ax 3 + 3 bx 2 + 3cx + d,