288 
ON THE INFLEXIONS OF THE CUBICAL DIVERGENT PARABOLAS. [345 
I remark that the equation x 2 + 2bx + c = 0 gives the level points (i.e. the points 
where the tangent is parallel to the axis) of the cubic parabola. In the complex case, 
where c < § b 2 , then a fortiori c < b 2 or the values of x are both real, one of these 
values gives y 2 positive, and we have thus the maximum ordinate of the oval; the 
other value of x gives y 2 negative. In the simplex case, where c > §b 2 , we may have 
l°.c<6 2 , and the two values of x give each of them y 2 positive; the least value of 
x corresponds to a maximum ordinate, the greatest to a minimum ordinate of the 
cubic parabola, and between these we have the ordinate through the two real inflexions, 
the tangents at the inflexions meeting on the axis within the parabola. 2°. We may 
have c = b 2 , the two values of x here coincide, giving the ordinate through the two 
real inflexions, the tangents at the inflexions being horizontal. And 3°. We may have 
c > b 2 , the two values of x are then imaginary and we have no real level point. 
These are, in fact, Murdoch’s three forms, which he distinguishes as the ampullata, 
media, and campaniformis. 
2, Stone Buildings, W.C., June 2, 1863.