345 | ON THE INFLEXIONS OF THE CUBICAL DIVERGENT PARABOLAS. 285 
is not without interest. The result which should be obtained is, by the general theory 
of cubic curves, known to be as follows : there is always an inflexion at infinity, in 
the point where the line infinity is met by the line x = 0 (or, what is the same thing, 
by any ordinate of the curve) ; but disregarding altogether this inflexion at infinity, 
then in the general case where the curve is without singularity, the remaining eight 
inflexions (two of them real, six imaginary) lie in pairs on four ordinates of the curve: 
if however the curve has an acnode, the six imaginary inflexions coincide with the 
aenode, viz. the three ordinates corresponding to these pass through the acnode, but 
there are still two real inflexions ; if the curve has a crunode, four of the imaginary 
inflexions and the two real inflexions coincide with the crunode, viz. the three ordinates 
corresponding to these pass through the crunode, and there is not any real inflexion, 
although there are still two imaginary inflexions; finally, if the curve has a cusp, then 
the eight inflexions coincide with the cusp, viz. the four ordinates corresponding to 
these pass through the cusp, and there is no inflexion real or imaginary. 
Proceeding now to the analytical investigation, if in order to form the Hessian we 
introduce the new coordinate 2 = 1, the equation of the curve becomes 
U = — y-z + ax? + 3b xz- + 3cxz 2 + dz'\ 
and tbence forming the second differential coefficients, and ultimately replacing z by its 
value, = 1, we have 
HU = 
6 (ax + b), 0 , — 6 (bx + c) 
0 , 2 , 2y 
6 (bx + c), 2y, — 6 (cx + d) 
= 0; 
whence, developing and dividing by 24, we find 
•3 {(ax + b) (cx + cl) — (bx + c) 2 } + (ax + b) xf = 0, 
or, what is the same thing, 
3 {(ac + b 2 ) x 2 + (ad — be) x + (bd — c 2 )} + (ax + b)y- = 0, 
as the equation of the Hessian curve, meeting the given cubic curve 
ax 3 + 3 bx 2 + 3 cx + d — y 2 = 0, 
in its points of inflexion. Multiplying the last-mentioned equation by b, and adding it 
to the equation of the Hessian, we obtain 
aba? + 3acx 2 + 3 adx + 4 bd — 3c 2 + axxf- = 0, 
or, what is the same thing, 
xxf 
bx 3 — 3cx 2 — 3 dx + 
3c 2 — 4 bd 
a