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ON THE INFLEXIONS OF THE CUBICAL DIVERGENT PARABOLAS. [345 
as the equation of a curve meeting the given curve 
у 2 = аж 3 + 3 bx 2 + Sex + d, 
in its points of inflexion; and if for greater simplicity we assume a = 1, d = 0 (the 
latter equation means obviously that the origin is taken at one of the three inter 
sections of the curve with the axis of x, say the real one, if the intersections are 
one real, two imaginary), then the equation of the curve is 
y 2 = x (x 2 + Sbx + 3c), 
and the inflexions are given as the intersections of the curve with the curve 
xy- = — Ъх 3 — Зсж 2 + 3 c 2 . 
There is, it is clear, an inflexion at the point at infinity on the line x = 0; and 
eliminating y 2 we find 
x x + Sbx 3 + Sex 2 = — bx 3 — Sex 2 + 3c 2 , 
or, what is the same thing, 
x x + 4 bx 3 + 6cx 2 — 3c 2 = 0, 
a quartic equation giving the four ordinates through the remaining eight inflexions. 
If the curve has a cuspidal point, then the origin will be at the cusp, and we 
have b = 0, c = 0, and the quartic equation becomes X х = 0; that is, the four ordinates 
pass through the cusp. 
If the curve have a node* then taking the origin at the node we have c = 0; 
the equation of the curve is 
y 2 = x 2 (x + 3b), 
and the curve has a crunode or an acnode according as b is positive or negative; 
the quartic equation becomes 
x 3 (x + 4b) = 0, 
and the factor x 3 — 0 gives three ordinates through the node; the remaining factor 
x + 46 = 0 gives the ordinate through the two inflexions; and substituting this value 
of x in the equation of the cubic, we find 
y 2 = — 166 3 , 
and the resulting values of у (consequently also the inflexions) are imaginary if b be 
positive, that is, for the crunoclal form; but real if b be negative, that is, for the 
acnodal form. It is to be observed that the indefinite ordinate x + 46 = 0 or x = — 46 
is real in each of the two cases: in the crunoclal case, the ordinate lies outside the 
curve, that is beyond the loop; in the acnodal case inside the curve, that is on the 
opposite side to the acnode in regard to the vertex ; and using 3b to denote the distance 
(taken positively) of the vertex from the node, (that is, in the crunodal case changing 
the sign of b), the distance (taken positively) of the ordinate from the vertex is 
= 46 — 36, =6, = ^. 36, that is, it is one-third of the distance of the vertex from the 
node.