MISS 0. A. SCOTT ON PLANE CUBICS. 
261 
Arranging the coordinates so as to give actual distances, with x y z — 1 ^ov 
fundamental identical relation, we wish to determine the various points on h, i.e., on 
y = z', we have therefore 
X 2y =■ 1 . 
For the cubic, 
1 — QXxif = 0 , 
i.e., 
3\a;(a; - 1)2 - 2 = 0.(1). 
For t, T, points on the Hessian, 
X(a;—1)2-|-G(a?—l) + 2=0. (2). 
For S, the cusp on the Ca}deyan, 
x:y ’.z = 2\ ~ 8:1:1; 
therefore 
4 - A 
t c 
(\-Z)(x-l)+l=0 .(3). 
For B, the intersection of inflexional tangents to the Hessian, 
2v — \y = 0, 2v — \z = 0 ; 
therefore 
4 
X = 1 -— ! 
\ 
t 6. 
\(a: - 1) + 4 - 0.(4). 
For P, the intersection of h with the inflexional tangent to the Hessian, 
2v — \x = 0, 
Xcc - 2 = 0. : . (4'). 
For z, points on the Cayleyan, most simply determined as the foci of TP, WH, 
Xx(x — 1) + 1 = 0 
(5). 
By means of these six curves, all of which can easily be drawn with a considerable 
degree of accuracy, we have a diagram (fig. 14), in which for any arbitrarily chosen 
ordinate X the abscissse* give the positions of all the points required in constructing 
the selected cubic, its Hessian, and its Cayley an. It will be noticed that the curves 
(P) and (tr) touch at x = ^, X = 4 ; that (K), (P), and (S) meet where X = 3 fl: \/3, 
and that (P) and (B) meet where X = 6, agreeing with the conclusions of §§ 14-16, 
* For the sake of distinctness, in fig. 14 the abscissa x is measured on a scale three times that of the 
ordinate \. 
