250 
MISS 0. A. SCOTT ON PLANE CUBICS. 
4. The two points /i, k, will be real or imaginary according to the position of K; 
they will coincide, so giving the acnodal cubic, when Y comes at 0, i.e., when Iga 
goes through 0. Thus the position of K for the acnodal cubic is the intersection of 
li with LJ, where J is the intersection of IgO, HgD ; call this j)oint Kg. If now we 
take K a very little furtlier away from D, Y is no longer at 0, but is between O and 
T ; thus the involution OG, DY, being overlapping, has imaginary foci, and the cubic 
is unipartite ; and similarly taking K a little nearer to D, we see that the cubic is 
bipartite. 
Now suppose that 
K travels from Kg towards H, 
then 
Y travels from O through T towards H, 
and 
G travels from Gg towards H. 
Thus G is initially beyond Y (estimating from O on the symmetrical diagram) 
(fig. 2), and travels at the same rate as Gj, which travels at the same rate as a, and 
therefore at the same rate as Y; consec|uently G remains beyond Y, i.e., the involu¬ 
tion remains overlapping, and the foci are imaginary. Thus when K is anywhere 
between H and Kg the cubic is unipartite. 
Now let 
K travel from Kg through D, O, T, towards H, 
then 
Y travels from 0 through D, . . . towards H, 
and 
G travels from Gg through . . O, D, towards H. 
The cubic is initially bipartite, and the segments OG, DY keep clear of one 
another until G comes at D, i.e., until K is at T; thus the cubic is bipartite when K 
is anywhere in KgOT. Similarly taking K in TH, we see that the cubic is unipartite. 
5. We next consider the Hessian and the Cayleyan. The Hessian has the same 
inflexions and harmonic polars, and passes through T^, Tj, Tg; let the triangle formed 
by the inflexional tangents be B^B^Bg, the sides of this meeting the harmonic polars 
in P^, Po, Pg. We have to determine B and P, which can be done by a linear 
eonstruction ; and t, r, the remaining points in which h meets the Hessian, are found 
as the foci of a certain involution. As regards the Cayleyan, we know that T is 
again a point, and that the harmonic polar h is a cuspidal tangent ; we arrive at a 
linear construction for the cusp S ; and z, the remaining two points in which h 
meets the Cayleyan, present themselves as the foci of an involution. 
6. Both the Hessian and the Cayleyan are explicitly dependent on the system ol 
