258 
MISS C. A. SCOTT ON PLANE CUBICS. 
therefore the Hessian is three straight lines, and the inflexional tangents to the 
cubic are concurrent in threes. 
Now for an equianharmonic cubic, the three points K, h, k are not differentiated 
as they are for a harmoDic cubic; therefore they cannot be found by linear and 
quadratic constructions. But plainly they cannot all be real, and the cubic is there¬ 
fore unipartite. 
16. Other sjDecial cubics might be considered, as for instance the one for which B 
and P coincide; this coincidence is necessarily at O, and thus the Hessian is equiau- 
harmonic. In the general case, BQ, OS, 1)T are in involution, thus in this case 
OQ, OS, DT are in involution, and therefore S comes at Q. Moreover, s, I, the foci 
of TP, WH are now the foci of TO, WH, and are therefore real, giving a bipartite 
Cayleyan. 
Again, the three cusps on the Cayleyan may be colliuear, i.e., S may be at H. In 
this case B is conjugate to Q in the involution OH, DT, and therefore comes at L; 
and t, T are now the foci of OD, TH, and are therefore real; thus the Hessian is 
bipartite. In both these cases K cannot be found by linear or quadratic constructions. 
HI. Variation in the Hessian and Cayleyan as the Cubic varies. Figs. 6-13. 
17. The cubics just considered are of interest in studying the variation of the 
Hessian and Cayleyan as dependent on the variation of the original cubic. Figs. 6-13 
exhibit this variation ; the cubic is represented by the heavy lines, the Hessian by the 
faint lines, and the Cayleyan is dotted. For these figures the point K was assigned, 
and the points h, k-, t, t ; S; B; z, determined by the constructions of § 12; for 
figs. 7 and 11 the position of K was determined by approximation and trial. 
K starts from D, and describes the segment DHT, the segment TOD being 
described by the complementary h, k for the bipartite cubic. The inflexional triangle 
for the Hessian (fig. 6) is at first turned the same way as that for the original cubic, 
but then by transition (fig. 7) through the form for which the Hessian is equian- 
harmonic, it turns the other way. The tricusp of the Cayleyan shrinks up, until the 
cusps, initially outside the oval of the cubic, are on the cubic (fig. 8), which is now 
harmonic, and accompanied by a unipartite harmonic Hessian, The tricusp is now 
inside the oval, and both shrink up to the point 0, giving the acnodal cubic, with an 
acnodal Hessian, and a degenerate Cayleyan composed of the point 0 and a conic, 
which for the symmetrical diagram is the circle inscribed in the triangle D^D^^Dg, 
At this stage all trace of the oval is lost, but the oval of the Hessian makes its 
appearance. The tricusp of the Cayleyan cannot disappear, so it now expands from 
the point form, reversed in position (fig. 9) as compared with its original form. The 
cusp and the point B approach T together, and we have the equianharmonic cubic. 
