MISS C. A. SCOTT ON PLANE CUBICS. 
259 
with degenerate Hessian and Cayleyan. Through the degenerate three-point form 
the Cayleyan passes from bipartite to unipartite (fig. 10). The cusps recede from T 
through H towards D, passing through the form for which they are at H (fig. 11), 
and therefore collinear on the line infinity. After this, we have the unipartite 
harmonic cubic, with a bipartite harmouic Hessian (fig. 12) ; the infinite branches of 
the Hessian are outside the limits of the diagram, but fig. 8 represents, on a smaller 
scale, the relation of the cubic (fine line) to the Hessian (heavy line). As K still 
recedes towards H, the cusp approaches Kf,; when K reaches H, the series gives a 
degenerate cubic; .but if we substitute for this the one that belongs to the series of 
proper cubics (see No. 351, in vol. 5, of Professor Cayley’s collected papers) viz., the 
one with the real inflexional tangents concurrent,* we have the change as in the case 
of the other equianharmonic cubic—the Hessian is three straight lines, and the 
Cayleyan changes from unipartite to bipartite through the three-point form. We 
then have (fig. 13) the quadrilateral unipartite cubic, with the bipartite Hessian and 
Cayleyan, these, as K approaches T, tending to coincidence with the sides and 
vertices of the triangle D^D^Dg. 
IV. Analytical Expi'cssion. Fig. 14. 
18. In considering the appearance of the cubic and its derived curves, the 
equation 
[x + y -p zY — (^\xyz — 0 
(discussed and compared with Hesse’s form by Professor Cayley, loc. cit.) appears 
more convenient than Hesse’s canonical form. It postulates only three inflexions, so 
excluding only the cuspidal form, and is therefore more comprehensive ; it relates 
only to elements all of which may be taken real, except for two special cubics, and is 
therefore convenient when diagrams are required. 
The invariants for this form are 
S = - V (4 _ X) ; T = - 8X^ (6 - 6\ + X^) ; 
A = T^ — 64S3 = — 4 X 64 X X8 (2X - 9) ; 
* In order to deal with the cubic whose real inflexional tangents are concurrent, while preserving 
the distinction between real and imaginary, suppose the lines (I), /tj, ha, to remain fixed, while the 
triangle formed by the inflexional tangents changes, D approaching 0 and then passing through it, so 
that the segments ODH, OTH are interchanged. The point Kq is indefinitely near to O, so K is 
beyond Kq, and the cubic is unipartite. Let K remain fixed, and let it be initially in the segment 
ODH, then by the interchange of segments it is finally in OTH ; consequently B, initially in OTH, is 
finally in ODH ; and (§ 13) the Cayleyan changes from unipartite to bipartite. 
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