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VII. On Plane Cuhics. 
By Charlotte Angas Scott. 
Communicated hy Dr. A. R. Forsyth, F.R.S. 
Received Sept. 9,—Read November 2S, 1893. 
No systematic investigation by simple geometrical methods of the variation of the 
Hessian and Cayleyan as dependent on the variation of the fundamental cubic 
appears to have been undertaken hithei’to, though the general relation of the three 
curves has been thoroughly studied both geometrically and analytically. This 
investigation however appears desirable, not only for itself, but also for the sake* of 
the explanation it offers of the importance and interest of some special cuhics. 
In the following pages the first few sections are devoted to certain constructions 
for the three curves, which are then applied to special cuhics, among these the erpri- 
anharmonic cubic, whose known properties present themselves very sim})ly by means 
of the preliminary constructions. The cubics here considered are, as appeal's in the 
next section, the critical ones when we follow out the variation of the Hessian and 
Cayleyan. In conclusion, the results are compared with those derived by analysis, 
and are exhibited graphically by means of a single diagram. 
I. Construction of the Cubic, its Hessian and Cayleyan. Figs. 1-3. 
1 . Let three collinear inflexions of a cubic be I^, L, I 3 (fig. l); call the iutersections 
ol the tangents at these inflexions Dj^, D^, the points in which they meet the 
harmonic polars T^, T.,, T 3 , the points in which the harmonic polars TjD^, TolL, 
T 3 D 3 , i.e., hy, h. 2 , A 3 meet the line of inflexions Hj, Hg, H.j, and the intersection of the 
harmonic polars O, so that 0 and the line (I) are pole and polar with regard to the 
triangle D^DoD,,. 
Let the points of contact of the three tangents from I^, which are necessarily 
on the harmonic polar hy, be K^, hy, Ky, &c. The arrangement of the K s is deter¬ 
mined by a consideration of the sixteen lines that have (I) for satellite. These 
sixteen lines are 
(1.) 
( 2 .) I^Ko, which must pass through one of the thi'ee points Kg, Ag, Kg; call this 
point Kg, and similarly select K^ by means of IgK^; then will I.iK^Kg be collinear. 
For {IiI.HJg] is harmonic, as also (I^KgV^K,], being the point in winch IiK 3 K 3 
meets the harmonic polar hy ; hence I^Kg, I 3 K 0 must meet on LI^V^, i.e., on hy, and 
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