252 
MISS C. A. SCOTT ON PLANE CUBICS. 
We found that h, k must be the foci of OG, DY, and therefore t, r are the foci of 
OG', D'Y', But K' and G' are respectively T and D, therefore t, t are the foci of 
OD, D'Y' ; also they are known to be harmonic to TS. Now in the original cubic 
(fig. 1), HgD, IgY meet on the tangent at K ; hence, referring this to the Hessian, HgD', 
IgY' meet on the tangent at T, i.e., on IT; call their point of meeting X (fig. 3); we 
have to determine X. Since OD, D'Y', TS are in involution, 
(D'ODT) = (Y'DOS}. 
Project the left-hand side through Ho, and the right-hand side through Tg, on to IT, 
we then obtain (the points M, N, p being as shown in fig. 3 ) 
i.e., 
{XDoMT} = [XDoNp], 
{XD^MT} = {D^XpN} ; 
therefore X, Dg are conjugate in the involution Mp, NT. Hence by means of the 
quadrilateral IgDJS, we see that JS goes through X; and then XHg goes through D', 
i.e., through B. Thus the infiexional tangents to the Hessian are found. A more 
convenient construction may be deduced ; from the identity 
{XDoTM] = {DoXMT}, 
there follows, by projection on to h from Ho and J, 
[BOTD] = {QSDT}, 
i.e., BQ, OS, DT are in involution. Thus to find B, let IgT meet HgS in p ; then by 
means of the quadrilateral HgTgL^/x, we see that Lop. goes through B. 
11 . The points 2, ^ on the Cayleyan are its points of contact with the conic polar 
of T. Now the inflexional tangent to the Hessian, i.e., IP, is known to be the line 
polar of T with regard to the original cubic ; it is therefore the line polar of T with 
regard to the conic polar of T ; and consequently T, P are harmonic with regard to z^. 
Also lo, Tg are conjugate poles, and are therefore conjugate with regard to the conic 
polar we are considering, viz., with regard to Iz, I^; therefore projecting from I on 
to h (fig. 3 ), we see that W, H are conjugate with regard to zt,. Thus 2, ^ are the 
foci of the involution TP, WH. 
] 2. The constructions are therefore :— 
(l.) IK meets H^D in a ; Iga meets li in Y ; k, k are the foci of OG, DY (fig. 2.) 
