MISS C. A. SCOTT ON PLANE CUBICS. 
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conic polars, which is constructed from three independent ones. The collinear 
inflexions give three known conic polars, bub these being sjzygetic, amount only to 
two independent ones, leaving one to be determined; the one that is most easily 
found is the conic polar of K ; let this meet h in K'. Since the conic polar of a point 
on a cubic divides any chord through this j)oint harmonically, K, K' are harmonic 
with regard to Jck, and are therefore conjugate in the involution OG, DY ; K' is 
therefore determinable by a linear construction as follows :— 
By harmonic symmetry, Hga, Hoa' meet on h, at e (figs. 1 and 2). Consider the 
triangles aDHg, OG^K ; DHg, Hga, aD meet G^K, KO, OG^ in a', e, Hq, three 
collinear points; the triangles are therefore in perspective, and aO, DGo, HgK meet 
in a point ; by means of the quadrilateral IgGoa/S we see that determines the 
conjugate to K in the involution OG, DY. K' is shown in fig. 2. 
7 . Now I, T being conjugate poles, we know that t, r are also conjugate poles, and 
are therefore conjugate with regard to every conic polar ; t, t are thus conjugate w^ith 
regard to KK', and also with regard to OD (since the conic polar of I3 is the line pair 
T2D, T3O), i.e., t, T are the foci of the involution OD, KK'. 
8. For a certain choice of K, will go through 0, i.e., K' will come at 0, and 
then t, T coincide, at 0 ; but Io/3 can go through D only if Go be at Dg, which makes 
K come at T, an impossible arrangement unless the cubic, and therefore also the 
Hessian, should degenerate ; [or if K be at D, wdiich has the same effect.] Thus 
the Hessian has a double point when I3/3 goes through O, i.e., when Iga/SO are 
collinear, i.e., w^hen a is the intersection of I3O and HgD, the condition already found 
for the occurrence of a double point on the cubic. Now when K is in the segment 
TH, K' is in DH ; when K is in HKq, K' is in HTO ; the foci of OD, KK' are real, 
and to the unipartite cubic corresponds a bipartite Hessian. When K is in KqD, 
K' is in OD; when K is in DO, K' is in DHO; and when K is in OT, K' is in OD ; 
thus the bipartite cubic gives a unipartite Hessian ; and for both cubic and Hessian, 
the transition from the one form to the other takes place through the nodal form. 
9 . As regards the Cayleyan, the cusp which has ^ as a tangent being at S, we 
know by the ordinary construction for the point of contact of a tangent to the 
Cayleyan that T, S are harmonic with regard to tr, and are therefore conjugate in the 
involution OD, KK'. Let IgT meet DG3 in 0 (figs. 2, 3 ), and let IgO meet 0 K in rj ; 
by means of the quadrilateral we see that /S17 goes through S. 
10. The inflexional tangent to the Hessian is determined when S is known ; let JS 
meet IT in X (fig. 3 ), then XHg goes through B. For the proof of this compare the 
Hessian, qud cubic, with the original cubic, and apply to it the proj^erties of the 
diagram for the cubic; for comparison, points on the Hessian may for the moment be 
denoted by the same letters as corresponding points on the cubic, accented. 
2 K 2 
