MISS C. A. SCOTT ON PLANE CUEICS. 
255 
the inflexions whose tangents are concurrent in T ; this line polar is therefore the 
tangent to the Hessian at each of the three inflexions, i.e., it forms a part of the 
Hessian, Thus the line T 3 T 3 joins three inflexions, and the tangents at these three 
inflexions pass through ; i.e., the Hessian is composed of the three lines joining the 
inflexions whose tangents are concurrent. 
Conversely, if the Hessian be composed of three straight lines, the inflexional 
tangents to the cubic (if a proper cubic) are concurrent in threes. For these nine 
inflexional tangents have to “ touch ” the Hessian; they must therefore have improper 
contact, i.e., they must pass through the three double points T^, Tg, Tg of the Hessian, 
and there being nine of them, three must go through each point T. 
Thus the two conditions, “ the inflexional tangents are concurrent in threes,” and 
“the Hessian is three straight lines ” are coextensive ; and there is plainly no need to 
exclude the degenerate cubics from this enunciation. 
The two points t, t now come at T, W ; therefore T, W are the foci of OD, KK', TS ; 
i.e., S must come at T, and therefore the Cayleyan is composed of the three points‘T. 
For P is at W, therefore z, I, are the foci of an involution which degenerates into 
TW, WH, i.e., they are at W, and consequently double points and double tangents 
(at W) are introduced on the Cayleyan. But it has already its maximum number, 
and therefore it is now a degenerate curve. Being a class-cubic, and preserving its 
triangular symmetry while degenerating so as still to pass through T\, Tg, Tg, it can 
only degenerate into these three points. 
Conversely, if the Cayleyan split up into three points, since the cusps cannot 
disappear, and the points T are in all cases points <311 the Cayleyan, we know that the 
three points are the points T, and that the degeneration is brought about by the 
coincidence of S with T. Now T, S have been proved conjugate in OD, KK', hence 
in this case T is a focus of OD, KK'; but t, t are the foci of this involution, and 
therefore one of the two points t, r, must come at T; and thus the Hessian has 
a double point at T^, and similarly at T, and Tg. 
The condition therefore that “ the Cayleyan splits up into three points ” is 
equivalent to those already discussed. 
We have now to show that if three inflexional tangents be concurrent, the cubic is 
equianharmonic. Referring tbe diagram to the concurrent tangents, a comes at G 3 , 
Y at G, and thus the construction requires modification. In the general case T, I, 
and therefore in the present case O, I, are conjugate poles on the Hessian, and are 
therefore conjugate with regard to any conic polar ; similarly for O, I 3 and for O, I 3 . 
Thus the line (I) is the polar of O with regard to every conic polar; t.e., 0, H are 
conjugate with regard to the conic polar of K, and therefore with regard to K K'; 
thus K' is known. 
Now [KGHO] by projection tbrough Gg (fig. 5) 
= {IlI^H^Hg} 
