254 
MISS C. A. SCOTT ON PLANE CUBICS. 
but these are equal, and therefore 
[To.GoySDW] = {K.G,/3DH2]; 
hence TgW, KHj must meet on the line Go^SD, at . 9 -. 
Projecting [DWOK} from - 9 ^ on to Ih, it becomes = [GoToOHg}, which by projec¬ 
tion from I on to ^ = [KWOH], therefore 
[DWOK] = [HOWK], 
therefore K is self-conjugate in the involution HD, OW ; i.e., for a harmonic cubic 
the point K is a focus of HD, OW. Hence there are two such cubics, one with K 
as in fig. 4 , giving a unipartite cubic ; one with K between O, W, giving a bipartite 
cubic. These points are at once found in the symmetrical diagram ; for H being at 
infinity, D is the centre of the involution ; and since Dl^ = DW. DO, we must have 
DK = DTo. Thus the two positions of K are as in figs. 8, 12. 
15 . The Equianliarmonic Cubics. —In special cases three inflexional tangents may 
be concurrent, this being allowed by the class of the cubic being = 6 ; but not more 
than three. Further, the three will be tangents at collinear inflexions ; for the line 
polar of the intersection of two inflexional tangents is the join of the inflexions, and 
thus if a third inflexional tangent pass through this point, the third inflexion must 
be the one that lies on this line. We can certainly find a line of inflexions for which 
the tang-ents are not concurrent, and therefore if we disreo^ard the distinction 
between real and imaginary, we can still use the symmetrical triangular diagram; the 
three concurrent tangents cannot meet in O (for the polar line of O is the line (I), 
which joins inflexions having non-concurrent tangents), therefore by triangular 
symmetry there must be three sets of concurrent tangents ; plainly if one of these 
be composed of the three real tangents, the other two must be composed of 
imaginary ones; in the other possible arrangement, the sets are composed each 
of one real and two imaginary tangents. 
Considering the two tangents that are concurrent with IT, we know that these two, 
being tangents at inflexions collinear with I, must meet on h ; their intersection is 
therefore at T, Now the Hessian has to touch each of these inflexional tangents, in 
addition to cutting it at the inflexion ; passing through T, it cannot meet the 
inflexional tangent again so as to touch it, consequently for every one of these three 
inflexional tangents the “ contact ” has to be at T; there can therefore only be 
improper contact, i.e., the Hessian must have a double point at T^, and similarly at 
Tg and Tg; it is therefore composed of the three lines T3T3, TgTj^, T;lT3. Now we 
know that the line polar of T is the inflexional tangent to the Hessian at I; and we 
have seen that, for the case we are considering, the line polar of T is the line joining 
