MISS C. A. SCOTT ON PLANE CQBICS. 
249 
(b.) Let IK meet H^D in a (fig. 1 ), and let Iga meet h in Y^, i.e. Y. Then k, k are 
harmonic with regard to DY. 
By harmonic symmetry, constructing a' by means of HgD, passes through Y ; 
let Hga meet in sr, and similarly for cr', then toct' passes through I; hence the 
quadrilateral has I, Y for two of its vertices. We have to show that Igts-, 
Lot', which by harmonic symmetry meet on h, actually meet at K. 
We have 
{I.HYtiT'K} = [IsYt^'a] = {I3T0H3IJ 
[by projection through a on to the line (I)], and is therefore harmonic; be., K is the 
intersection of the diagonals. 
Now consider the triangle Yaa', and determine the polar of D. Y, a, a, projected 
through D on to the sides, give K, 7jS y TIT j TiTTIT j To K, Kot meet aa , aY, Ya at 
Ij, Ig, I 3 ; hence the line (I) is the polar of D, and estimating on the transversal h, 
we have 
J 2 _ 3 
1)K 1)Y 1)H. 
Now the line (I) is the polar of D with regard to the cubic, and therefore 
From (iii.) and (iv.). 
DK ^ m ^ B/c DH 
1 
hk 
(iv.) 
i.e., h, K are harmonic with regard to DY. Thus b, k are the foci of the involution 
OG, DY, and are therefore given when K is given. 
3. Now the IDH scheme depends on a triangle and one other straight line. Thus 
any two such schemes can be projected into one another ; i.e., excluding for the 
present (1) the cubic with three real concurrent inflexional tangents, (2) the crunodal 
cubic, (3) the cuspidal cubic, we may say “ all cubics have the same framework.” But 
in connecting projectively the frameworks of two cubics we have exhausted the possi¬ 
bilities of projection, and so have no means of bringing the K’s of the two cubics to 
coincidence ; thus different positions of the three K’s on Ji give essentially distinct 
cubics, so exhibiting clearly the known fact that the essential nature of the general 
cubic depends on one parameter only. 
Since we can project so that the triangle D^DgDg becomes equilateral, while the 
line (I) goes to infinity, we can always use a symmetrical diagram. This simplifica¬ 
tion is adopted for most of the diagrams here given. 
MDCCCXCIV.— A. 2 K 
