248 
MLSS C. A. SCOTT ON PLANE CUBICS. 
therefore necessarily at K^. Similarly the three points /t^, are grouped, and 
also the remaining three k^, k.^, thus giving nine of tlie sixteen lines. 
(3.) For the remaining six ; must go through one of the points on Ag ; now this 
cannot be Kg or Ag, hence it must be K-g; thus these six lines are of the type KjA^/Cg. 
Now let the tangents at K^, Kg meet at G^, which, by harmonic symmetry, is of 
course on We have thus three groups of G’s, viz.: — G^, Go, Gg; g^, g^, g^\ 
7v 72 ) Tsj arranged in triangles, corresponding to the K’s, and, moreover, collinear in 
threes, again corresponding to the K’s. The proof of this last statement depends on 
a property proved in the next paragraph, that are harmonic with regard to 
O, G| ; for then 
aacHiHjj = 
l,C, 
Therefore the three points (OHo) (I 1 A 4 ), (OHj (I^Gj), (OHg) he., go, Gj, yg, 
are collinear. 
2 . The three collinear inflexions with their tangents amount to eight conditions ; 
thus any one of the nine points K com]:)letes the determination of the cubic ; conse- 
queidly the tw-o points A, k, must be determinable from K'”' ; as a matter of fact they 
present themselves as the foci of a certain involution. 
(a.) A, K are harmonic with regard to OG. One of the four poles of the line (I) 
(flg. 1 ) -with regard to the cubic is 0; hence, estimating on the transversal A, 
we have 
A. 4. 1 . 1 _ A r >1 
0K“^0/A O/C OH. 
Now consider the triangle GG 0 G 3 ; OG, i.e., A, meets GoGs K; ^ 2^3 
meets GoGg in I, &c., therefore the line (1) is the polar of O with regard to this 
triangle. Hence, again estimating on the transversal A, 
From (i.) and (ii.). 
- + — = — 
OK ^ OG OH 
A 
OA 0/c OG ’ 
(ii.), 
i.e.. A, K are harmonic with regard to OG. 
* Points on the three harmonic polai’s arc natni-ally distinguished by suffixes 1, o; but as tlie con¬ 
clusions are applicable indifferently to the points on any one harmonic polar, though all the constructions 
start from Aj, the suffix 1 is in general dropped in the text, while the suffixes 2, 3 are retained. The 
poiiits K|,, vJu, in § 4 are special positions of Kj, Gj. 
