Egberts — On Binary CiiUcs and their Associated Forms. 527 



The forms A and v have been discussed before, and it remains to 

 attach a geometrical meaning to ®J' = {aaya^a^. E'ow, from any given 

 point on the curve can be drawn two chords which will meet the line 

 00', the above equation in X : //, determining the points of meeting on 

 0' ; if, then, the co-efficient of A/x is zero, the chords drawn from the 

 given point meet the line 00' in two points harmonically conjugate 

 with regard to and 0' . 



Hence 0^^ is represented geometrically by the two points on the 

 curve ; the chords from which, to meet the line 0', divide it har- 

 monically. 



IX. 



Let us now determine the tangent lines to the curve which meet a 

 line through X, Y,Z, W, and a point on the curve the parameter of 

 which is Xi : Xn. 



The co-ordinates of a line through X, Y,Z, W, and a point x-^ : x-^ 

 on the curve, are 



a = xr~ xJZ - x^xo!^ Y, /= x^^ W - Xxo!\ 

 h = x{ x.r-X- x(^Z, (J - x'-xJ TF - Xx,'\ 



c = x,'^Y -x,'-x^X, h = x^x.!^W-Zx^\ 



forming the condition that this line may meet a tangent line, the co- 

 ordinates of which are given by Y., and dividing by a factor 



we find 



x^^ ( JTx^' - Zx.^) + 2xiX2{ YxJ - Zx{) + ( Yx( - Xxi) x\ = 0. 



Suppose now the point X, Y,Z, TF to be the point 0, and the point on 

 the curve, the point j^^ or (Aa)-a^, the above equation becomes 



Xi~. ittax' + a^x-l) + 2^1:^2. {a^xl + aox-l) + [a^x-l + a^iXn^j x^ = Q\ 



or {ap)a^ = 0. "We can easily express this in terms of Clebsch's forms 

 as follows : — 



hence 2(®A)®,- («a)%ffl;^(aA) + a^(«A)}; 



therefore 2 (0 A )®^^^ = a^{a^){aa){aa) H^^ 



+ («a)^(flA)aj.A^ = 



«^(fla)(aA){(?^(Aa) + a^(«A)} + (aa)^(«A)a,A, 



= - a,\aa) (Aa)2. (See Clebsch, § 34.) 



Hence {pa) a/ = 2 (0A) 0^ A^. 



The three remaining quadratic covariants are constructed as fol- 

 lows : — 



