CoNKAN — Some Theorems on the Twisted Cubic. 251 



Let x = \^; y = \'{l-v\f', z = [l + Xy; 



the tangent at X is x [\ +\Y - y - zX^ = ; 



this meets the curve at fx if 



fx' (1 + A)^ - 11^ (1 + pif - (1 + At)- V = 0. 



Dividing by [fx - Xf we find 



Ail = - (1 + XJ + v/l + X + X^ 



//3 = - (1 + X) - v/1 + A + X\ 

 Harmonic conjugates to fxi and /lo are 



i"!^ + K:A'2^ iUi^ (1 + jUi)" + Kfxz^ (1 + AX2)-, (1 + jUi)' + (c (1 + A*,/; 

 and 



fJ-l' - KfX2^, jUl^ (1 + A«l)^ - KfX2^ (1 + fXzf, (1 + Atl)^ - K (1 + /Xi}-. 



If the first point satisfies x + y + z =^ 0, 



(I + Ati + liii~)\ 



(1 + A^2 + iUj')' 



.*. The second point is 



X = Ati' (1 + At2 + iU3^)' + A^2^ (1 + Afi + f^i'Y ; 



2/ = Ail' (1 + fxiY {1 + fX2 + fX2^y + fxz (1 + A<2)' (1 + A^i + A'l')' ; 



z = {I + fXiY (1 + At2 + fJiz'f + (1 + A^2)' (1 + A^i + lui-y. 



Now 



1 + At2 + i«2' = 2 (1 + X + X=^) + (1 + 2X) yi +X + X'. 



Ati (1 + iU2 + A^2-) = - (1 + X + X=^) + (1 - X) yi + X + X^ 



(1 4 Ati) (1 + iU2 + luz') = 1 + X + x^ + (2 + X) yi + X + x^ 



Ail (1 + Ail) (1 + iU2 + iU2') = 1 + X + X^ - (1 + 2X) yi +X + X'. 



a; = 2 (1 + X + X^)^ + 2 (1 - X)^ (1 + X + X^) ; 



y = 2(1 + X + X'y + 2 (1 + 2X)- (1 + X + X^) ; 



s = 2 (1 + X + X')^ + 2 (2 + X)2 (1 + X + X-). 

 X : y : z 



: : 2X^- - X + 2 : 5X' + 5X + 2 : 2X^- + 5X + 5^ 



8x-y-z = K.9{X-lf; 

 8y - z - X = K.9 (2X + 1)^ 

 8z - X - y = K.9{X + 2)-; 



ySx - y - z ■{- ySy - z - x + y^z - x - y = 0, 

 which proves the theorem. 



[36*] 



