100 
[819 
819. 
ON TWO CASES OF THE QUADRIC TRANSFORMATION BETWEEN 
TWO PLANES. 
[From the Johns Hopkins University Circulars, No. 13 (1882), pp. 178, 179.] 
Seeking for the coordinates x 3 , y 3 , z 3 of the third point of intersection of the 
cubic curve a? + y 3 + z 3 + Qlxyz = 0 by the line through any two points (x x , y x , z x ), 
(x 2 , y. 2 , z 2 ) on the curve, the expressions present themselves in the form 
x 3 : y 3 : z 3 = P + 21A : Q + 2IB : R + %IC, 
where 
P = x x y x yf + z 1 x x z? - yrx,y, - z x 2 z, x,, A = x 2 y 2 z, - y 2 z x xi, 
Q = yiZ 2 zi + X 2 y x Xi - Z?y,Z, - xjx,y 2 , B = y, 2 Z, X, - Z\ X x yi, 
R = z x x x x£ + y x z 2 yi - x 2 z,x, - y?y»z 2 , C = zfxpy. - x x y x z.?\ 
but it is known that, in virtue of 
U x = x x + y x 3 + z x 3 + 6lx x y x z x = 0, U. 2 — x<? + y 2 3 + z 3 + Qlx 2 y 2 z 2 = 0, 
which connect the coordinates (x x , y x , z x ) and (x. 2) y. 2 , z. 2 ), we have P : Q : R — A : B : C*, 
so that the coordinates (x 3 , y 3 , z s ) of the third point of intersection may be expressed 
indifferently in the two forms 
0C 3 : y s : Z 3 = P : Q : R, and x 3 : y 3 : z 3 = A : B : C. 
But these considered irrespectively of the equations U x = 0, U 2 = 0, are distinct 
formulae, each of them separately establishing a correspondence between the three 
points (x x , y x , z x ), (x 2> y. 2 , z 2 ), (x 3 , y 3 , z 3 ), or if we regard one of these points as a fixed 
point, then a correspondence between the remaining two points, or if we consider these 
as belonging each to its own plane, then a correspondence between two planes. 
See Sylvester on Rational Derivation of Points on Cubic Curves, Amer. Jour, of Math. vol. in. p. 62.