Full text: The collected mathematical papers of Arthur Cayley, Sc.D., F.R.S., sadlerian professor of pure mathematics in the University of Cambridge (Vol. 2)

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MEMOIR ON CURVES OF THE THIRD ORDER. 405 
y 3 -^ = (yi - z i) (yi + ^i 3 + 6^x2/^! - 8^! 3 ) 
= (yi - zi) X - (1 + 81 s )xi, 
since (x x , y x , £1) is a point of the cubic; and forming in like manner the values of 
£3 — p and £ 3 — ?7 3 , we obtain the theorem. 
31. The preceding values of (x 3 , y 3 , z 3 ) ought to satisfy 
Ox 2 + Zly x zi) x 3 + (2/1 2 + 2^x«x) 2/3 + (^x 2 + 2lx x yi) *, = 0, 
# 3 2 + yi + ¿3 2 + Qlx 3 y 3 z 3 = 0; 
in fact the first equation is satisfied identically, and for the second equation we 
obtain 
xi + yi + *s 2 = x i (yi ~ zi) 3 + yi (zi - x 1 3 ) 3 + zi («x 3 - 2/j 3 ) 2 
= - Xx 9 (yi ~ zi) - yi (zi - xi) - zi (xi - yi) 
= (xi + yi + zi) (yi - zi) (zi - xi) (xi - yi), 
x 3 y 3 z 3 = x i y 1 z l (yi - zi) (zi - xi) (xi - yi), 
and consequently 
xi + yi + zi + 6lx 3 y 3 z 3 = (xi + yi + zi + Qlx^^i) (yi - zi) (zi - xi) (xi - yi) = 0, 
which verifies the theorem. It is proper to add (the remark was made to me by 
Professor Sylvester) that the foregoing values 
x 3 : 2/3 : z 3 = x^(yi - zi) : y 1 (zi - xi) : z x (xi - yi) 
satisfy identically the relation 
xj + yj 4 zj __ xj + yj + zj 
x 3 y 3 z 3 x x y x z x 
Article Nos. 32 to 34.—Formulae for the Satellite line and point. 
32. The line %x + yy + %z = 0 meets the cubic 
x 3 + y 3 + z 3 + Glxyz = 0 
in three points, and the tangents to the cubic at these points meet the cubic in 
three points lying in a line, which has been called the Satellite line of the given line. 
To find the equation of the satellite line; suppose that (x 1} y x , z x ), (x 2 , y. 2 , zi), 
(x 3 , y 3 , zi) are the coordinates of the point in which the given line meets the cubic; 
then we have, as before, 
(1, 1, 1, Z$/3£- yy, 7! - a£, a v - ££) 3 = (ax x + fiy x + 7zi) (ax 2 + fiy 2 + 7zi) (ax 3 + /3y 3 + yz 3 ). 
146] A 
These values give
	        
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