390 
A MEMOIR ON CURVES OF THE THIRD ORDER. 
[146 
Hence the «-coordinate is proportional to 
(3№ - TTWZX) (3№ - l + WX'Y') - (3l 2 Z 2 - 1 + 2PXY) 3l 2 Y' 2 - T+WZ'X'), 
which is equal to 
9l 4 {Y 2 Z' 2 - Y*Z 2 ) + 3P (1 + 21*) YY'{XT -X'Y) + 31 2 (1 + 21 s ) ZZ' (ZX' - Z'X) 
- (1 + 21J XX' {YZ' - Y'Z); 
or introducing, as before, the quantities f, y, £, a, ß, y, to 
- 9M + №(1 + 21 s ) (ߣ + yy) - (1 + 2l 3 ) 2 a%, 
= (- 1 - 13I s - H 6 ) a| + 21 2 (1 + 21 s ) (ßS + yy). 
But by the first of the equations (A) ß^ + yy = 2la%, and the preceding value thus 
becomes ( — 1 — *71 3 + 8l 6 ) a%. Hence throwing out the constant factor the coordinates of 
the point 0 are found to be 
ßv, 7^ 
12. The points G, 0 are conjugate poles of the cubic. 
Take a, b, c for the coordinates of G, and a', b', c' for the coordinates of 0, we have 
a, b, c = yy- ß& <-7£> ßZ-ay, 
a', b', c' = ag , ßy , y£ 
These values give aa' + l (be' + b'c) 
= «£ (yy -ߣ) + l [ßv (ߣ - ay) + 7^ «- 71)} 
= %y ( a 7 + ¿ß 2 ) + y 2 (- laß) 4- £ 2 (lay) + (- aß - ly 2 ); 
or substituting for %y, y 2 , £ 2 , their values in terms of a, ß, y, this is 
( 
~h 2 
- \ a ß) («7+m 
+( 
ß 2 
i 2 
— ( 
-laß) 
♦( 
7 2 
l 2 
- ( 
lay) 
+( 
— aß — ly 2 ). 
which is identically equal to zero. Hence, completing the system, we find 
aa' + l {be' + b'c) = 0, 
bb' +1 (ca' + c'a) = 0, 
cc' + l {ah' + a'b) = 0, 
equations which show that 0 (as well as G) is a point of the Hessian, and that the 
points G, 0 are corresponding poles of the cubic.