388 A MEMOIR ON CURVES OF THE THIRD ORDER, 
[146 
and thence 
r 
v 2 
r 2 
vt 
£ 
Zv 
and conversely 
^ 2 (l + 8Z 3 )a 2 = 
^ (1 + 8Z 3 ) /3 2 = ?? 2 -4^, 
p (1 + 8Z 3 ) 7 2 = £ 2 -4^, 
-i(l+8P)/8 7 = 2ip+ > 
-^(l + 8Z 3 ) 7 a =2l V >+ & 
-j 2 (l+8l*)a/3 = 2l?+ % V . 
= ^ a 2 — 4/3 7 , 
= /3 2 - 4 7 a, 
= 7 2 - 4a/3, 
= - - a 2 _ I /3 7 
Z Z 2P7, 
=-l *-*#■’ 
(C) 
(D) 
8. It is obvious that 
f x + yy + & = 0 
is the equation of the line EF joining the two conjugate poles, and it may be 
shown that 
(xx + /3y + yz = 0 
is the equation of the line IJ, which is the polar of E with respect to a conic 
which is the first or conic polar of F with respect to any syzygetic cubic. In fact, 
the equation of a syzygetic cubic will be a? + y 3 + z 3 + QXxyz = 0, where A, is arbitrary, 
and the equation of the line in question is 
(Xd x + Yd y + Zd z ) (X'd x + Y'd y + Z'dz) O 3 + y 3 + z* + QXxyz) = 0; 
or developing, 
XX'x + YY'y + ZZ’z 
+ \{YZ' + Y'Z) x + {ZX' + Z'X) y + (XT + X'Y) z] = 0 ;