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 ;