ME. A. CAYLEY’S MEMOIE ON CUEVES OE THE THIED OEDEE. 
421 
1?= — 7“’— 
2 ^2^/3 
lri = — -i7—pap', 
and conversely, 
=r (D) 
i(l + 8?F=,' -il%l 
i(l+8P)/=^' -4f5, 
-i(l+8?)(3y=2P+ 
-p(l + 8P)7«!=2Z,'+ II 
-i(l + 8Z»)»(3=2Z^”+ I,. 
8. It is obvious that 
lx-\-r,y-\-^z=^ 
is the equation of the line EF joining the two conjugate poles, and it may be shown 
that 
oa-\-^y-\-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 where X is arbitrary, and the equation of 
the line in question is 
(Xb,+Yb,+Z^JX'B,+Y'B,+Z'B,)(a;H/4-^*+6x^7^)=0; 
or developing. 
and the function on the left-hand side is 
{\-^{ax-\-^y-\-yz), 
which proves the theorem. 
9. The equations (A) by the elimination of (|, n, Q, give 
- +i3^+y^) H- (- 1 + 4^>l3y = 0, 
which shows that the line IJ is a tangent of the Pippian ; the proof of the theorem is 
given in this place because the relation just obtained between a, (3, y is required for the 
proof of some of the other theorems. 
