426 
ME. A. CAYLEY’S MEMOLB OX CTJEYES OE THE THIRD OEDEE. 
but the equation 
gives 
and we have 
-Z(«^+^3^+/)+(-l + 4Z>,Sy=0 
3Za^=-3Z(a/3*+a/)+(-3+12Z>^/3y, 
L=(-4+4^V|3y+(-8^+8Z^)(a/3^4-a/)+(-16Z^+16r)|3y 
=(_4+4P)(a’|3y+2Z(«(3’+o!/)+4;‘/3y) 
=(_4+4i»)(a7+2Z(3’)(a(3+2y) ; 
or, in virtue of the equations (D), 
L=(_4+4^^)Z^r?.^^e^=(-4+4^^)Zt^;jr=(-4+4^y«.?. 
Hence, omitting the common factor, we find L : M : N = ^ : 2^, and the equation 
L;r+My+Nz=0 becomes 
which is the equation of the line EF, that is, the line through two consecutive positions 
of r is the line EF ; or what is the same thing, the line EF touches the Pippian in the 
point r which is the harmonic of G with respect to the points E, F. 
18. The lineo-polar envelope of the line EF, with respect to the cubic, is the pair of 
lines OE, OF. 
The equation of the pair of lines OE, OF, considered as the tangents to the Hessian 
at the points E, F, is 
{(3^^X^-r+2^^YZ> +(3Z^Y^-iq:^/^ZX)y +(3Z^Z2-T^^^XY>} I 
X {(3Z^X'^-l+2Z^Y'Z')a;+(3^^Y'^-T+2^^ZX')3/+(3^=Z'^--T+2^^XY>}i” ’ 
And on the left-hand side the coelRcient of is 
9Z^X^X'^ - 3^^(1 -f 2Z^)(X^Y'Z'-f X'^YZ) + (1 ■^21JYY'ZZ\ 
which is equal to 
9ZV-3^^(l-f2Z^)(^^i3y-Fla^)+(l + 2Z^f/3y, 
that is, 
l(-Z+?<){3;a>+2(l+2?),8r}; 
and the coefficient of yz is 
9ZXY^Z'^+Y'^Z^)-3^^(l-f2Z^)(YY'(XY'+X'Y)+ZZ'(XZ'+X'Z))-f(l-f-2^^)^XX'(YZ'+VZ}, 
which is equal to 
91*^06^— 2(3'/^ — 3Z^(l-l-2^^)^ — |/3y^ + 
7(-^+^^){(l-4?V-6ri3y}. 
that is, 
