442 
:MTI. a. CAYLEY’S MEMOIR CEEYES OF THE THIED OEDEE. 
=(Y^-Z^)(Z^-X^)(X^-Y-’) 
r Z^(X^+Y^+Z^)XYZ 
+(1-Z^)X^Y^Z^ 
[-/(Y^Z^+Z^X^H-X=’Y^) ; 
and thence recollecting that 
HU=/^(X®+Y^+Z^)-(1+2^^)XYZ, 
we find 
and the equation of the envelope is 
-^(r+^^+r)+(-i+4^=‘)i;?2:=o, 
which is therefore the Pippian. We have thus the theorem, — 
The envelope of the polar of the satellite point in respect to the Hessian of the 
tangent at any point of the Hessian, such polar being in respect of the conic which is 
the first or conic polar of the point of the Hessian in respect of the cubic, is the Pippian. 
Article Nos. 37 to 40. — Investigations and theorems relating to the first or conic polar of a 
point of the cubic. 
37. The investigations next following depend on the identical equations 
{a(X^+2WZ)+|3(Y^+2fflX)+y(Z^+2^XY)} 
X { — ^KY7i{x^-\-'if-{-z^)-{-(X.^-\-Y^-\-7i^)xyz} 
= {X.{cif-Y2lyz)-\-Y{y^-^2lzx)-\-7A{z‘-\-2lxy)] 
X{X(W-Z^)(y3/-|32)+Y(Z*-X^)(«2-y^)+Z(X^-WX|3a’-a3/}} 
+ {^(X^+2WZ)+y(W+2^ZX)+2(ZH2^XY)} 
X { -(aYZ+^ZX+yXY)(X^^+Y/+Z2^)+(aX^+/3W+yZ^)(X3/s+Yz.T+Z.r^)}. 
which is easily verified. 
I represent the equation in question by 
KT=WL+P0; 
then considering {x, y, z) as current coordinates, and (X, Y, Z) and (a, j3, y) as the 
coordinates of two given points 2 and fl, we shall have U=0 the equation of the cubic. 
W = 0 the equation of the first or conic polar of 2 with respect to the cubic, P=0 the 
equation of the second or line polar of 2 with respect to the cubic. The equation T=0 
is that of a syzygetic cubic passing through the point 2 : the coordinates of the satellite 
point in respect to this syzygetic cubic of its tangent at 2 are 
X(Y^-Z^) : Y(Z^-X®) : Z(X^- Y*) ; 
and calling the point in question 2', then L=0 is the equation of a line through the 
points 2', n. The equation 0=0 is that of a conic, viz. the first or conic polar of 2 
with respect to a certain syzygetic cubic 
-2(aYZ+f3ZX+yXY)(^^+3/^+5:^) + (aX^+SW+yZ=^).ri/^=:0, 
