ME. A. CAYLEY’S MEMOIE ON CUEVES OF THE THIED OEDEE. 441 
The coefficient of on the right-hand side will vanish if (l-f-2Z^)A-l-Z^B = 0, 
or, what is the same thing, if A=^^, B=— (l-}-2Z^); and substituting these values, we 
obtain 
+(-4^+4?^)(f +»?3+r)i^2: 
or, what is the same thing, 
+ 2 ®) — (1 + 2 l^)xyz D (r — f )(f — '<t) 
X { -/(r+;?^+r)+(--H-4Z^)|pja^ 
Hence the left-hand side vanishes in virtue of the relation between or we have 
which proves the theorem. 
36. Suppose that (X, Y, Z) are the coordinates of a point of the Hessian, and let 
(P, Q, R) be the coordinates of the point in which the tangent to the Hessian at the 
point (X, Y, Z) again meets the Hessian, or, what is the same thing, the satellite point 
in regard to the Hessian of the tangent at (X, Y, Z). And consider the conic 
X(.r^ + 2 lyz) + Y(y^ -f 2 Izx) -j-Z(sf-\-2 lxy\ 
which is the first or conic polar of the point (X, Y, Z) in respect of the cubic. The 
polar (in respect to this conic) of the point (P, Q, R) will be 
lx-\-riy-\-tz=^, 
where 
l=PX+Z(RY+QZ), 
;,=QY+Z(PZ+RX), 
^=RZ+/(QX+PY); 
or putting for (P, Q, R) their values, 
l=(Y^-z^)(x^-r^z), 
>?=(Z^-X*)(Y^-/ZX), 
2:=(X^-Y^)(Z^-/XY). 
And if from these equations and the equation of the Hessian we eliminate (X, Y, Z), 
we shall obtain the equation in line coordinates of the curve which is the envelope of 
the fine ^x-\-’/iy-{-tz-=^. We find, in fact. 
;3 + ;,3^^3^(Y^_2^3^^2^_X3)(X^_y3) 
^ Z*(X^+W-|-Z7 
J _3Z(X’+W-1-Z^)XYZ 
[4-(1_4^3)(y^Z^^Z^X^H-XW*), 
3 M 
MDCCCLVII. 
