444 ME. A. CAYLEY’S JVIEMOIE OX CUEVES OE THE THERE OEDEE. 
or, if we please, considering X as arbitrary parameters, then the four points lie in the 
conic 
(2;sX, 2«Y, 2«Z, — xX, — XY, —XZJjr, zy=0, 
or, what is the same thing, they are the points of intersection of the two conics 
= 0 , 
^yz-\-Yzx-\-7ixy=(). 
39. Considering the four points as the angles of a quadrangle, it may be shown that 
the three centres of the quadrangle lie on the cubic. To effect this, assume that the 
conic 
( 2 ; 4 X, 2;sY, 2kZ, -XX, -XY, -XZX^’, y, zf={) 
represents a pan of lines ; these lines will intersect in a point, which is one of the three 
centres in question. And taking x, y, z as the coordinates of this point, we have 
\y^ \z^ \yz\zx\ xy=^i^YTj — X^X^ 
:4«^ZX-X^Y^ 
: 4;s^XY-X^Z^ 
: X=*YZ+2;cXX2 
:X^ZX+ 2 ;fXY^ 
: X^XY+2;tXZ2; 
and we may, if we please, use these equations to find the relation between «, X. Thus 
in the identical equation x'^.y'^—{xyy=.Q, substituting for of, xy, y'^ their values, and 
throwing out the factor Z, we find (4«®— X^)XYZ— «X^(X®4-Y^+Z®)=0, and thence, in 
virtue of the equation X^+Y^+Z®+ 6 ZXYZ = 0 , we obtain 
4«^+6Z«X=*-X^=0. 
But the preceding system gives conversely, 
X 2 . : YZ : ZX : XY=4«y-XV 
: ^x^zx—X^y'^ 
: 4:%,^xy—Xh^ 
: X^yz-\-2x.Xx^ 
: X^zx-\-2xXy^ 
:X^xy-{-2x7iz^. 
Hence from the identical relation X^. Y^—(XY)®=0, substituting for X^ XY, Y* their 
values, and throwing out the factor z, we find (4«®— + 3 /^+ 2 ;®) = 0 , and 
thence, in virtue of the equation 4;^^— X^= — 6 Z;>jX^, we obtam 
q_^3 _|_ ^,3 _J_ 0 _ Q ^ 
which shows that the point in question lies on the cubic. We have thus the theorem, — 
The first or conic polar of a point of the cubic touches the cubic at the point, and 
