445 
ME. A. CAYLEY’S IMEMOIE ON CUEVES OE THE THIED OEDEE. 
meets it besides in four points, which are the angles of a quadrangle the centres of which 
lie on the cubic. In other words, the quadrangle is an inscribed quadrangle. 
40. To find the equations of the axes of the quadrangle, that is of the lines through 
two centres. 
We have 
(4«WZ - + (X^XY+ + (?i^ZX+ 2;f?vY=*)2= 0 
(X"XY+2;sXZ^>4-(4;£^ZX-?.W%+(AWZ + 2;s?X:">=0 
(X^ZX+2;sXY>+(aWZ+2^aX%+(4«^XY-?i^Z^> =0 ; 
or arranging these equations in the proper form and eliminating x?, kX, we find 
YZ.r, 2?y-\-Y^z , X(— Xr+Yy+Z^) 
ZX?/, Y( Xi’-Yy+Z^) 
XY2, Y^x+'K^y, Z( X^^+Y^-Zz) 
or multiplying out, 
XYZ { (Z* - Y^K + (X' - ’^^)f + ( Y" - } 
+.'r^^ZY^ (-2X*+Y^+Z^)+z.rWZ*(2X*- Y^-Z®) 
+/2XZ^ (-2W+Z^+X^)+^/ZX(2Y^-Z^-X®) 
+2^.rYX^(-2Z^-fX+Y®)+yz^XY=*(2Z^-X^-Y^)=0. 
We may simplify this result by means of the equation X*+Y^+Z®+6fXYZ = 0, so as 
to make the left-hand side divide out by XYZ : we thus obtain 
(Z*-Y>"+(X-Z^)?/«+(Y^-X^)z^ 
+( - 3XW- UY^Zyy +( - 3Y^Z - 6ZZ^X)/z +( - 3Z^X- 6/XW>=*^ 
+( 3XY^+6ZX^Z>'i^^-f( 3YZ^-f6W=*X)j/.s*+( 3ZX*+6/ZW>a;^=0 ; 
or in a difierent form, 
[y^ — 2^)X® + ( 2 ^ — Y^ + {x^ —y^y^ 
-\-{—2>x^y—^ lzyX.^Y + ( — Sy'^z — 6 lafyJY^Z + ( — Sz^x — 6 yzyL^'X. 
-l-( 3^3^*+6fy0*)XY^+( 2>yz^ -\-^I zci^)Y7a^ Zzx^-\-Qlxy^)7i'K?=-(i, 
as the equation of the three axes of the quadrangle. 
Article No. 41 . — Recajyitulation of geometrical definitions of the Pippian. 
In conclusion, I will recapitulate the difierent modes of generation or geometrical 
definitions of the Pippian, obtained in the course of the present memoir. The curve in 
question is — 
1. The envelope of the line joining a pair of conjugate poles of the cubic (see 
Nos. 2 and 13). 
2. The envelope of each line of the pair forming the first or conic polar with respect 
to the cubic of a conjugate pole of the cubic (see Nos. 2 and 14). 
