396 
A MEMOIR ON CURVES OF THE THIRD ORDER. 
[146 
We have 
X : Y : Z = l% 2 —: Irf — ££ : ttf — £77, 
and X, Y, Z are to be eliminated from these equations, and the equation 
¿2 (X s + Y 3 + Z 3 ) - (1 + 2Z 3 ) XYZ= 0 
of the Hessian. We have 
X 3 +Y 3 + Z 3 = l 3 (£ 3 +v 3 + £ 3 ) 2 
-3Z 2 (f + 
+ 9Z fy£ 2 
-(1 + 2Z 3 ) + + fV)> 
ZF^= ZCf + v’+HW 
4 (- 1 + l 3 ) £ 2 
- Z 2 <y£ 3 + +1 8 77 3 ), 
and thence 
HU = Z 5 (f 3 + i? 3 + £ 3 ) 2 
-(Z+5Z 4 )(f + 77 3 + ^ 3 )^^ 
+ (1 + 10Z 3 -2Z 6 )fV^ 2 ; 
and equating the right-hand side to zero, we have the equation in line coordinates of 
the curve in question, which is therefore a curve of the sixth class in quadratic 
syzygy with the Pippian and Quippiau. 
Article No. 21.—Geometrical definition of the Quippiart. 
21. I have not succeeded in obtaining any good geometrical definition of the 
Quippian, and the following is only given for want of something better. 
The curve 
T.PU{P6H(olU + 6/3HU)} - P (6HU) {T(aU + 6/3#U). P (aU + 6/3#U)}= 0, 
which is derived in what may be taken to be a known manner from the cubic, is in 
general a curve of the sixth class. But if the syzygetic cubic aU + 6/3HU = 0 be 
properly selected, viz. if this curve be such that its Hessian breaks up into three 
lines, then both the Pippian of the cubic a U + 6/3#U = 0, and the Pippian of its 
Hessian will break up into the same three points, which will be a portion of the 
curve of the sixth class, and discarding these three points the curve will sink down 
to one of the third class, and will in fact be the Quippian of the cubic. 
To show this we may take 
aU + 6/3 HU = x 3 + y 3 + z 3 , = 0