428 
ME. A. CAYLEY’S MEMOIE OX CCEYES OE THE THIRD OEDEE. 
Article No. 20 . — Theorem relating to the curve of the third class, mentioned in the 
preceding article. 
20. The consideration of the curve 3T.PU — 4S.QU=0, gives rise to another 
geometrical theorem. Suppose that the line (^, r\, t), that is, the line whose equation is 
is with respect to this curve of the third class one of the four polars of a 
point (X, Y, Z) of the Hessian, and that it is required to find the envelope of the line 
We have 
X : Y : Z^ir-nl : Ir^-l^ : IV-ln, 
and X, Y, Z are to be eliminated from these equations, and the equation 
^2(X3_pY^+Z3)-(l+2/^)XYZ=0 
of the Hessian. We have 
X^+Y^+Z®= 
XYZ= 
and thence 
HU= 
+(l4-10^*_2Z«)?Vr; 
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 mth 
the Pippian and Quippian. 
Article No. 21 . — Geometrical definition of the Quippian. 
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(aU-f6i3HU)}-P6HU{T(aU+6,3HU).P(aU-f6/3HU)}=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 alJ 6|3HU = 0 be pro- 
perly selected, viz. if this curve be such that its Hessian breaks up into three Imes, then 
both the Pippian of the cubic aU-4-6/3HU=0, and the Pippian of its Hessian aHII 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 on the third class, and 
will in fact be the Quippian of the cubic. 
