A MEMOIR ON CURVES OF THE THIRD ORDER. 
as the equation of the syzygetic cubic satisfying the prescribed condition, for this value 
in fact gives 
H (a U+ G/3H U) = - xyz, = 0, 
a system of three lines. We find, moreover, 
P (all + GfiHU) =P (x s + y 3 + z 3 ), = — 
and 
P [GH(aU + 6/3HU)} = P (— Gxyz), = — 4^^, 
the latter equation being obtained by first neglecting all but the highest power of l in 
the expression of PU, and then writing ¿ = —1: we have also T{olTJ + GfiHU) = 1. 
Substituting the above values, the curve of the sixth class is 
№ { - 4P. P U + P (GHU)} = 0 ; 
or throwing out the factor £77 we have the curve of the third class, 
— 4P. PU+P {GHU) — 0. 
Now the general expression in my Third Memoir, viz. 
. P (aU + G/3HU) = (a 3 + ma/3 3 + 4P/3 3 ) PU + (a 2 /3 -4£/3 3 ) QU, 
putting a = 0, /3 = 1, gives 
P(6HU) = 4T.PU-4S.QU, 
or what is the same thing, 
- 4P. PU + P (6HU) = - 4S .QU\ 
and the curve of the third class is therefore the Quippian QU = 0. It may be remarked, 
that for a cubic U = 0 the Hessian of which breaks up into three lines, the above 
investigation shows that we have PU = — ^77^, P (GHU) = — and T= 1, and conse 
quently that —4 T. PU + P (6HU) ought to vanish identically; this in fact happens in 
virtue of the factor S on the right-hand side, the invariant S of a cubic of the form 
in question being equal to zero; the appearance of the factor S on the right-hand 
side is thus accounted for a prion. 
Article No. 22.—Theorem relating to a line which meets three given conics in six points in 
involution. 
22. The envelope of a line which meets three given conics, the first or conic 
polars of any three points with respect to the cubic, in six points in involution, is 
the Pippian. 
It is readily seen that if the theorem is true with respect to the three conics,