ME. A. CAYLEY’S MEMOIE ON CIJEVES OE THE THIED OEDEE. 
429 
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To show this we may take 
aU+6^HU=a;^+^^+2^=0 
as the equation of the sy 2 ygetic cubic satisfying the prescribed condition, for this value 
in fact gives 
H(aU+6^HU)= -xyz=^, 
a system of three lines. We find, moreover, 
P(aU+6|3HU)=P(.a7^+^^+2^)=-.^;?2: 
and 
P{6H(aU+6^HU)}=P(-6ary2)=-4^;j2:, 
the latter equation being obtained by first neglecting all but the highest power of I in 
the expression of PU, and then writing I— — 1: we have also T(aUd-6/3HU)=l. 
Substituting the above values, the curve of the sixth class is 
^;j^{-4T.PU+P(6HU)}=0; 
or throwing out the factor we have the curve of the third class, 
-4T.PU+P(6HU)=0. 
Now the general expression in my Third Memoir, viz. 
P(aU+6i8HU)=:(a^+12Sa|3^+4T|3^)PU 
+(a^j3-4S/3^)QU, 
putting a=0, /3=1, gives 
P(6HU)=4T.PU-4S.QU, 
or what is the same thing. 
-4T.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 P(6HU)= — 4 ^j 3^, and T=l, and conse- 
quently that — 4T.PU-j-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 ’priori. 
Article No 22 . — Theorem relating to a line which meets three given conics in six jgoints in 
involution. 
22. The envelope of a line which meets three given conics, the first or conic polars of 
any three points -svith 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, 
d\5 
— 1 1 
dy 
it is true with respect to any three conics whatever of the form 
^_0 — — 0 ——0 
dx—^' dv~^' dz—^' 
rfU , d\] d\] . 
dz 
