418 
ME. A. CAYLEY’S MEMOER OX CLETES OE THE THIED OEDEE. 
obvious that the tangents to the Pippian through the point P are in like manner the 
line GFE, and the pair of lines FBA, FBC, and that these lines are harmonically 
related to FO the tangent at F of the Hessian. And similarly, the tangents to the 
Pippian through the point O are the line GO and the lines AOD, BOC, and the tangents 
to the Pippian through the point G are the line GO and the lines GFE and Gf ef^. 
Thus aU the hues of the figure are tangents to the Pippian except the lines EO. FO, 
which are tangents to the Hessian. It may be added, that the lineo-polar envelope of 
the line EF with respect to the cubic is the pah* of lines OE, OF. 
It will be presently seen that the analytical theory leads to the consideration of a line 
IJ (not represented in the figure) : the line in question is the polar of E (or F) with 
respect to the conic which is the first or conic polar of F (or E) with respect to any 
syzygetic cubic. The line IJ is a tangent of the Pippian, and moreover the lines EF 
and IJ are conjugate polars of a curve of the third class syzygetically connected with 
the Pippian and Quippian, and which is moreover such that its Hessian is the Pippian. 
Article Nos. 3 to 19 . — Analytical investigations, comprising the proof of the Theorems, 
Article No. 2. 
3. The analytical theory possesses considerable interest. Take as the equation of the 
cubic, 
U = + 6 = 0 . 
Then the equation of the Hessian is 
']lG=-l\oi^-\-y^-\-z^) — {l-\-2l^)xyz=0 ; 
and the equation of the Pippian in line coordinates (that is, the equation which expresses 
that is a tangent of the curve) is 
PU = - ^(f ^^) + ( - 1 + 0. 
The equation of the Quippian in line coordinates is 
QU=(I-I0Z*)(f+;?H^^)-6ZX54-4Z^)|r,^=0; 
and the values of the two invariants of the cubic form are 
S = — l-\'l*f 
T=I-20Z^-8Z«, 
values which give identically, 
p_64S'=(I + 8Z^)*; 
the last-mentioned function being in fact the discriminant. 
4. Suppose now that (X, Y, Z) are the coordinates of the pomt E, and (X'. Y', Z') 
the coordinates of the point F ; then the equations which express that these points are 
conjugate poles of the cubic, are 
XX'-F^(YZ' +Y'Z)=0, 
YY'-t-/(ZX' + ZX)=:0, 
ZZ' +/(XY^'+X'Y)=0; 
