aiE. A. CAYLEY’S MEMOIE ON CUEYES OF THE THIED OEDEE. 
417 
The point E is a point of the Hessian ; this being so, its first or conic polar, with 
respect to the cubic, will be a pair of lines passing through a point F of the Hessian ; 
and not only so, but the first or conic polar of the point F, with respect to the cubic will 
be a pair of lines passing through E, The 
pau’ of lines through F are represented in the 
figure by FBA, FDC, and the pair of lines 
through E are represented by EGA, EDO, and 
the lines of the one pair meet the lines of the 
other pair in the points A, B, C, D. The point 
O, which is the intersection of the lines AD, 
BC, is a point of the Hessian, and joining EO, 
FO, these lines are tangents to the Hessian at 
the points E, F, that is, the points E, F are cor- 
responding points of the Hessian, in the sense 
that the tangents to the Hessian at these points 
meet in a point of the Hessian. The two point 
nition, conjugate poles of the cubic. 
The line EF meets the Hessian in a third point G, and the points G, O are conjugate 
poles of the cubic. The first or conic polar of G, with respect to the cubic, is the pair 
of lines AOD, BOG meeting in O. The first or conic polar of O, with respect to the 
cubic, is the pair of lines GEF and Gf'efe’ meeting in G. The four poles of the line 
EO, with respect to the cubic, are the points of intersection of the first or conic polars 
of the two points E and O, that is, the foirr poles in question are the points F, F, e, e'. 
Similarly, the foui’ poles of the line FO, with respect to the cubic, are the points E, E, f \f'- 
The line EF, that is, any line joining two conjugate poles of the cubic, is a tangent to 
the Pippian, and the point of contact F is the harmonic with respect to the points E, F 
(which are points on the Hessian) of G, the third point of intersection with the Hessian. 
Gonversely, any tangent of the Pippian meets the Hessian in three points, two of which 
ai’e conjugate poles of the cubic, and the point of contact is the harmonic, with respect 
to these two points, of the third point of intersection Avith the Hessian. 
The line GO in the figure is of course also a tangent of the Pippian, and moreover 
the lines FBA, FDG (that is, the pair of lines which are the first or conic polar of E) and 
the lines EGA, EDB (that is, the pair of lines which are the first or conic polar of F) are 
also tangents to the Pippian. The point E represents any point of the Hessian, and the 
three tangents through E to the Pippian are the fine EFG and the lines EGA, EDB ; 
the line EFG is the line joining E Avith the conjugate pole F, and the lines EGA, EDB 
are the first or conic polar of this conjugate pole F Avith respect to the cubic. The 
figure shoAvs that the line EO (the tangent to the Hessian at the point E) and the before- 
mentioned three lines (the tangents through E to the Pippian), are harmonically related, 
viz. the line EO the tangent of the Hessian, and the line EF one of the tangents to the 
Pippian, are harmonics Avith respect to the other tAvo tangents to the Pippian. It is 
MDCCCLA^II. 3 I 
