384 A MEMOIR ON CURVES OF THE THIRD ORDER. [146 
F are represented in the figure by FBA, FDG, and the pair of lines through E are 
represented by EGA, EF)G, and the lines of the one pair meet the lines of the other 
pair in the points A, B, G, D. The point 0, which is the intersection of the lines 
AD, BG, 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 corresponding 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 points E, F are, according to a preceding definition, 
conjugate poles of the cubic. 
The line EF meets the Hessian in a third point G, and the points G, 0 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 0. The first or conic polar of 0, 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 0, that is, the four poles in question 
are the points F, F, e, e'. Similarly, the four 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 T 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. Conversely, any tangent of the Pippian meets the Hessian in three 
points, two of which are 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 with 
the Hessian. 
The line GO in the figure is of course also a tangent of the Pippian, and more 
over 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 line EFG 
and the lines EGA, EDB; the line EFG is the line joining E with the conjugate 
pole F, and the lines EGA, EDB are the first or conic polar of this conjugate pole 
F with respect to the cubic. The figure shows 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 
with respect to the other two tangents to the Pippian. It is obvious that the 
tangents to the Pippian through the point F are in like manner the line GFE, and 
the pair of lines FBA, FBG, 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 0 are the line GO and the lines AOD, BOG, and the tangents to the 
Pippian through the point G are the line GO and the lines GFE and Gfefe'. Thus 
all the lines 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 pair of lines OE, OF.