I) 
A MEMOIR ON CURVES OF THE THIRD ORDER. 
Article No. 41. Recapitulation of geometrical definitions of the Pippian. 
In conclusion, I will recapitulate the different modes of generation or geometrical 
definitions of the Pippian, obtained in the course of the present memoir. The curve 
in question is: 
1. The envelope of the line joining a pair of conjugate poles of the cubic (see 
Nos. 2 and 13). 
2. The envelope of each line of the pair forming the first or conic polar with 
respect to the cubic of a conjugate pole of the cubic (see Nos. 2 and 14). 
3. The envelope of a line which is the polar of a conjugate pole of the cubic, 
with respect to the conic which is the first or conic polar of the other conjugate pole 
in respect to any syzygetic cubic (see Nos. 2 and 9). 
4. The locus of the harmonic with respect to a pair of conjugate poles of the 
cubic of the third point of intersection with the Hessian of the line joining the two 
conjugate poles (see Nos. 2 and 17). 
5. The envelope of a line such that its lineo-polar envelope with respect to the 
cubic breaks up into a pair of lines (see No. 24). 
6. The envelope of a line which meets three conics, the first or conic polars of 
any three points in respect to the cubic, in six points in involution (see No. 22). 
7. The envelope of the second or line polar with respect to the cubic, of a point 
the locus of which is a certain curve of the sixth order in quadratic syzygy with 
the cubic and Hessian, viz. the curve —S. U 2 + (HU) 2 = 0 (see No. 27). 
8. The envelope of a line having for its satellite point a point of the Hessian 
(see No. 35). 
9. The envelope of the polar of the satellite point with respect to the Hessian 
of the tangent at a point of the Hessian, with respect to the first or conic polar of 
the point of the Hessian in respect to the cubic (see No. 36).