A MEMOIR ON CURVES OF THE THIRD ORDER. 
The expression, a syzygetic cubic, used in reference to two cubics, denotes a curve 
of the third order passing through the points of intersection of the two cubics; but 
in the present memoir the expression is in general used in reference to a single cubic, 
to denote a curve of the third order passing through the points of intersection of 
the cubic and its Hessian. As regards curves of the third class, I use in the memoir 
the full expression, a curve of the third class syzygetically connected with two given 
curves of the third class. 
It is a well-known theorem, that if at the points of intersection of a given line 
Avith a given cubic tangents are drawn to the cubic, these tangents again meet the 
cubic in three points which lie in a line; such line is in the present memoir 
termed the satellite line of the given line, and the point of intersection of the two 
lines is termed the satellite point of the given line; the given line in reference to 
its satellite line or point is termed the primary line. 
In particular, if the primary line be a tangent of the cubic, the satellite line 
coincides Avith the primary line, and the satellite point is the point of simple inter 
section of the primary line and the cubic. 
Article No. 2.—Group of Theorems relating to the Conjugate Poles of a Cubic. 
2. The theorems which I have first to mention relate to or originate out of the 
theory of the conjugate poles of a cubic, and may be conveniently connected together 
and explained by means of the accompanying figure. 
The point A is a point of the Hessian ; this being so, its first or conic polar, 
Avith 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 Avili be a pair of lines passing through E. The pair of lines through