416 
lilE. A. CAYLEY’S MEMOIE OX CUEYES OE THE THERD OEDEE. 
Next, considering, in connexion with the cubic, a line — 
(c) The first or conic polars of each point of the line meet in four points, which 
are the four poles of the line. 
[d) The second or line polars of each point of the line envelope a conic, which 
is the lineo-polar envelope of the line. 
And reciprocally considering, in connexion udth a curve of the thu'd class, a line, we 
have — 
{e) first or conic pole of the hne. 
[f) The second or point pole of the line. 
And considering, in connexion with the curve of the thu’d class, a point — 
[g) The first or conic poles of each line through the point touch four lines. Avhich 
are the four polars of the point. 
[h) The second or point poles of each line through the point generate a conic, 
which is the point pole locus of the point. 
But I shall not have occasion in the present memoir to speak of these reciprocal 
figures, except indeed the first or conic pole of the line. 
/ 
The term conjugate poles of a cubic is used to denote two points, such that the first or 
conic polar of either of them, with respect to the cubic, is a pair of lines passing through 
the other of them. Keciprocally, the term conjugate polars of a curve of the thhxl class 
denotes two lirres, such that the first or conic pole of either of them, with respect to the 
curve of the third class, is a pair of points lying in the other of them. 
The expressiorr, a syzygetic cubic, used in reference to two cubics, denotes a cmve of 
the third order passing through the points of iirtersection of the two crrbics ; but in the 
preserrt memoh’ the expressiorr is iir general used irr reference to a surgle cirbic, 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 hr the memoh the full expression, 
a curve of the third class syzygetically connected -with two giverr curves of the thhd 
class. 
It is a well-krrown theorem, that if at the points of intersection of a given Ihre vith a 
giverr cubic tangents are drawrr to the cubic, these tarrgents again meet the cubic hr 
three points which lie hr a Ihre ; such hrre is in the preserrt memoh termed the satellite 
line of the giverr Ihre, and the point of intersectiorr of the two lirres is termed the satellite 
point of the giverr lirre ; the given lirre irr reference to its satellite hrre or pohrt is termed 
i\\Q primary line. 
In particular, if the primary lirre be a tangent of the cubic, the satelhte hrre coincides 
with the primary hrre, arrd the satelhte pohrt is the point of sirrrple irrtersectiorr of the 
primary hne arrd the cubic. 
Article No. 2. — Group of Theorems relating to the Conjugate Poles of a Cubic. 
2. The theorems which I have first to merrtiorr relate to or origirrate out of the 
theory of the corrjugate poles of a cubic, arrd rrray be converrierrtly corrrrected together 
and explahred by means of the accompanyhrg figur’e. 
