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ME. A. CAYLEY ON THE ANALYTICAL THEOEY OF THE CONIC. 
Article Nos. 1 to 17, relating to a single conic. 
1. The point-equation of the conic is 
(a, b, c,f, g, hjx, y, 
which expresses that the point [x, y, z) is an ineunt of the conic. 
The line-equation of the same conic is 
1, 
1 
a, 
A, 
9 
k, 
f 
9^ 
c 
or putting 
(A, B, C, F, G, H)=(5c— ca—g^, ab—}f, gJi—af, hf—hg, fg — cJi), 
the line-equation is 
(A, B, C, F, G, HXI, 0^=0, 
which expresses that the line (|, Q (that is, the line the point-equation whereof is 
is a tangent of the conic. We are thus in the analytical theory of the 
conic concerned with the quadrics (a, b, c,f, g, Ji^x, y, 2 )* and (A, B, C, F, G, HXI, ^5 
which are the characteristics or nilfactums of these equations respectively. 
2. I put also 
K= 
«, h, g , 
A, b, f 
or, what is the same thing, 
9 ^ / 
K =r abc —af^—bg^— clf + 2 /^A. 
3. Tt may be convenient to notice that when (a, . .\x, y, zY breaks up into factors, 
the conic the equation whereof is (a, ..X^? y? 2 )^= 0 , becomes a pair of lines; and that 
when (a, . .X^, y-, zf is a perfect square, the conic becomes a pair of coincident lines, or 
say a twofold line. But a pair of lines, distinct or coincident, cannot be represented by 
a line-equation. The analytical formulae presently given show that in the former case 
(A, . .XI 5 is the square of a linear function, which equated to zero gives the line-equa- 
tion of the point of intersection of the two lines, or node of the conic ; and the equation 
(A, ..Xi? accordingly represents such point considered as a pair of coincident 
points, or say a twofold point. But in the latter case, where the conic is a twofold line, 
(A, ..Xii ^5 KY identically equal to zero, and the line-equation (A, ..X ^5 ^5 ^Y — ^ ^ 
mere identity 0 = 0, thus ceasing to have any signification at all. And the like remarks 
apply to the conic as represented by the line-equation (A, ..XI, >1, the conic here 
breaking up into a pair of distinct or coincident points, &c. 
4. It is proper to remark also that 
{a, ...Xx',y',z'Xx,y, z)=0 
