650 
MR. A. CAYLEY ON THE ANALYTICAL THEORY OE THE CONIC. 
where the Product part may also be written 
Product [^(A, ..Xr, ?'IX, ft, r).(A, ..X?', V, ?'X^, f) 
-i (A, . .xr, ?'f • (A, . -x X, fo, > XI, ?) 
±y-(A,..xi',ti',?7 
21. And also 
^5 fl, K 
r, V, r 
X, (A, V 
]• 
IV. 
where 
(A, , . 3 ( 1 , rj, Quotient by [a, . .yaf, y\ z'f of 
+Quotient by K(A , . .yny' —mz ’ , . .f of Product 
K(l)+^/— K(o!, ..yaf, y\ ^)\K, ..yny'—mz’,..yi, n, 
0 = ax'+hy'+gz', haf -]-hy' +fz' , gx' +fy'+cz' 
ny'—mz', Iz'—nx^ , ms(f—ly’ 
which may also be written 
= {a, ..yx',y\ z'yi,m, n){x'l-\-y'ri->rZ%) 
— {a, ..yx\ y, z'f . {II J^7nn^-nl^). 
22. The geometrical signification is obvious. The formulae 1. and II. each of them 
show that the equation 
{a, ..yx, y, zf={) 
of the conic may be written in the form 
W’+iQE=0, 
where Q=0, R=0 are any two tangents of the conic, and W=0 is the line joining the 
points of contact, or chord of contact corresponding to the two tangents; viz., in the 
formula I. we have 
W =(a, ..X^, y, ^yx, y, z\ 
|=(A, . .ym^—ny\ ..yyz'—y'z, . .)±\/— K(a, . .yx, y, zf 
Ql- 
R 
X, y, z 
x\ y, 2' 
I , m, n 
(or for a different form of Q, R see the formula). The quantities (^, y\ z') are the 
coordinates of the point of intersection of the two tangents, or pole of the chord of 
