ME. A. CAYLEY ON THE ANALYTICAL THEORY OF THE CONIC. 
G 45 
viz. See . 
Also 
(<Bc-4^^ m-(B\ m-w, ijr-m 4r<a-ci) 
z =( x , y , zy . K(«5, b , c , f , g , hyx , y , zf ; 
that is, 
BC— jr"=^"K(a, b , c , f , g , hXx , y , z )\ See . ; 
whence also 
(^C— 4^^ vf ={' Kx -^ gjy + vzy . K { a , ..J^, y, zf , 
( 33 C— .. XK , [ Jj , vX ^ J , gJ , v ’)={ Xx ^ g . y -\- vz ){}! x -\- gJy -\- v ' z ). K .{ a , ..J^, y . 
and moreover 
The last equation shows that (A, . . Xyz ' — y ' z , zx '— z ' x , xy '— x ' yf , considered as a 
function of ( x ', y \ z '), breaks up into factors, or (what is the same thing) this expres- 
sion, considered as a function of ( x , 3/, z ), breaks up into factors ; we may in all the 
fonnulae interchange ( x , 3/, z ) and ( x ', y', z '), writing ( 91 ', 33 ', C', ^ff', ( B ', W ) in the place 
of (a, B, 1). 
Article Nos. 18 to 28 , relating to a single conic in connexion with a point or line. 
18 . I apply the decomposition formula to the function (A, .. Xn ^' — •• f -> which, 
considered as a function of (^, 3/, z ), breaks up into factors. We have 
(A, .. Xyz !- y ' z ,.. f ={ 9 !, y , zf 
=7 ^ Product of 
(H', ..X, m, nf 
TI m nTr v .. Xn ^— ny , .- Xk 3) 
But we have 
( 91 ', .-X^, m , nf ={ X .. Xmz '— ny \ ..)^ 
(91', ..X^, m, nXx, 3 /, z)=(A, ..Xmz'—?iy',..Xyz’—y'z, 
(33'C'-Jf'^ ..Xnz—ny, ..XK >') 
= [ x '( mz — ny )-\- y '{ nx — lz )-\- z '{ ly — mxy \{‘^ x '+( jjy '-\- vz ') K ( a , .. X ^\ X 
( 3 S'C'— 4^'^ . - X ^, |W-, t / f = ( Xx ’ gjy ' + vzyK ( a , . . Xx ', ij , z ' f , 
and thence 
( 33 'C'— ..X?^-g— wy? --X^^ 0 
s/-(B'C'- .. X \ ,f 
X, 
ryJ 
\Aj ^ 
I, 
z 
z ' 
y-> 
y'^ 
w, 
n 
y-K(«, ..I*', y, if. 
