646 
ME. A. CAYLEY ON THE ANALYTICAL THEOEY OE THE CONIC. 
whence we have 
(A, ..lyz'-yz, ■■y= (A, ■■XW-.y'/ Product of 
(A, ..Xyz'—y'z, ..)+ 
X, y, z 
x\ y\ z’ 
I , m, n 
\/— K(«, ..Jx’, y'^zf-, 
And the identical equation 
(«, ..Xx, y, z)\{a, ..Xx\ y\ zj-[{a, y, zXx', y', z')J={A, ..Xyz'—y'z, ..f 
now gives 
(a, ..X^\ y, zf= Quotient by [a, ..X^^ y? 
I. 
i [(«. 
1 
1 1 
! + Quotient by (A, ..X'^^z'— 
ny\ 
of Product 
(A, 
.X^nz' —ny' , . -Xy^'~y’^ 
+ 
X, 
y, ^ 
x\ 
y', 2' 
\ 
1 , 
n 
x/— K(«, ..x^, y, z')\ 
where the Product part may also be written 
(«, ...XI, 'ni, nXx\ y, .-X-^. y, y\ z') 
— (a, ... X^', y, z'f .(a, .. X^, y, zXl , m, n) 
±V^-K(«,..X^',y, zj 
X, y, z 
x', y\ z' 
I ^ m., n 
19. Writing in the formula I. 
a, h, g X^', y^ z')={^’, >]', ^), 
li, h, f 
y-> ^ 
we have 
and thence 
(.r',y, 2 ')=^ 
H, B , F 
G, F, C 
Assume 
K(«,..x^',y,2') -(A,..xr,^', cr 
(«, ..X^, y, zX^', y, z') = ^'x+?i'y-\-^z. 
{I, m, n) — {vrl — gj!;\ — — 
then from the foregoing values of (x', y', z') 
'£')(G?+F,'+CC)-(;<.r-W)(Hr+B,'+FC)| 
mz 
