ME. A. CAYLEY ON THE ANALYTICAL THEOEY OF THE CONIC. 
649 
the right-hand side is 
= Quotient by (A, ??', of ^ 
K{^'xx-'^',j+y'zy 
' +Quotient by (a, . .Xvfl ' — ^'5 D" of 
z).{K -ir, ri, r)?+mA, ..ir, n\ KJ). 
where 
n=K 
X, y , z 
x\ y, z' 
Z , n 
or, what is the same thing, 
n= 1 X , 
y ^ 
z 
— 
X 
y 
z 
Kx', 
Ky, 
ILs' 
Al'+HVd-G^', 
HI'-f-BV+F?', 
GI'd-FV-f-C^' 
1 , 
m , 
n 
More simply, the right-hand side is 
= Quotient by (A, . .^1^ n\ of] 
, j 
K(l'x-j-^'y+^'zy 
d-Quotient by (a, of 
. {[{a, ..X>''^'-i^y',..Xx, y, z)-]\K, ..jr, ; 
Or restoring the left-hand side, and resolving into its linear factors the function in { } , 
we have 
II. 
(«, . .X^-> y-> 2 )*=Quotient by (A, . .X^\ K'f of j 
-{-Quotient by («, .•X '/^' of Product 
. n±x/— (A, . .Jl', rl, . .Xr/i—^^\ ..Xx, y, z), 
where 11 has the value given above, which may also be written 
n= (A, ..XI', >}', ^'XK vX^'x+'^y+^'z) 
—(A, ..XI', ??', ^'y(xx-i-{^y-yvz). 
20. We deduce at once the inverse or reciprocal formulae 
(A, ..XI, fi, Quotient by (A, ..XI', V, K’f of 
III. 
[(A, . .XI, ai', .', nr 
-{-Quotient by (a, ••X*'^’ — of K into Product 
(«, ..1.,'-^', ■■)±\/-(A, -X? O’ 
V 
§, 
1', 
4 T 2 
