648 
ME. A. CAYLEY ON THE ANALYTICAL THEOEY OE THE CONIC. 
Similarly, 
(A, . . —mj\ . .Jp' —'ijz, . .) 
=^|(A, vXy^—y'z, ..).(A, ..Jl', KJ 
-(A, ..ir, riy^'-y «,••)• (A, -ir, >/, ?'p.. »)}. 
But 
{A, y.,vXyz!— 7 /z, ..) 
= (AX+HjM.+Gj'Xy^'— y^) 
+(HX+B|M/ -\-'Fv){zo(^ —^x) 
4-(Gx -\-Cv)(x^ — afy) 
= xy{Gx+Fi^ + &)-z'(H>.+ Fv}] 
■Fy[z’(Ax+Fiy.+Gi>)-x'{Gx+F(ju-\- C.^)] 
+ 2 [^(HX+B^ +Fv)-y(AX+H,!/,+Gv}], 
which, substituting for x', y, z' their values 
becomes 
(^, y, z')~ ^ 
H, 
G, 
H, 
B, 
F, 
G ir, ri. rx 
F 
C 
4(BC-F X^^'-/^?')+(FG-CHX^r-‘'i')+(HF-BGX/-o|'-xV)] 
+2/[(FG-CHXv^'-f^r)+(CA -G^ X^?'-^s')4-(GH-AFX^r-W)] 
+z (HF-BGX^^'-i=4-r)+(GH-AFXxr-*'r)+(AB-H^ XA^i'->-^')}. 
which is 
=(65, ..Xa5, y, zXvr!—^K\ ••)> 
and by merely writing (|', V, ?"') in the place of (X, |M/, i'), we have 
(A, . .XI', ??', K'Xyz'-y'z, ..)=o ; 
so that we find 
(A, ..Xmz'—ny', . .Ji/z' —y'z, ..) 
=g(A, ..XI', V, ?'X.(a, .-X^? zXvri'—y'^', J'l', 
Now, writing the formula I. in the form 
(«, ..X^? 2 :X=Quotient by K(65, ..X^', y, ^'X j 
K[(«, . .X^, y\ z')J 
+Quotient by K(A, ..Xjn^—ny^ ..^ of 
“ K"{[(A, ..Xmz'—ny', ..Xyz'—y'z, ..)]"+ 
i 
X, y , z 
y ' , 2 ' 
l , m, n 
'K(a, ..X^r,y, ^X), 
