652 ME. A. CAYLEY ON THE ANALYTICAL THEOEY OE THE CONIC. 
This gives 
(A, H, Gym^ —ny\ ..) 
— Iz' )-\-G{iy —mod) 
=ad{ILn-Gm)y-y {Gl-An)-\-z'{Am-m ) 
= 1- ^[H(GX+F;a,+Cv)-G(H?i+Bja,+F.^)] 
+y[G(A?i.+H|a/+Gt') — A(G?\.+F|«;+C j')] 
+2'[A(m4-B/^+Fv)-H(m+B,!>o+Fv)] 
We have thus the system 
(A, H, GJm^—ny\ . )—ih{god ■\-gz')—v {liod -\-hj -\-f^ \ 
(H, B, Yym^—ny\ ..)=v {a:^ y-hy -\-gz')—'k{cjx' 
(G, F, Cym^—ny, . .)=x{]iod -{-hy’ -{-fz')— g.{aod -\-hy’ -\-god), 
and thence 
{A, .Jmz’—ny\ ..Jjjz'—y'z, ..) 
= — y^—y'z , zod—z'x , xy'—x'y 
acd+hy'+gz\ hod -^hy’ -\-fz' , god -{-fy’ -^cz' ; 
X , {/j , V 
or observing that the term in X is 
— (zx' — z'x){gx’ -\-fy' -\-cz’)-\- {xy' — x'y) {hod + by' -\-fz'), 
which is 
= x{od{ax'-\-hy' +gz') -\-y'{hx' -\-fy' -{-fz') +^' {god -\-fy' + c£)) 
— X . od{ax' + hy' + g^) 
—y.od{hx'-\-by'-\-fz') 
— z.x'{god+fy'-\-cz') 
= —x'{a, . .y^, y, zyx', y, z')y-x{a, ..yx', y, z')\ 
with similar expressions for the terms in g/, v, we have 
(A, ..ymz'—ny', ..yyz’-y'z, ..) 
= —{Xx'+gjy'y-vz').{a, ..yx, y, zyod, y', z')-\-{Xx-{-gjy-\-vz) .{a, ..yod, y', z !)^ ; 
and so also 
(A, ..ymz'—ny' , ..f 
= —{Xod+iJ.y'-\-vz!).{a, ..yi, m, nypd, y', z!)-\-{Xl^gjm-\-m).{a, . .yx', y', z'f, 
{a, ..yi, m, nyx’, y', z')=Xx' +g.y' -]rvz', 
Xl-\-g.m-\-m =^(A, . .X>^, iM-, f'f, 
where 
