ME. A. CAYLEY ON THE ANALYTICAL THEOEY OF THE CONIC. 
653 
so that 
(A, ..Xmz'—ny\ — ..^X, yj, v)\{a, y', z'f. 
Moreover, 
X, y , z 
X ^ y 1 z 
I , m, n 
which is 
K 
'^A.X-\-'U.yj-\-Gv)[yz' —y' z)-{-{'HX-\-'Qyj-\-Yv){zx' — z' x)-\-{GX-^Yyj-]rCv){xy' — x'y) 
=-(A, ..J\, (M, vXyz'-y’z, 
and hence instead of the formula I. we have 
(a, . .yjo, y, 2 )^= Quotient by (a, . y\ z'Y of 
I. (bis) 
[(a, ..Jx, y, zjx\ y\ z')J 
+ Quotient by +(A, ..yx, (x, vy(a, ..yx', y\ z'f — K.{Xcy-\-yjy'-\-vz'yoi K Product 
'{Xx'-\-y.y'-\-vz').{a, ..yx,y, zyx', y\ z') — {Xx+iJ.y+vz).{a , . .yxf,y\ zj 
±-^\/-K(a, ..yx\ y\ z'y{K, ..JX, vyyz'—y'z, 
25. If, in like manner, in the formula II. we introduce, instead of (X, yu, v), the new 
quantities {I, m, n), where 
{X, yj, i/)z=( a, A, g yi, m, n), 
h, h, f 
or, what is the same thing. 
K 
then we have 
9^ /’ 
c 
( A, 
H, 
G 
''h 
H, 
B, 
F 
G, 
F, 
C 
(a, h, ••)=«(He+B,' +F?')-m(Gr +F>,' +Cf'), 
(A, ..)=« (G5'+F,' +Cf')-?»(Ar +H,'+Gr), 
(g,f, ..)=m(Ar+H,'+Gf')-« (Hr+BV+Ff); 
and thence 
{a, ..yv/i—u.^’, ..yx, y, z) = 
X 
’'J I VT,.' 
, y , 2 
Ar+W+G^', Hr+B;?'+F^', Gr+Fpy'+C^' 
^5 m , n 
= (A, . .ymz—ny, . .Jl', ?!, ^'), 
(a, ••)’=^{(A, ..B, g,, ..xr, f7-[(A, ..XA, g., •J,?, n', ?')]’} 
= (a, ..Jl, m, «)».(A, ..Xr, f7-.K(;r+W+0>; 
