644 
ME. A. CAYLEY ON THE ANALYTICAL THEOEY OF THE CONIC. 
and from these we deduce 
( a, h, g Q=(0, 0, 0), 
h, b, f 
viz. a^+h;j+g^=0, «&c. 
Also 
(be— ca— g% 
that is, 
Whence also 
ab-h^ gh-af, hf-bg, fg-ch)=(|,>?,^7.(A,B,C,F,G,HXI,^,^r, 
bc-P=r(A, B, C, F, G, HXI, l)\ &c. 
(bc-P, . .Jl, m, nf (A, . .XI, n, 
and 
(be— P, ..\l, m, n\l', m\ . .XI, n, ; 
and moreover 
abc — af ^ — bg^ — ch^ + 2fgh = 0 . 
16. The last equation shows that (a, ..X^^^ — ?l^ — ^^1, I??' — considered as a 
function of (^', ;j', X), breaks up into factors. Or since the expression is not altered by 
interchanging (|', yj, ^') and (|, ^), the same expression, considered as a function of 
(I, ??, ^), breaks up into factors. It is in fact easy to see that any quantic whatever, 
(*X^?^ — ^^5 considered as a function of (|, ^), breaks up into linear 
factors; for in virtue of the equation any one 
of the quantities yj^' — ;jX, ^I^— ^^I, I>?' — 1'^ can be expressed as a linear function of the 
other two ; so that the quantic can be expressed as a linear function of any two of the 
three quantities ; and qua homogeneous function of two quantities, it of course breaks 
up into factors, linear functions of these two quantities. 
We may in all the formulae interchange (x', y\ z') and (x,y, z), writing (a', b', c', f', g', h') 
in the place of (a, b, c, f, g, h). 
17. Putting, in like manner, 
(A, B, C, F, G, H'^yz' —y'z, zx'—z'x, xy’—x'yY 
=(a, 33, c, #, <B, y, z')\ 
so that 
91= Cy+BP-2Fy^, 
33= AP+CA’^-2G2.r, 
C = B^^ + Ay — 2 ^xy, 
4r= — Ayz — F^ + G(xy + lAzx, 
<B = — ^zx-\-Yxy — Gy +H^^, 
^=—Cxy-\-Yzx-\- G^2: — HP, 
we obtain 
( 91, il, X-^> = 0, 0), 
3$, f 
C 
