660 
ME. A. CAYLEY ON THE ANALYTICAL THEOEY OE THE CONIC. 
and the equation in (6, &) is 
35a+C5'=0; 
so that, writing 5 = C, 93, the equation of the pair of lines is 
? , '5' , 7" 
it', r' 
{qx-]r(!y-\-rz){q’x-^(j'y-\-T^z) — 
X, g^, » 
X', g.', .' 
X, g., V 
X, 
gj, V 
§ , 0 - , r 
f , <7 , r 
- / 1 t 
A, V 
X', 
gl, .' 
? 7"' 
f ', , r 
and it is easy to see that the left-hand side does in fact break up into factors, and that 
the equation is 
, y , 
z 
X 
y ^ 
gur' — m' , — Xr', 
X.t' 
g.r—vq , 
, 
X(7 — 
<7.' — rgJ, rX' — 
^gJ 
— ffX' 
g'v’ — r’gJ, 
A' — 
^'gJ-ff’X' 
which of course might have been obtained at once by means of the four points which 
are the intersection of each component line of the first conic by each component line of 
the second conic. 
37. Suppose that the first conic is 
(a, b, c,f, g, hjx, y, zf—^, 
wdiile the second conic is the pair of lines 
2{\x-\-gjy-\-vz){l!x-\-gJy-\-v'z) = 0 ; 
then putting, as before, 
2ax', ..X^? y^ 
we have 
{% 93, C, BIS, SI=0, 
where 
a =K, 
B = 2(A, B, C, F, G, HIa, y., .Xa', 
C = — {a, b, g, h\g>v' — gJv, fX' — v'X, \gJ — 
B= 0 ; 
and the equation in (5, S') is 
KS"-1-2(A, ..Jx, j^XV, gJ, v')Qb'—(a, ..Xg^v'—g^'v,..fQ'^=zO, 
which may be written 
{KS + (A, ..XX, (X, .XX', gJ, .')S'}^={[(A, ..XX, .X^', g.', .')X+K(«, ..Xp'-A 
= (A,..XX,^, .)^(A,..XX',/A',.')^S'^ 
that is. 
KS=[±^/ (A, . .XX, g^, vf^ (A, . .XX', gj', v'y—(A , . .Xx, .X^', g^', v')]S' ; 
