ME. A. CAYLEY ON THE ANALYTICAL THEOEY OF THE CONIC. 
659 
The condition for this is 
Disct. ..yjc, y, 2 )^= 0 , 
which is 
(^, <B, C, 
where 
a=K, 
B =Aa'+B5'+Cc'+2F/'-f 2G^'+2H7f, 
C=A'a+B'54-C'c+2F/+2G'^+2ra, 
B=K' 
(the significations of K', A', B', C', F', G', H' being of course analogous to those of 
K, A, B, C, F, G, H). The three roots 5 : & correspond, it is clear, to the three pairs 
of lines which can be drawn through the intersections of the two conics. 
35. The equation 
Disct. (9[, 33, C, 5')^=0, 
which is of the fourth order in 91, B, C, B, and of the sixth order as regards [a, c,f\ g, h) 
and (a', h\ d^f \ g\ h!) respectively, is the condition in order that the two conics may 
touch each other. Assuming that it is satisfied, the cubic equation in has a pair 
of equal roots ; or say there is a twofold root and a onefold root ; the twofold root gives 
the pair of lines drawn from the point of contact to the other two points of intersection, 
the onefold root gives the pair made up of the common tangent and the line joining the 
other two points of intersection. 
36. In particular, suppose that the two conics are 
2{qx-\-(ry+rz){^x-{-dy +r' 2 :) = 0, 
2{'Kx-]rg.y-{-vz){dx-\-gJy-]rv'z)=0-, 
so that 
{a,h, c,f, g, 1i)z={2q^, ffr'+ff'r, §ff'+§'<r), 
(ft', J', (f, g^ A'j = (2XX', ^gigj , ‘2ivv' , giv' -\-gJv, j'X' 
(A, B, C, F, G, H)=— (<rr'— (7V, r§'— r'§, 
(A, B', C', F', G', H')=-(p'-^V A -A, XgJ-dg.f; 
and thence also 
^=K = 0, 
33=Aft'+ &c. = — 2 
X, V 
X', gJ, 
§ , O' , r 
f , 
t', d 5 d 
J J 
f , ff , T 
C=A'ft4-&c.=— 2 
? , , r 
§5 ^ ^ 
X, g., V 
X, V 
X', gJ, v' 
X', g], v' 
B=K' = 0; 
