[325 
325] NOTE ON A THEOREM RELATING TO A TRIANGLE, LINE, AND CONIC. 
101 
LINE, AND 
and the equation of the line joining this point with the vertex (y = 0, z = 0) is 
(ay — hX) y = (gk — av) z. The equations of the three joining lines therefore are 
(ay — hk) y = (gk — av) z, 
(bv — fy) z = (hy — bk) x, 
(c\ - gv) x = (fv - cy) y, 
lines which will meet in a point if 
(ay — hk) (bv — fy) (ck — gv) — (gk — av) (hy — bX) (fv — cy) = 0, 
or, multiplying out and putting as usual 
K = abc — a/ 2 — bg 2 — ch 2 + 2fgh, 
21 = be -f\ &c., 
if 
2 (abc — fgh) kyv 
-183.] 
+ a®yv 2 + af>y 2 v 
+ bfevk 2 + b%v 2 k 
+ c%ky 2 + c®Xy J 
of Steiner’s,” an 
that is, the line must touch a curve of the third class. 
triangle join the 
n regard to the 
le line must be 
up into a pair 
break up into 
If this equation break up into factors, the form must be 
(cCk + (3y + <yv) (Ayv + Bvk + Cky) = 0; 
that is, we must have 
Aa + B/3 + Cy = 2 (abc — fgh), 
Ba = bS$ , Ca = c©, 
C/3 = c § , A@ = a^Q, 
A<y = a®, B<y = b$ ; 
and the last six equations give without difficulty 
. ka 1 ^ * 
A ~ g ’ a ~ k 
a -®. 
C=|, 7 = >, 
where k is arbitrary ; the first equation then gives 
“| S + -f + C f = 2 (abc-fgh);