308 Proceedings of Boyal Society of Edinhurgh. [sess, 
so that the result of the elimination should be 
(ABC - AF2 - BG2 - CH2 + 2FGH)2 = 0 . 
Here considering in connection with the triangle x = 0, y = 0y z = 0 
(say the vertices hereof are the points A, B, C), the conic 
(a, A c,f g, hfx, y, z)^ = 0 
the first equation represents the pair of tangents from the point A 
to the conic, the second the pair of tangents from the point B to 
the conic, and the third the pair of tangents from the point C to 
the conic. The first and second pairs of tangents intersect in four 
points, and if one of the third pair of tangents passes through one 
of the four points, then it is at once seen that the conic must touch 
one of the sides x = 0, y = 0, z = 0 of the triangle, viz., we must 
have hc-f^ — 0, ca-g‘^== 0, or ab - Jd = 0. But we have a — BC - F^, 
&c., or writing = ABC - AF^ - BG^ - CH^ + 2FGH, then these 
equations are A = 0, K^B = 0, K;^C = 0, all satisfied by = 0. 
We may regard Kj = 0 as the condition in order that the conic 
(a, h, c, f g, h){x, y, zf — 0 may be a point-pair : the analytical 
reason for this is not clear, but we see at once that if the conic be 
a point-pair, then the three pairs of tangents are the lines drawn 
from the points A, B, C respectively to the two points of the point- 
pair, so that the three pairs of tangents have in common these two 
points. Regarding = 0 as the condition in order to the existence 
of a single common point, and recollecting that the true result of 
the elimination is = 0, the form perhaps indicates what we have 
just seen is the case, that there are in fact two common points of 
intersection : but at any rate the foregoing geometrical considera- 
tions lead to Ki = 0, as the condition for the coexistence of the 
three equations. 
I remark in conclusion that I do not know that there is any 
general theory of the case where a result of elimination presents 
itself in the form q, as distinguished from the ordinary form 
fi = 0. 
