100 
[325 
325. 
NOTE ON A THEOREM RELATING TO A TRIANGLE, LINE, AND 
CONIC. 
[From the Philosophical Magazine, vol. xxv. (1863), pp. 181—183.] 
I FIND, among my papers headed “ Generalization of a Theorem of Steiner s,” an 
investigation leading to the following theorem, viz.: 
Consider a triangle, a line, and a conic; with each vertex of the triangle join the 
point of intersection of the line with the polar of the same vertex in regard to the 
conic; in order that the three joining lines may meet in a point, the line must be 
a tangent to a curve of the third class; if, however, the conic break up into a pair 
of lines, or in a certain other case, the curve of the third class will break up into 
a point, and a conic inscribed in the triangle. 
Let the equations of the sides of the triangle be 
the equation of the conic 
and that of the line 
x = 0, y = 0, z = 0, 
(a, b, c, f g, h\x, y, zf= 0, 
Xx + y,y + vz = 0 ; 
then the polar of the vertex (y - 0, z=0) has for its equation 
ax + hy + gz = 0; 
it therefore meets the line Xx + py + vz = 0 in the point 
x : y : z = hv — gfji : g\ — av : ap—hX,