102 
NOTE ON A THEOREM RELATING TO A TRIANGLE, LINE, AND CONIC. 
or, reducing by the equations = 21$ + ciK, &c., this is 
which, substituting for 21, 23, (5 their values, becomes 
Hence if K = 0, that is, if the conic break up into a pair of lines, or if 
in either case the equation of the curve of the third class becomes 
that is, the curve breaks up into a point, and a conic inscribed in the triangle. 
In the case where the conic breaks up into a pair of lines, then we have 
(a, b, c, f g, h\x, y, z) 2 = 2{px + qy+ rz) {p'x + q'y + r'z), 
and thence 
(21, 23, 6, $, @, #$>, y, zj = - {(qr' - qr) x + {rp - r'p) y + (pq' - p'q) z} 2 
so that the equation in (X,, p, v) is 
{(qx’ — qr) X + {rp' — r'p) p + {pq —p'q) v} 
[pp {qr — q'r) pv + qq {rp' — r'p) v\ + rr {pq' —p'q) X/x} = 
where the point represented by the equation 
{qr — qr) X + {rp' — r'p) /i + {pq' — p'q) v = 0 
is, of course, the intersection of the two lines.