34 
ON SOME GEOMETRICAL THEOREMS &C. 
[165 
and 
— PQR — (f + v + £) A, 
it is easy by means of these relations to verify the identical equation 
3^^(« 2 + y 2 + z 2 — 2yz — 2 zx — 2 xy) + (ax + by + cz)(^+^ + 
- (Pgx + Qrjy + jR&) (Px + Qy + Rz) = A {— - yy 2 - + (f +1) + £) (yz + zx + xy)}, 
or, writing for A its value A = 0, the equation of the conic takes the form 
(an +by + cz) (~ + \ + ■) “ ( P & + Qyy + R&) ( Px +Qy + Pz ) = °> 
which is as it should be. 
It may be added, that the common point of intersection and the points in which 
the harmonic lines (r), (s) meet a side of the triangle lie in a conic passing through 
the points /, J, such that with respect to this conic the point of contact is the pole 
of the line (o). Thus for the side x = 0 the equation of the conic in question is 
0x(x + y+z) + (l -^-fyiP + iby + cz) (|+^)= 0, 
where 
^ _ a (b — cf (b + c) _ (b + c)' 2 
bc(a — b)(c — a) 2 be ’ 
and similarly for the other two conics. 
In the case where the triangle is an equilateral triangle and the line (o) is the 
line at oo, the conic becomes a circle, the points I, J, are the circular points at infinity, 
and lines harmonic in respect to the points I, J, become lines at right angles to 
each other: the foregoing results agree, therefore, with Sir F. Pollock’s Theorem.