An Extension of the Harmonic Relation between Conics

If there exists one self-polar triangle of a conic S inscribed in a second conic S', then there exists an one-fold infinity and S' is out-polar or ex-harmonic to S. On the other hand it is easily seen that there always exists a finite number of self-polar triangles of S with their three vertices situated severally on three assigned conies S₁, S₂, S₃. The object of the present note is to shew that under certain circumstances the conies S₁, S₂, S₃ admit an infinity of such triangles. For the symbolic method which is used throughout, reference may be made to Grace and Young : Algebra of Invariants (Ternary Forms).