In the Proceedings of the National Academy of Sciences of the U.S.A., Vol. II. (1916), page 171, Professor F. Morley has established a theorem which both extends and simplifies the theorem of Feuerbach, viz., All curves of class three which (i) touch the six lines OP, OQ, OR, QR, RP, PQ joining four orthocentric points, O, P, Q, R, and (ii) pass through the circular points, also touch the common nine-points-circle of the triangles PQR, OQR, ORP, OPQ. Sixteen of these curves of class three break up into one of the four points aud a circle touching the sides of the triangle formed by the other three. Thus the sixteen instances of Feuerbach's theorem derivable from the four triangles are included as special cases, in Morley's theorem. A purely geometrical proof of the theorem may be worth consideration.
(Received September 09 1919)
(Revised November 14 1919)