  First examples of symmetric Hadamard matrices of orders 508 and 764 are
constructed. The method used is known as the propus construction. A conjecture
regarding this method is formally proposed but it appears implicitly in three
previous joint papers with N. A. Balonin et alias. The conjecture has been
verified for Hadamard matrices of order 4v with odd v<54. In this note the
conjecture is verified for v=55,57,59,61,63.
