1 Kings 1:32-33:
"2 And king David said, Call me Zadok the priest, and Nathan the prophet, and Benaiah the son of Jehoiada. And they came before the king.
33 The king also said unto them, Take with you the servants of your lord, and cause Solomon my son to ride upon mine own mule, and bring him down to Gihon:" (KJV)
If a public ceremony was the only point of Solomon's anointing as the next king, why would he have to be taken out of the city to Gihon? What was there about Gihon that was necessary for the anointing of the king?