What is the special appointment in "After Twenty Years"?
The "special appointment" is fully explained by Bob when the uniformed patrolman whom he doesn't recognize as his old friend Jimmy Wells stops in front of him at the doorway of the closed hardware store.
“Twenty years ago tonight,” said the man, “I dined here at ‘Big Joe’ Brady's with Jimmy Wells, my best chum, and the finest chap in the world. He and I were raised here in New York, just like two brothers, together. I was eighteen and Jimmy was twenty. The next morning I was to start for the West to make my fortune. You couldn't have dragged Jimmy out of New York; he thought it was the only place on earth. Well,we agreed that night that we would meet here again exactly twenty years from that date and time, no matter what our conditions might be or from what distance we might have to come. We figured that in twenty years each of us ought to have our destiny worked out and our fortunes made, whatever they were going to be.”
Bob expected to meet his old friend in front of a big, busy restaurant. He doesn't like having to stand in front of a closed hardware store because he knows he is conspicuous. But the restaurant was torn down five years ago and replaced by some shops. Bob has no choice but to wait on that spot because that is the only place Jimmy would know where to look for him.
O. Henry invented the "special appointment" because it was a good way to bring two old friends together in the great city of New York after they haven't seen each other in twenty years. The author didn't want the two men meeting at a big, crowded, well-lighted restaurant, because Bob would not feel nervous and compelled to explain what he was doing there. In other words, it would not look suspicious to be meeting a man at a restaurant, but it does look suspicious for a man to be standing "In the doorway of a darkened hardware store."
The "special appointment" leads to Bob being in this dark doorway in this darkened neighborhood. This leads to Bob's nervousness and confusion when the cop stops right in front of him, and this leads to Bob's providing all the necessary exposition to the reader in the form of dialogue to the cop. Jimmy doesn't introduce himself immediately because the doorway is dark and he hasn't seen his old friend in twenty years. "After Twenty Years" is a story in which O. Henry wanted to be totally objective and not explain anything to the reader in the form of prose exposition by a anonymous omniscient narrator.