The peacock feather murders

The murderer sent a formal invitation to Scotland Yard telling them the time and place of the murder. Incredulous, and astounded at the audacity of such a note, the Yard recalled a similar, and still unsolved, case of two years previous. Sir Henry Merrivale and Chief Inspector Masters accepted the invitation and had the house surrounded. Upstairs in an otherwise empty house was a furnished room. A man entered the house. Promptly at the time set by the murderer a shot rang out. The police rushed in and discovered that same man on the floor with a bullet through the back of the head and another in his spine... but no one else had entered the house! It was an impossible situation, but it DID happen.