The use of so awkward and difficult a weapon implies that the crime was hastily arranged that evening. The location of the crime and the search of club files imply that the murder was prompted by club business, which narrows the likely suspects to those at the board meeting the night of the crime and to the niece of the Fitzgeralds.
The killer knew how to fire the harpoon gun, and so must have been connected by blood or marriage to the Fitzgerald family. This narrows the suspects further to Scotty Fitzgerald and the Hemenway couple. Florence Hemenway, who worked late in a public restaurant, had a verifiable alibi. Either Scotty Fitzgerald or Doug Hemenway, both of whom were short, would likely have used the wooden chair to reach and fire the harpoon gun. Fitzgerald's crippling arthritis, however, prevented his climbing a chair. The prime suspect was therefore Doug Hemenway.
Jackson eventually found that Gordon Fitzgerald had suspected his brother of embezzling club funds. Fitzgerald had asked Hemenway just before the board meeting to meet him secretly back at the club after the meeting. When they returned and reentered the club (using Fitzgerald's key), they began to examine club records. Gordon's suspicions about Scotty were correct, but he did not know that Hemenway was in cahoots with the treasurer.
When Hemenway realized that a review of the records would reveal his complicity, he left the office as if to use the bathroom, scrambled to the harpoon gun, called Fitzgerald out of the office and, relying on the victim's limited mobility, harpooned him. The fact that he had stayed at a distance meant that he had avoided blood stains on his clothes that other more confrontational murder methods would have caused.
Hemenway wiped off the gun, hastily replaced the files, and arrived home just before his wife.