Back then it was a large saltwater indoor swimming pool. As related by Alex Castles in his fascinating book The Shark Arm Murders, business had been so bad that proprietor Charlie Hobson placed a four-metre, one-tonne Tiger shark in the pool. The monster had been caught off Sydney eight days earlier, and Hobson hoped people would pay to come in and have a look.

On Anzac Day some of them saw more than they expected. At 4.30pm the shark went into a frenzy and finally vomitted copiously. Among the contents of its stomach were a rat, a bird, and a human arm.

Officers attended from Randwick Police Station and the limb was pulled out of the water. It was a left arm with a length of rope attached to the wrist, and on the inside forearm a tattoo of two figures boxing. The tattoo was about 20 centimetres high. Interest in the find heightened when it was realised the arm had not been chewed off, but cut.

Fingerprints obtained from the hand indicated it belonged to a former boxer named Jim Smith. The rest of him was never seen again. Smith was an England-born criminal who had lived in Balmain with his wife and child. He’d managed a billiards saloon, a seedy type of place where vice flourished. Either there or at Tattersalls Club, where he’d also worked, he’d met a major criminal named Reginald Holmes.

Holmes was an interesting fellow, an apparently respectable boat-builder with premises on the shore of Lavender Bay and a prestigious residence at McMahon’s Point. He was married, with two children, and dined at the Royal Sydney Yacht Club and was a pillar of the local Presbyterian Church. He also employed men to drive fast motor boats out to sea and collect packages of cocaine and other goods thrown overboard by seamen on incoming ships. He made more money from this and his other criminal activities than from boat-building.

Police eventually decided that Smith had done some criminal tasks for Holmes, and later been killed by a criminal named Patrick Brady, probably near Cronulla. The exact relationship between the three men remains uncertain, but it seems Brady cut off Smith’s arm and took it in a taxi to show Holmes, in order to extract money from him in some way. The arm was later thrown into the sea at Maroubra, presumably off one of its cliffs. The rest of Smith’s body was given a classic “Sydney Send-off”, weighed down and dumped into the ocean off a boat.

Brady was arrested and charged with murder. However, police had no firm evidence, and Brady denied the charge, although he did admit to having met with Holmes around the time of Smith’s disappearance.

The boatbuilder said he’d never met Brady, but something was not quite right. One day in April he took out his high-speed motor boat and had a nervous breakdown. He stopped off the shore at Lavender Bay and tried to shoot himself in the head, fell into the water and clambered back on board. He proceeded to zoom around Circular Quay and disrupt the morning ferry services, and then led the Water Police on a four-hour chase around the Harbour and beyond. Finally, two kilometres out to sea, he stopped and allowed the police to take him. “Jimmy Smith is dead and there is only another left,” he’s reported to have said. “If you leave me until tonight I will finish him.” There was a brandy bottle in the bottom of the speedboat, empty.

When Holmes was finally released from hospital some weeks later, he told police a rambling and not always convincing story pointing the finger of blame at Brady, and saying he himself had nothing to do with Smith’s death. Police hoped the matter would finally be resolved at the coronial inquest.

On the morning of the inquest, in Hickson Road in the Rocks, Reginald Holmes was found dead in the driver’s seat of his Nash sedan. There were three gunshot wounds in his chest. The inquest came to no conclusion and the case against Patrick Brady was dismissed. In a final bizarre twist, police concluded Holmes had probably been killed by a hitman hired by himself, in a strange act of suicide.