The AutoGlove, the Akins accelerator, and the Bump Fire Stock share much in common. All three devices were designed as work-arounds to federal law. Each, when used with a semiautomatic weapon, allows the user to fire at rates that can exceed 400 rounds per minute.

The first two, the Bureau of Alcohol, Tobacco, Firearms, and Explosives has determined, can’t be manufactured or sold in the United States. The third, the Bump Fire Stock, is legal, the agency has said.

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That determination is now drawing scrutiny. On Tuesday night, the Associated Press reported that the suspect in the Las Vegas mass shooting had equipped a dozen rifles found in his hotel room with bump-stock. A New York Times analysis of the sound of the event suggests that the gun used in the shooting was augmented with such a device.

Mark Jones, a retired supervisory special agent at ATF and head of the Police Foundation’s National Law Enforcement Partnership to Prevent Gun Violence, told The Trace that ATF officials knew that if they did not approve such a product, they would get grief from lawmakers and gun industry executives.

“The industry knows it can push ATF around,” Jones said.

The ATF did not respond to requests for comment. But there are technical, but key differences between the AutoGlove, Akins accelerator and bump-stock that help explain why the agency banned the first two, and allowed sale of the third.

The Akins Accelerator was designed to let a rifle’s recoil bounce its functional parts like the trigger group and barrel back and forth within its stock, so that the trigger could be pulled ever faster. When the inventor first submitted his device to the ATF for approval, the agency granted its blessing, only to decide in 2007 that because Akins functioned with the aide of a spring, it converted weapons into machine guns.

In September, the Firearms Technology Industry Services Branch of the ATF put the kibosh on the AutoGlove, a wearable device that combines a battery pack, a glove, and an electric plunger at the end of the trigger finger.

Firearms experts at the ATF strive to stay within the strict letter of federal law, which defines a machine gun as a weapon that can shoot more than one round with a single, unaided action. Because the Autoglove and the Akins augment the shooter’s trigger finger with mechanical parts, they broke the law, but bump-stocks don’t, so they’re legal, the agency determined.