The .351 was part of a series of Winchester Self Loading (WSL is a commonly seen acronym) recoil operated rifles in the first decade of the 20th century.
The design was essentially the same; as heavier calibers were added, the gun was just beefed up. The same basic design also saw use in at least one Winchester .22 rifle.
The first in the series was the 1905, chambered in .32 and .35 WSL cartridges.
From all accounts, they were pretty worthless cartridges that never developed a niche, with one interesting exception, which I'll note later. Both were under powered, and never caught on. They were both made obsolete with World War I.
In 1907 Winchester beefed up the gun and the cartridge and offered the .351 WSL. This gun caught on. It was just powerful enough for deer-sized game (although there were a lot better choices).
Where the 1907 really caught on, though, was with law enforcement. I think something like 80% of all of the 1907s went to law enforcement, both public and private, prison guards, etc.
Some of these guns, believe it or not, are still in inventory with some small police forces.
Some 1907s even saw combat with French forces in World War I, and a very interesting hybrid, the 8mm Ribeyrolle, an 8mm Balle D bullet from the French Lebel rifle and the .351 WSL case, was developed for an experimental semi-auto rifle project that didn't go anywhere.
In 1910 Winchester brought out the last in the series, the .401 WSL. It was a lot more powerful than the other rifles in the series, and was a decent short-range deer, bear, and even jaguar rifle. It was, in some ways, the early ancestor to Ruger's .44 Mag. semi-auto carbine.
Ammo production for the .351 and the .401 survived WW I, but when the ammo companies realigned for WW II, the .401 was dropped and never brought back.
The .351 survived as a production item until sometime in the late 1970s early 1980s.
Ok, I mentioned that the .32 WSL had a very interesting side note associated with it.
In 1940, while Winchester was pitching its version of a military carbine to replace the 1911A1 pistol in many applications, Winchester dusted off the .32 WSL casing, did some modifications, and developed what would become the .30 M1 Carbine round.
A redesigned .351 would have, in my opinion, been a MUCH better choice all around, but the military wanted a .30-cal. weapon.