The cartridge is most likely a .38 S&W, which is substantially identical to the .38 Colt New Police, and the British .38/200.
In Europe and South America it is also called the .380 Rim and .38 S&W Corto.
It is a very old cartridge, being introduced in 1877.
It has been used in dozens of models of S&W, Colt, Webley, and Enfield revolvers.
It is documented in Wikipedia at:
According to Wikipedia:
The .38 S&W is a revolver cartridge developed by Smith & Wesson in 1877. Though similar in name, it is not interchangeable with the later .38 Smith and Wesson Special due to a different case shape and slightly larger bullet diameter.
The British military adopted a loading of this cartridge as the Cartridge, S.A., Revolver Ball, 380 in, MkI .38-200, with the "200" referring to the weight of the bullet in grains. In 1937, this cartridge was replaced in British Service by the Cartridge, S.A., Revolver Ball, 380 in, MkII. The main difference between it and the previous round was* that it had a 178 gn. FMJ bullet.
The .38 Colt New Police was Colt's Manufacturing Company's proprietary name for what was essentially the .38 S&W with a flat-nosed bullet.
The U.S. .38 S&W Super Police cartridge was nearly identical to the British .38/200 Mk I, using a 200 grain (13 g) lead alloy bullet with a muzzle velocity of 630 ft/s (189 m/s) and a muzzle energy of 176 ft·lbf (239 J), and was supplied by several U.S. manufacturers to the British government as equivalent to the Mk I loading.
The .38 S&W is also called the .380 Rim and .38 S&W Corto.
I have several .38 S&W handguns, including a Smith Victory model, and an Enfield which are a lot of fun to shoot.
Ammo is still manufactured, and old and surplus .38 S&W smmo in its various iterations can sometimes be found at bargain prices.