It is the .32 Safety First Model Double Action. S&W made c. 91,417 of them between 1888-1902. Yours would be a bit over half way through the production run. All that First Model should be considered black powder guns, though current .32 S&W ammunition is held to very low pressures and that gun should be safe with modern factory ammo (if you can find it), at least for a limited number of shots.
The .32 Safety or "Safety Hammerless" was one of the most popular revolvers of the period; they were made up to 1937 with a total production of almost 243,000 for the three models. Due to the effect of corrosive primers and the fact that almost all were nickel plated and the plating is peeling, value is pretty low. But they were well made guns (unlike many top-break revolvers of the time) and if they had any kind of care they will still work OK. The .32 S&W cartridge is, of course, about the bottom level of revolver power, and few would recommend it for serious purposes, but it would (IMHO) beat throwing stones.
FWIW, I don't think any of the .32 Safety had the S&W logo on the frame.