I have a few things to add:
This gun MAY be a Spanish copy. Many of these were sold in the USA prior to WWII. A bona fide S&W will
usually have an intertwined S&W trademark on the LH side of the frame behind the cylinder, and "MADE IN U.S.A." lettering on the RH side above the trigger guard; this area is washed out by glare in the picture. Spanish copies of S&W .32's have very little value- typically in the $50-$100 range- and are sometimes unsafe to shoot, or wear out very quickly if you do so.
The serial number is on the butt. This number, or the last few digits of it, should be repeated on the underside of the barrel above the ejector rod, and on the rear cylinder face OR under the ejector star on some older versions. If the numbers do not match, these parts are not original. Ignore any numbers on the frame under the cylinder yoke; these are meaningless assembly numbers.
I've seen a few of these guns that had damage to the ejector star, ejector rod, or cylinder as a result of trying to eject stuck .32ACP aka .32 Auto cases. This ammo is much more widely available than .32 S&W and .32 Long, and some shooters unfortunately believe that it's interchangeable, but it has a much smaller rim than the correct cartridges and can get stuck in the chambers when it gets whacked by the firing pin.
Some of the prewar versions of the .32HE are not safe to carry with a live round under the hammer, and/or are very hard to get parts for. As James K said, a serial number is needed to correctly determine its age.