The revolver in your pictures has a double-action trigger, so it's really in a different ballpark than a single-action No. 3. If it's chambered in .44 Russian, it's a
.44 Double Action First Model aka "New Model No. 3 Navy" (which IIRC was not an actual factory designation); if it's chambered in .44-40 WCF, it's a
.44 Double Action Frontier Model. Two rare and valuable variants exist: the
.38 Double Action was offered in .38-40 WCF, and the
.44 Wesson Favorite was a .44 Russian variant with lightening cuts on the barrel and frame and extra lettering on the barrel and cylinder. This gun does not appear to be a Wesson Favorite because I don't see the frame lightening cuts.
The .44DA and .44DA Frontier aren't particularly valuable; in this condition, I would put the value at $400-$475 for the former and $450-$525 for the latter, depending on local market conditions. On the plus side, it appears to have the original bluing, and blue finish seems less common on these guns than nickel; however, the gun is pretty rusty, and the S&W DA large-caliber top-breaks don't usually get into 4 figures unless they have unusually nice original finish. OTOH if the gun is indeed a .38-40, it could be worth ~$2,000, although .38 values are hard to estimate because only about 300 were made.
Interesting historical footnote: The large-frame DA top-breaks were never particularly good sellers for S&W, even when they were introduced in 1881. Sales nosedived at the turn of the 20th century with the introduction of large-caliber swing-out-cylinder models from both Colt and S&W, but the company had stockpiled several thousand frames in the mid-1890s to meet anticipated future demand, and 19th-century New Englanders were famously loathe to throw away anything useful; consequently, the guns lingered in the catalog until 1913, and were assembled from existing frames and parts whenever an order was received. As a result of having been built from pre-1899 parts, ALL large-frame S&W DA top-breaks are legally considered to be antiques and can be bought and sold without an FFL.
