Howdy
The Serial Number on yours is fairly early. The SN on this one is 24XX and it shipped in 1881. This one has been refinished, and I replaced the front sight with an old dime.
If your cylinder is 1 7/16" long it is most likely a 44 Russian, as 44-40 would not fit in a cylinder that long. After about SN 15340 cylinders were 1 9/16" long.
44 Russian was a straight case, 44-40 has a slight taper or bottleneck.
The cartridge third from the left in this photo is a 44 Russian, the round all the way on the right is a 44-40. A little bit difficult to tell, but the taper is there on the 44-40.
So if you look into the charge holes (S&W Speak for chambers) from the rear, a 44-40 cylinder will have a slight taper to accommodate the shape of the cartridge, the charge hole for a 44 Russian will be shorter, and it will be straight.
Of course both charge hole shapes will have a narrower 'throat' at the front.
P.S.lebomm is correct, the 44 Double Action was not chambered for 44 Special. 44 Special is basically a stretched version of the 44 Russian cartridge, stretched about 1/8 of an inch. The first firearm chambered for 44 Special was the 44 Hand Ejector, 1st Model, also known as the New Century, and dubbed Triple Lock by collectors. This revolver was Smith and Wesson's first large frame revolver with a cylinder that swings out to the side, and it did not appear until 1908.