Boothroyd's: "Directory of British Gunmakers" may offer some light on "W. Harris & Co.". Unfortunately, my copy is on loan...so no info readily available. There may be a link through gunmakers: "Harris & Trulock". An amalgamation of the two firms may have occurred at a latter time. Not sure, but may be of some use in researching the gun.
An internet site that may prove of value, is: "Internet Gun Club".
internetgunclub.com/general/history.php
As previously noted, this gun appears to have been built prior to 1875, as the pinned forearm and proof marks on the barrel (Birmingham) more certainly appear to indicate. As a matter of interest, all British made choke-bored guns manufactured between 1875 and 1887 were required to be stamped "not for ball". So, this would also help indicate a manufacture date prior to 1875. The "13" stamped on the barrels indicates the gauge: 13-gauge. This was normally measured 9 inches form the breech end of the gun and indicates the gauge of the barrel at its widest point, excluding the chamber.
DO NOT fire modern smokeless cartridges in this gun! Have it checked by a competent gunsmith before even thinking about firing it. Then, should it pass muster, shoot only cartridges loaded with black powder. While you're at it, be sure to have the chamber dimensions checked, as it will likely be a 2 1/2-inch chamber(fairly standard for British game guns).
Hope this helps. It can be difficult and frustrating trying to find info on many of these small gunmaking shops, as little or no information often exists.