The R & S may be Rogers & Son, patent holders of #397. The respective names being John Thomas Rogers & John Rogers who I assume were father and son. Given that Patent # 397 became the standard British trade sidelock action (according to Crudgington & Baker) then #81 would be the Patent Use Number.
Crudgington & Baker also provide an illustration from Clabrough's San Francisco catalogue of 1881 showing a gun built on this patent, the lock plate shape being the same as that on the original poster's gun.
