I'm afraid W&J Davis appear to be an economy gun retailer. You almost certainly have one of their guns as described in the info below. The proof mark of 13 is not the indication of a'13 bore' as such.
This is what IGC has on them:
Name W & J Davis
Address1 Unknown address?
Address2 31 Newton Road, Stratford Road
City/Town Birmingham
Country United Kingdom
Trade Gunmakers
Dates 1876-1928?
Notes
W & J Davis claimed establishment in 1876. They were recorded in about 1900 at 31 Newton Road, Stratford Road, Birmingham.
The firm's catalogue dated to 1901 reveals that by that time they supplied the public direct with claims of the least expensive guns and best value for money. Most barrels were twist or Damascus, only one or two models were available with steel barrels. No calibres were stated for their rook and rabbit rifles, no mention was made of .22 calibre. They supplied muzzle loading guns as well as boxlocks, but most of the models available were hammer guns with very few sidelocks. The catalogue suggests they bought guns from the trade as cheaply as possible. The firm appears to have closed in about 1928.
Tim
