Hi Ike,
Not a BSA but a mid-1920s Rudge. I'm pretty sure it's a 350 (there was a 500 as well), which was built in 1924 and 25 only. Rudges of this period were a little different from most other bikes, with a 4-valve engine and 4-speed gearbox. The rim brakes are probably best forgotten.
The bike has a few modifications: not surprising since it was 8 or more years old when the photo was taken. The headlight has been changed, tool box and twist-grip throttle added, and the mufflers replaced by straight-through exhausts.
Leon