The only S&W .38 breaktop with those patent dates is the Perfected model, which is a breaktop but also has a push type lock like the swing cylinder guns. As a further identification point, it is the only S&W top break with the trigger guard integral with the frame.
The serial number would be one of those (I suspect the 3350) and would be in tiny numbers. The other number is probably a Pinkerton inventory number.
The Perfected Model is a highly desireable collectible. 59,400 were made from 1909 to 1920, numbered in their own series. Yours was made early in production no matter which number is the serial number.
Value depends on condition. In top condition, it would bring $900+, less if worn, rusted, or missing parts, or with cracked or missing grips. I don't know if Pinkerton marked their guns, but if so such a marking would bring a premium. If possible, try to obtain an affidavit or some other paper documenting the Pinkerton connection. Also, get a S&W letter; it may have been shipped directly to the Agency.
Jim