The Colt Police Positive made about that time would have been .38 S&W caliber, or .38 Colt New Police (.38 Colt NP). With a short (2") barrel it would have been known as the Banker's Special. (The .38 S&W and .38 Colt NP are the same cartridges, different names.
If memory serves me correctly, the frame was lengthened to accommodate .38 Special cartridges and the gun was designated as the Police Positive Special, and the snub version known as the Detective Special.
Did I help, or just further cloud the issue?