my 2 cents worth,,,
I think the first regular issue New York City Police weapon was the Colt .32 New Police, in the year 1895 by them Police Commissioner Theodor Roosevelt. Before that year, Policemen were responsible for supplying their own arms. As you might imagine, each Policeman had a very different view as to which weapon was best. A few still had and liked the cut down 4 inch barreled 1851 Colt Navy cap and ball; but many other types were used.
The .32 New Police was chosen because it was noted for accuracy, and very light weight. At that time most Police were walking a beat, and a small light pistol and a very large night stick seemed preferable.
Not long after 1895, more and more cars made the scene, while the .32 seemed adequate for occasionally chasing a suspect on foot, such a small pistol was entirely inadequate for a car chase, even when chasing Model "T" Fords. So before long the ,32 lost favor and the .38 was selected.
Interestingly, in the 1930's, cars became faster and the metal on cars grew thicker. and to many Police Departments, the .38 Special became inadequate, so the .38/.44 and the .357 Magnum made the scene. Then in the 1970s/1980's cars grew smaller and metal covers thinner, and the 9mm auto's seemed best.
??? What next?