The .38 Police Positive and Police Positive Special are almost identical, except the "Special" has a longer frame and cylinder, and is usually found chambered in .38 Special or .32-20 Win.
The Police Positive has a shorter (1.250") cylinder and is usually found chambered in .38 Colt New Police (identical to .38 S&W) or .32 Colt New Police (identical to .32 S&W Long).
The .38 Police Positive is far less popular today, because of the expense and difficulty in finding ammunition.
In both cases, finding repair parts for the pre-World War Two models is not easy, nor is it any easier to find a gunsmith that's qualified to fix them.
Therefore I strongly advise that one avoid the "beaters," unless in perfect mechanical shape, because if something is wrong (or goes wrong) you may pay as much to get it fixed as you paid for the gun.