Sounds like the Colt Police Positive Target.
Flat top frame with a small drift adj rear site w/ set screw lock.
Front site blade pinned into a base on the bbl with a small set screw adj for elevation coming in from the front. Same setup as on the Woodsman.
Early guns had the black HR grips,,later walnut checkered w/medallions.
Made from before WW1 till the start of WW2.
32Colt P/P is the smaller dia 32 rd using the healed bullet.
32 Colt New Police is the chambering that is the same as the 32 S&W.
Both chambering were available in the fixed site model as well as the Target version
The fixed site model was also made in 38 Colt New Police, but not the Target IIRC
Target was made in 22RF and 22WRF as well as was the fixed site edition.
They bring some decent $$ if in collectible condition.,,$1200/$1500
Shooter grade guns are around $300 to $600 again the condition is everything.
The .22rf cal are usually popular for obvious reasons and the ones in 32NP and 38NP as you can shoot 32S&W and 38S&W ammo.
They suffer from the usual problems of poor timing and cylinder rocking, ect.
Standard Police Positive & Police Positive Special parts if you can find any worth using.
added
(Police Positive Special is the same pistol with a slightly longer frame and cylinder to accomodate the 38spec and 32-20 ctgs).