Will a Pietta 1851 or 1861 cylinder fit a Pietta 1862 Police?

[sharps54]

I'm pretty sure one of them should but I wanted to make sure. I have found a great deal on a Pietta .36 1862 Police (which is basically a shortened 1861 with a fluted cylinder    ) and I want to replace the fluted one with a smooth or engraved one. The seller tells me a Kirst conversion for the 1851/1861 Pietta fits but I figured I would check about the C&B cylinders before sending Cabelas any money.
Thanks in advance!
Mild Myles

[sharps54]

Thanks Seth! I knew the Ubertis are more historically correct (5 shot, smaller frame) but I just wanted to check before I spent the money. To my eyes a fluted cylinder just looks wrong on a cap and ball revolver. The only bad thing I see about this is if the 1862 fits my had better then my .36 Remingtons I might be looking for a brace of 1851s!