Here's an old post from 12/27/12 by Alnamvet68 in a thread entitled "S&W 360 .38 Special:
[T]he Model 360J is chambered only for the 38 special. This is one unique handgun in that it is built with a scandium frame, and depending on whether it is an earlier model, or a most recent production, comes with either a carbon steel cylinder, or stainless steel cylinderl, all finished in melonite. Of course, if one was to change out the cylinder for one that accomodates a 357 mag, then it would be safe to use 357, since the only difference in the 360J and the 360 or 360 PD is the cylinder...the 360 uses a stainless steel cylinder and accomodates the 357, and the 360PD uses a titanium cylinder which accomodates the 357.
That said, you will not find anything on the S&W site, archived or otherwise, and you will get a variety of responses when you call and speak with S&W as to why the 360J 38 cal only does not appear anywhere on their web site. I am of the opinion that this is a handgun that was built on contract for a foreign LE/military, and for whatever reasons, either they built more then they needed, or the contract went *****...S&W will not say yay or nay on either theory. One last thing, you will notice the serial number is on the left side of the revolver, just under the cylinder release latch, instead of on the butt, where all the J frames have their serial numbers. If you take off your grips, you will notice a hole in the butt which appears to be for a lanyard...
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I like the revolver because of its shrouded barrel and durable melonite finish, not to mention its light weight...oh, and its a S&W that most "experts" have no clue as to its origins and purpose.