The thought is that each rifled barrel – the tube that the slug flies through has twists that spins the slug along its longitudinal axis so they fly through the air predictably – will be unique to that one barrel. If you recover a slug that hasn’t been really deformed, between the slug and the shell (if you have it), you can reasonably conclude it was fired by the same weapon. There are flaws to this, do we really know two brand new barrels are unique? If you fire a soft tip slug, a frag, or most hollow points, the bullet will be so deformed that a match is essentially impossible.
Having the discarded casing is helpful, first of all, it could have fingerprints on it. Secondly, the hammer strike is specific to the weapon type. If I have a pistol I took off a suspect and I can verify the shell came from that weapon type, the slug matches the grooves well, I can place that weapon at the scene. It doesn’t *necessarily* place the suspect at the scene, as someone else could have taken the weapon and used it, but it is pretty helpful.
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