They would cut of the limp that had the frostbite in order to stop it from hurting the victim and to stop it from spreading to the rest of the body. This is also known as decapitation.
I'm afraid I couldn't give you a complete answer, or the answer that you're looking for, for this question regarding the Native-born Americans in 1860's since there is no map provided with this question. Try re-posting this question with the map on it.
