I checked my copy of Charles R. Suydam's book, "The American Cartridge: An illustrated study of the rim fire cartridge in the United States". On page 90 appear photos of 4 cartridges nearly identical to yours identified as a ".41 Derringer" round (aka ".41 Short Rimfire"). It was identied as having a: .405-.407" bullet dia, a .460-.468" rim dia, a case length of .453-.493", a .870-.924" cartridge overall length, case mouth dia of .405-.406", head dia of .405-.406". Range of measurements reflect products from different manufacturers and different shaped bullets giving different overall length. If these measurements are close...then .41 Derringer/.41 Short Rimfire should be correct. Original loadings ranged from 10 to 13 grains of blackpowder and typically a bullet of 130 grains. Head stamp is not referenced in Suydam's book.
A different reference: "The Cartridge Guide" by Ian V. Hogg, identifies several "M" head stamps, the most likely being one by a gunsmith: P.J. Medicus of NY, NY. Medicus only used this headstamp on .41 Short Rimfire/.41 Derringer. 2 other European manufacturers used "M" as well. Medicus contracted with Remington-UMC to make cases. Should be of reasonable collector value. Hope this helps.