Hishergarret... as BigCypressHunter would say, please put a green check on your post.
My old memory that your cartridge-case is Swiss was correct. Now I've found the exact info, by following memory-jogging advice from my fellow researcher Dave Poche, and a friend of his in Canada. Your cartridge is a .41 Swiss Rimfire, first manufactured in 1866, all the way into the early 1890s, when the Swiss rifle they were made for was phased out. Specifically, your cartridge is a 10.4millimeter (.41-caliber) Swiss Rimfire, length 38 milimeters long, for rifle use. There was also a 10.4mm Swiss revolver (pistol) of the same era, but it used a centerfire cartridge, which excludes it from being your catridge.
To see photos of an Original (1867-1890s) copper-case cartridge (not a centerfire brass-casing modern Reproduction), complete with the Original lead bullet, go to:
http://castboolits.gunloads.com/showthread.php?t=21917
and
http://www.swissrifles.com/ammo/index.html
To learn about the Verletti rifle which fired your cartridge, go to:
http://www.swissrifles.com/vetterli/
Also see:
http://www.militaryrifles.com/Switzerland/SwissVet.htm
which says:
"The decision to use the 10.4 millimeter rim fire cartridge in the Veterrli rifle was apparently motivated by considerations of economy since the cartridge was already in existence and proven in the Milbank-Amsler conversions of the Federal rifle. Notwithstanding that it is a rimfire, this round was a relatively high velocity, flat trajectory load, far ahead of the short range, large caliber rounds used in the other converted breach loading cartridges. The Vetterli striker has a forked firing pin which passes through two firing hole pins in the bolt face for double striking the rimfire cartridge."