Larry21556,
Your rifle is chambered for 8x57IR, as Raimey opined. In earlier times, cartridges with the head diameter(about.468")of the military cartridge for the Mod 88 commission rifle were often identified as 88 and then nominal bullet diameter, ie 88.9 or 88.8,or sometimes 8x57 m88, 9x57m88, etc.The 7.7 is the bore diameter(not groove or bullet dia.)measured with plug gauges.It passed a 7.7mm(.303")gauge but not a 7.8mm(.307")gauge.This is a little tight, but common.Sometimes 8mm barrels gauage 7.6mm. The 57 is the case length.It was not thought necessary to stamp a "R", because the gun is a "tip up" and the rimrecess can be seen.It won't hurt anything if you want to make a chamber cast and slug the barrel.You can use whichever bullet dia.that will fit freely into a fired case.
Mike