Yep; Per Macks "Cartridge Case Measurements" the 7.8x57 is for the 1888 Mauser.
Now maybe someone could clear up the history of the groove dimensions. I have read that the 88's used an oversized bullet
that wouls swage down. To aid this the forcing cones were tapered. I have seen 1888 Mausers with this. Later they increased the groove depth keeping the same bore size.
At this piont the designated the differences as S & J sizes.
I have also read that the Military used the same bullet with
both groove depths.
Any Mauser historians out there?.
Thanks!