That gun is a Stevens No. 360 as shown in the J. Stevens Arms & Tool Co. General Catalog No. 51. The No. 350 had "Special Smokeless Steel" tubes in the monobloc, The No. 360 had Twist tubes, and the No. 370 had Damascus tubes.
By General Catalog No. 52, said to be about 1906, they had dropped the underbolts in favor of a rotary bolt through a rib extension, and added a No. 380 which had Krupp Fluid Steel tubes.
By General Catalog No. 52 (Revised Edition), J. Stevens Arms & Tool Co. had gone to making their barrels by the chopper lump method, which they called Demi-Bloc.
All in all the gun in question looks to be a very nice example of a very short-lived varient.
