Sliding Gunter rule, improved by J. Robertson, as described in W. Mountaine, “A Description of the Lines drawn on Gunter’s Scale, as Improved by Mr. John Robertson”, London, 1778.
This type is also described in the following article, which mentions 4 other known examples:
R. Otnes, “Robertson’s Slide Rule with Runner”, Journal of the Oughtred Society, 8:2, Fall 1999.
This article explains that the “holes” at the right side cross-section were used to attach a fine adjustment screw system for precise positioning of the slide.