Both of the listed bayonets appear to be identical in construction and fully interchangeable.  Both have Collins Type 1 long hilts, an inside muzzle ring diameter of ~20.8 millimeters and a guide slot of ~25.5 millimeters.  The only difference is found with the way they are serial numbered. 

The Sharps bayonet usually has a small font 5 digit serial number stamped horizontally on the hilt flat.

The example listed below as a Sharps Model 1859 is believed to be correct but it is not serial numbered at all.