A reader recently asked about the benefits of the rising bite and how it works. I thought that the answer may be of interest generally, hence, its posting here.
I appreciate the compliment but I think of myself as much more a student than an expert (there are enough "experts" in this field already).
The rising bite (Rigby and Bissel's patent) is a so-called "third fastener" in which a vertically-oriented bolt engages a horizontally-oriented, "horseshoe" shaped rib extension. What this type of fastener provides, that no other "third bite" does, is positive axial restraint of the breech. Meaning, that it prevents the barrels and standing breech from moving "fore and aft" (in opposite directions, due to breech thrust) during firing.
Most break-action designs only restrain the barrels in the vertical plane and, while this keeps the barrels from pivoting open during firing, it also allows the standing breech to deflect axially, since this axial deflection is resisted only by the mass and strength of the material at the juncture of the standing breech and the frame flats. This is the reason that double rifles usually exhibit "bolsters" of added material at this juncture. Unfortunately, many of these bolsters are little more than ornamental by design.
Many third bites, such as Purdey's and Holland& Holland's hidden third bites, also allow this deflection to occur due to the fact that these "hidden bites" also act to restrain the barrels only in the vertical plane. The so-called "doll's head" rib extensions are supposed to prevent this axial deflection but, with the exception of Westley Richard's bolted doll's head, all fail in this regard. It must be noted here that Westley Richard's bolted doll's head was their primary (and only) fastener for quite some time and they later added the underbolt in order to make the gun have three bites, so WR's doll's head is not technically a "third bite". The Greener-type crossbolt has the potential to prevent this axial movement but only if fitted properly, which most are not. In fact, in most guns (other than Greeners) that employ this type of third bite, the bolt makes little, if any, contact with the rib extension.
The rising bite demands appreciable skill in its fitting during manufacture. The underbolt and the vertical bolt must fully engage their bites when closed, and the linkage that connects them must be designed so that both bolts clear their respective bites at the same time, in order to allow the barrels to open, while also translating the underbolt's horizontal movement to vertical movement of the rising bolt. The added time in design and manufacture obviously adds to the cost, and this is probably the primary reason that it's not seen more often.
All of the above applies to single-barrel and side-by-side guns, whose frames are inherently quite flexible. Over/unders (especially a Boss-type) are a bit of a different story since their frames are much stiffer (more resistant to deflection), due to their frame height and the deep sides of the frame acting as shear webs. The rising bite in the Boss guns is a very technically interesting design but, owing to the frame's inherent stiffness, it is little more than technical curiosa.
The over/under's inherent stiffness in the vertical plane, not only in the frame but also in the barrels, is one of the reasons that the O/U is technically superior to the SxS.
I hope this helps. If you have any further questions, feel free to ask.