The rising stud is actuated by moving the underlever and 'lifts' the barrels to start the opening motion. When closing the barrels, the depressed stud moves the underlever towards the center, bringing it within finger's reach. It is an assisted opening/closing mechanism developed by Beatus Beringer of Paris, in use on pin-fire actions from at least 1837. The stud could be centered, like on the Reilly, or slightly to one side, as in Beringer's diagram below (Journal Des Chasseurs, 1837).

[Linked Image from i.imgur.com]

The fact that there is a corresponding wear mark under the barrel shows the Reilly gun was opened and closed a great deal. By contrast, Casimir Lefaucheux used a fixed pin protruding from the barrel flats which cammed agaist the locking lug, which accomplished the same purpose.

The use of Beringer's rising stud on British pin-fires is interesting in that it shows that not all elements of Lefaucheux's action were copied. In fact, Beringer's action improvements and design choices (such as the single bite) were far more common in British pin-fires than Lefaucheux's, despite all British pin-fires being called 'Lefaucheux' at the time. Joseph Lang copied Beringer's single bite, but did not include the rising stud. Blanch, Reilly, and others, and Birmingham action makers such as Samuel Breeden and Joseph Brazier, did. While the pin-fire system of Lefaucheux, based on the cartridge, was adopted and correctly credited, Beatus Beringer had a greater impact on British designs, something which has not received the attention it deserves. Blanch bought a Beringer gun to study; what is not known if other gunmakers subsequently copied Blanch's copy, or obtained Beringer guns to dismantle on their own. I have no doubt that some Lefaucheux actions were brought in and retailed by British makers, and renamed. What I am unsure about is whether any British maker built copies of Lefaucheux's original double-bite design, I have yet to come across definitive proof.