  Following Sklyanin's proposal in the periodic case, we derive the generating
function of the Gaudin Hamiltonians with boundary terms. Our derivation is
based on the quasi-classical expansion of the linear combination of the
transfer matrix of the XXX Heisenberg spin chain and the central element, the
so-called Sklyanin determinant. The corresponding Gaudin Hamiltonians with
boundary terms are obtained as the residues of the generating function. By
defining the appropriate Bethe vectors which yield strikingly simple off shell
action of the generating function, we fully implement the algebraic Bethe
ansatz, obtaining the spectrum of the generating function and the corresponding
Bethe equations.
