  The Functional Bethe Ansatz (FBA) proposed by Sklyanin is a method which
gives separation variables for systems for which an $R$-matrix is known.
Previously the FBA was only known for $SL(2)$ and $SL(3)$ (and associated)
$R$-matrices. In this paper I advance Sklyanin's program by giving the FBA for
certain systems with $SL(N)$ $R$-matrices. This is achieved by constructing
rational functions $\A(u)$ and $\B(u)$ of the matrix elements of $T(u)$, so
that, in the generic case, the zeros $x_i$ of $\B(u)$ are the separation
coordinates and the $P_i=\A(x_i)$ provide their conjugate momenta. The method
is illustrated with the magnetic chain and the Gaudin model, and its wider
applicability is discussed.
