  We present a new explicit formula for the $m$-th Bernoulli number $B_m$,
which involves two integer parameters $a$ and $n$ with $0\le a\le m\le n$. If
we set $a=0$ and $n=m$, then the formula reduces to the celebrated Kronecker
formula for $B_m$. We give two proofs of our formula. One is analytic and uses
a certain function in two variables. The other is algebraic and is motivated by
a topological consideration of self-intersections of curves on an oriented
surface.
