  We sum in a close form the Sneddon-Bessel series \[ \sum_{m=1}^\infty
\frac{J_\alpha(x j_{m,\nu})J_\beta(y j_{m,\nu})}
{j_{m,\nu}^{2n+\alpha+\beta-2\nu+2} J_{\nu+1}(j_{m,\nu})^2}, \] where $0<x$,
$0<y$, $x+y<2$, $n$ is an integer, $\alpha,\beta,\nu\in \mathbb{C}\setminus
\{-1,-2,\dots \}$ with $2\operatorname{Re} \nu < 2n+1 + \operatorname{Re}
\alpha + \operatorname{Re} \beta$ and $\{j_{m,\nu}\}_{m\geq 0}$ are the zeros
of the Bessel function $J_\nu$ of order $\nu$. As an application we prove some
extensions of the Kneser-Sommerfeld expansion.
