Let $n$ and $k$ be fixed integers; $n$ even $\ge 4$; $1 \le k \le \frac{n-2}{2}$. Let $f(t)$, $0 \le t < \infty$, be differentiable and such that $\int_{0}^{\infty} |f(t)| \, t^{(n-2k)/2} \, dt < \infty$. We prove that, under these conditions, the integral equation (2) admits the solution (4). In the particular case $n = 4$, $k = 1$ (which is important in the quantum theory of fields) the reciprocal formulae (2) and (4) have already been obtained (on the basis of heuristical considerations) by Källen [1].