Geometric probabilities for a cluster of needles and a lattice of rectangles

A cluster of $n$ needles ($1\leq n<\infty$) is dropped at random onto a plane lattice of rectangles. Each needle is fixed at one end in the cluster centre and can rotate independently about this centre. The distribution of the relative number of needles intersecting the lattice is shown to converge uniformly to the limit distribution as $n\rightarrow\infty$.