The problem of paired spheres and of straight chains of spheres in slices or polished sections

SUMMARY The starting point is the problem of finding the percentage q2 of paired nuclei in liver cells, when the percentage p2 of twin cuts in slices is known. The general solution for unequal spheres becomes most simple for equal spheres and leads to an extrapolation for endless straight chains of equal spheres, at first for closed and then for open series with constant intervals. Three total sums of endless series of fractions are a welcome mathematical extra. The microscopist will prefer instructions which are generalized from counts in final rows of spheres. With them he can analyse the composition of a mixture of limited chains, with different numbers of spheres.