If El, E2, and G mR n then A(EI,E2;G) denotes the set of all closed paths joining E 1 and E_ in G (cf. [9, p. 21]). Let El, Bi(r) = : r E 2 cR n with 0 6 El' E2' let {x 6 R n Ixl 0, and let F r = A(EI,E2;R n ~ Bn(r)) for r > 0. In this note we shall study the problem of finding a lower bound for the n-modulus M(F r) of F r in terms of r and the densities of E 1 and E 2 at 0. One can derive estimates of this kind by means of the cap-inequality [9, 10.9] under the assumption that both E 1 and E 2 intersect ~Bn(r) for all r 6 (0,i]. Hence it is of interest to study the situation under weaker density conditions on E 1 and E 2. For this purpose we employ the lower capacity density cap dens (E,x) of a set E at x 6 R n (cf. Section 2 and Martio-Sarvas [6]). The main result of this note, which is given in Theorem 3.1, states (Ej = > 0 j : 1,2, then that if cap dens ,0) 63 ,