Let G be a molecular graph, n the number of its vertices and phi(G,x) its characteristic polynomial. Already in 1940 Coulson expressed the total pi-electron energy of conjugated unsaturated molecules in terms of the function F(x) = n - ixphi'(G,ix)/phi(G, ix). Recently. the Coulson function F(x) found applications also in modeling the structure-dependence of physico-chemical properties of alkanes. We now analyze the Coulson function and establish some of its hitherto unnoticed features, in particular its relations with the Hosoya index. (C) 2002 Published by Elsevier Science B.V.