By two methods, a linearization and a variational principle, the Born-Green-Kirkwood equation for the radial distribution function is solved approximately and the osmotic pressure of chain polymer solutions computed at arbitrary concentration. The gaussian intermolecular potential energy of Flory and Krigbaum is used, and this restricts the range of validity of the theory to volume fractions less than one-tenth. It is shown how the distribution of polymer molecules in the solvent becomes random as the concentration is increased. For good solvents, the quantity [(P/c2)—RT/Mc], where P is the osmotic pressure and M the molecular weight, is predicted to increase rapidly with concentration c, and then to level off rapidly, the whole effect being accomplished at quite low concentrations as the molecules are forced to overlap. Some experimental corroboration is displayed. Severe doubt is cast on the practicality of the virial expansion of P, and possibly on the validity, beyond quite low concentrations.