The Chapman‐Enskog approximation to the velocity‐energy distribution for a polyatomic gas is reinvestigated and found to yield unacceptable results for small amplitude sound waves. The solution to the linearized Boltzmann equation is then expanded in a set of orthogonal polynomials in the dynamic variables for translational and internal degrees of freedom and an iterative method of approximation is set up. This method of attack is a natural continuation of the work of Maxwell, Waldmann, of Ikenberry and Truesdell. Under some conditions the first iteration approaches the second Chapman‐Enskog approximation (or Navier‐Stokes approximation) ast→ ∞. The polynomial expansion was then truncated and the first iteration was used to calculate the heat flux, pressure tensor, and characteristic temperatures for a small amplitude sound wave. The second iteration was also investigated and was found to affect mainly only the heat fluxes. A useful formula is presented for the thermal conductivity of a polyatomic gas withNinternal degrees of freedom in the limit of zero frequency.