Both Hinch and Leal and Stewart and Sørensen solved the diffusion equation for the configurational distribution function of rigid dumbbells at high shear rates by several different numerical methods and obtained asymptotic expressions for the viscometric functions. In this note we have considered only a contracted distribution function which, however, can be calculated analytically. This contracted distribution function not only contains interesting information about the behavior of rigid dumbbells at high shear rates but can also be used to find an exact asymptotic expression for the first normal‐stress coefficient. By introducing modified time constants, the results of the present paper, which have been derived for rigid dumb‐bells with Oseen‐Burgers hydrodynamic interaction, can easily be generalized to rigid multibead rods and Rotne‐Prager‐Yamakawa hydrodynamic interaction.