This paper consists of two parts. The first is theoretical and extends Elsasser's theory of stress propagation in the upper mantle to an asthenosphere with nonlinear rheology. Exact solutions of the nonlinear equations are found for two geologically important problems. The second part uses these theoretical results as the basis for a measurement of the rheology of the asthenosphere. The seaward migration pattern of aftershocks from the February 4, 1965. Rat Island earthquake is analyzed, and strong evidence for a non-Newtonian stress-strain relation in the asthenosphere is presented. It is found that an individual large earthquake can influence the regional stress pattern only to a distance of about 300 km perpendicular to the line of rupture. Excellent agreement is found between the stress propagation coefficient calculated from the aftershock migration pattern and that calculated from laboratory measurements of high-temperature creep in olivine. We thus arrive at a picture of stress propagation in the uppermantle which is consistent both with theoretical expectation and with observational evidence.