ABSTRACT. For a glacier flowing over a bed of longitudinally varying slope, the influence of longitudinal stress graruents on the flow is analyzed by means of a longitudinal flow-coupling equation derived from the •vertically" (cross-sectionaUy) integrated longitudinal stress equilibrium equation, by an extension of an approach originally developed by Budd (196g), Linearization of the flow-coupling equation, by treating the flow velocity ii ("vertically" averaged), ice thickness h, and surface slope a in terms of small deviations t:lii, llh, and Aa from overall average (datum) values ii 0 , h 0 , and ~· results in a differential equation that can be solved by Green's function methods, giving t:lii(x) as a function of M(x) and Aa(x), x being the longitudinal coordinate. The result has the form of a longitudinal averaging integral of the influence of local h(x) and a(x) on the flow u(x):where the integration is over the length L of the glacier. The A operator specified deviations from the datum state, and the term on which it operates, which is a function of the integration variable x' , represents the influence of local h(x' ), a(x' ), and channel-shape factor f(x' ), at longitudinal coordinate x' , on the flow ii at coorrunate x, the influence being weighted by the "influence transfer function• exp(-lx' -xl/1) in the integral. The quantity I that appears as the scale length in the exponential weighting function is called the longitudinal coupling length. lt is deter~ned by rheological parameters via the relationship I = 2h n{Ti/3'fl, where n is the flowlaw exponent, ii the effective longitudinal viscosity, and n the effective shear viscosity of the ice profile. ii is an average of the local effective viscosity n over the ice crosssection, and (l'lr 1 is an average of n-1 that gives strongly increased weight to values near the base. Theoretically, the coupling length I is generally in the range one to three times the ice thickness for vaUey glaciers and four to ten times for ice sheets; for a glacier in surge, it is even longer, J -12h. It is distinctly longer for non-linear (n • 3) than for linear rheology, so that the flow-coupling effects of longitudinal stress gradients are markedly greater for non-linear flow .The averaging integral indicates that the longitudinal variations in flow that occur under the influence of sinusoidal longitudinal variations in h or a, with wavelength >., are attenuated by the factor 1/(1 + (2nl/ >.) 2 ) relative to what they would be without longitudinal coupling. The short, intermediate, and long scales of glacier motion (Raymond, 19gO), over which the longitudinal flow variations are strongly, partially, and little attenuated, are for >. s 21 , 21 s >. s 201 , and >. ;;:; 201.