Frequently, it was thought that frictional slip would occur in the direction of least resistance, which was unfortunately taken as the direction of the shortest normal to the free boundary. In this paper, the condition of least resistance is accepted, but the direction of resistance is properly determined without assumption. The result is "the rule of gredient", that is, at a given point on the contact surface, the direction of least resistance is the direction of the gredient of unit friction τ, which is related to the unit pressure P and the coefficient of friction, f, by τ=fP, the gredient lines of τ and P coincide with each other. Consequently, the family of least resistance lines of friction is exactly the family of curves orthogonal to the pressure contours, and can be determined from the experimental surface distribution of pressure. One case of such friction-lines in rolling is presented, the curves bear remarkable resemblance to the under-evalu-ted "probable" lines of friction derived by siebel from deformation meassurements. The way to consider change in direction of τ in one-dimensional theory of rolling is to take an average friction line whose direction cosine, cos φ, vanishes at the neutral section according to the gredient rule. By doing so, f cos φ corresponds to the "coefficient of friction" which vanishes at the neutral section according to Brown's theory. The Karman's equation is written in the mean value form by taking τx=fP cos φ instead of τx=fP. The modified equation yields solutions smoothly continuous at the neutral section, and two such continuous solution to Karman's equation for the case of solid friction are presented, detailed investigation is left to another paper. By simple arguement, it is thought that the boundary of no-slip region should be a crossed curve given by a pressure contour which is a roop according to experimental results.The rule of gradient has already led to three concequences, and is expected to be a very useful relation for plasticity under compression, because up to this paper the differential equation for slip direction remains unknown.
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