36 University of California Publications in Geology [Vol.9 



which is 0. For any point P on the curve, the areas ORE^P and 

 OSS\P are equal. From this it follows, by a mathematical treatment 

 which Professor T. M. Putnam has kindly carried out for me, that 

 the equation of the curve OPT is 



, nj? sin B 2 cos A 



;c(OR — y) — Ax- -=OSy-\-hv tV 



J 2 sin A cos B 



The curve OPT, for the assumed condition of a flat-topped mesa, is 



a hyperbola to which the limiting' surface of the fan is tangent at 



the point 7', where the hyperbola intersects the mesa surface. 



It is apparent from this, that if the mesa be very broad rela- 

 tively to its height, the suballuvial benches cut on both sides of 

 an elongated mountain block might intersect the siirface of the mesa 

 short of the median line, so that on the completion of the process of 

 degradation, the central part of the resulting stable profile would 

 be that of the original mesa. If the mountain block be relatively 

 narrow, and this appears to be the common case, then the suballuvial 

 bench will not intersect the mesa surface, but will meet the corres- 

 ponding bench approaching from the other side at some point below 

 the level of the summit. 



We may now vary our assumption as to the initial form of the 

 mountain. If instead of being a mesa its cross-sectional area from the 

 center to the edge be less than that shown in the diagram, then 

 the fan surface will rise more slowly and the bench will be flatter 

 without changing the character of its curvature. If, as is usually 

 the case, the mountain mass be higher at its center than on its margin, 

 so that the cross-sectional area be greater than that of the diagram, 

 then the area Offi^P of one stage of the process may become suffi- 

 ciently greater than the similar area of the next preceding stage to 

 cause each successive layer added to the surface of the fan to be 

 thicker than the preceding layer, and the curvature of the bench will 

 be concave upward. It thus appears that for the same width of 

 valley some suballuvial benches may be convex and others concave 

 upward ; and that the condition which determines the convex curve 

 may change to the one which determines the concave curve, so that 

 the resulting bench may be concavo-convex. In order that the bench 

 should be a straight slope, the fan would have to rise by equal 

 increments in equal times throughout its entire growth, and this is 

 a rare and accidental case. 



It is further apparent that the character of the suballuvial bench 

 is a function of the width of the valley, OS, in which the detritus 



