Problems of Denudation. 18o 



where B is a constant ; then 



~r =B(2 cot a coseca-f cos a), . ..... (3) 



act 



-j- = B(2 cot 2 a cosec a + coseca— sin a), . . (4) 



da. 



z = z Q — B k 2 cosec a— sin a), (5) 



#=# — B(cosec a cot a— cos a), (6) 



where # and z are arbitrary constants. 



When z = z Q — B, we have a = -j7r and # = # . For smaller 

 values of a, # and z become steadily smaller ; and when 

 «is small z behaves like z — 2B/a and x like «£ — B/a 2 . Thus 

 the surface is very steep near the top, the slope gradually 

 decreasing as we recede from this. Again, if the surface is 

 flatter at a distance from the ridge than the peneplain form 

 and steeper near the ridge, we see that f tan a is greater near 

 the ridge and less away from it ; thus the inside will tend to 

 sink more slowly than the outside, the peneplain conditions 

 being thereby restored. For displacements of this type the 

 peneplain is therefore stable. 



Except close to the ridge, where the surface is nearly 

 vertical, the form is practically parabolic, the axis being 

 horizontal and the latus rectum being 4B. The depth and 

 the mean velocity are both zero at the top, the water running 

 away as fast as supplied ; at other places both increase like 

 cot a, or practically like the square root of the horizontal 

 distance traversed. The amount of water crossing a contour 

 in unit time is proportional to the product of the depth and 

 the mean velocity, and therefore to the horizontal distance. 

 Thus it steadily increases the further we go down the slope, 

 the increase being supplied by the rain gathered on the way. 



Evidently the theory cannot be expected to hold close to 

 the ideal ridge; the removal of solid matter would cause the 

 surface atter a short time to be at a uniform normal distance 

 from the original surface, and not vertically below it. As 

 long as the slope is small this will not make much difference, 

 but when it is great there will be a considerable cutting back 

 in a horizontal direction ; any sharp angle will therefore be 

 exposed to denudation on two sides, and will tend to be 

 rounded off. This effect will be accentuated by the fact that 

 rain falling on a steep slope would retain for some time part 

 of the velocity acquired in its fall through the air, so that the 

 velocity would be greater than on the theory. As long as 

 we are concerned with only moderate gradients, however, 

 the theory will hold. In dealing with the summits of hills, 



