Problems of Denudation, 



187 



same direction 

 we assume X* 



as a 

 - 1 



fine soil. If as a preliminary estimate 

 a = 0'l cm. (corresponding to a sand), 

 & = 6'02 cm. 2 /sec, A =3 x 10 "~ 4 cm. /sec, we find B = 

 0(200) cm. Thus at a distance of 10 km. from the high 

 ground the depression of the surface would be of order 

 200 metres. With a finer soil it would be less, and the 

 scales indicated appear to be of the same order of magnitude 

 us those observed. 



III. The Stability of the Peneplain. 



It was shown in the last section that a peneplain with 

 uniform soil could retain parallel contours indefinitely if 

 there were no external disturbance. Suppose, however, that 

 on account of some local irregularity in the rainfall or the 

 soil or some other factor the perfect peneplain form were 

 slightly altered, would subsequent denudation increase or 

 decrease the alteration ? If it decreased it. the peneplain 

 would be stable, and would be expected to persist for long- 

 intervals without considerable change. If it increased it, on 

 the other hand, the peneplain would be unstable ; the par- 

 ticular type of variation that increased most rapidly would 

 become the most important, and in time would dominate all 

 others. 



Let the equation of the peneplain be z=z x , where z 1 is a 

 function of x only, and suppo.se it to be slightly disturbed, so* 

 that its equation is changed to 



s=si+<K^y)» W 



where </> and all its derivatives are small quantities of the first 

 order. Then neglecting second-order terms, we see that the 

 dip-lines satisfy the equation 



dp >- dj> I'dzi 

 dx dv Jdx' 



(2) 



Hence to this order 



y 



-b 





dx 



is constant 



long 



dip-line, and can be put equal to //,. The integral is to be 

 taken along the dip-line or (with a second-order error) along 

 a section of the surface by n plane parallel to the axis of x. 

 Now ds 2 is the element of length along a contour, and we 

 have 



ds2 2 = dx 2 + dy 2 . 



* We should expect X to be less than the ordinary coefficient of sliding 

 friction, for the motion is partly rolling, and the water must have some 

 lubricating action. Experiments on traction in channels concern tur- 

 bulent friction, and give no information on the present problem 



