184 Dr. H. Jeffreys on 



The equation of continuity therefore reduces to 



c^ OfCh 2 



-j^* • • • (12) 



gsm 



the integration being along the lines of greatest slope. At 

 the top of the slope V£ must evidently be zero, so that the 

 lower limit of the integral must correspond to the summit. 

 The problem is thus reduced to one of quadratures. 



II. Denudation by Surface Water. 



The friction of surface water on the ground tends to remove 

 the finer particles and carry them away. Solution also occui s 

 in certain cases, but the purely mechanical effect is always 

 one of the most important, and will be considered first. If M 

 denote the mass of a particle of density p', it will be detached 

 when the f rictional force on it reaches a certain proper fraction 

 of the normal force between it and the surface. The velocity 

 in its neighbourhood being u, and the linear dimensions of 

 order a, the f rictional force is 0(Jcpua) in the case of pure 

 viscosity. Now u is practically a~du/~dv evaluated for v=0, 

 and this is ag^sinajk. Thus the frictional force is 0(pa 2 g%sm a). 

 The normal force is 0{ (p'—p)a 3 g cos a}. The ratio of the two 

 being a definite number, say X, we see that the size of a 

 particle that would just be moved is given by 



Hence the mass of the largest particle that can be transported 

 by viscosity without sticking in the first small hollow it 

 comes to is proportional to £ 3 tan 3 u. The rate of denudation 

 is therefore a function of ftana, its form depending on the 

 distribution in the soil of particles of different sizes. It is to 

 be noted that when the depth and slope are the same the 

 limiting mass is independent of the kinematic viscosity. 



An interesting case arises if f tan « and a are constants, 

 ol being small. The surface sinks at a uniform rate all over, 

 retaining its size and shape, but progressively sinking. This 

 represents one case of the " peneplain." When the surface 

 is such that all the contour-lines are parallel to the axis of y, 

 its form can be found. For I. (12) when £ is proportional to 

 cot a and A to cos a, corresponding to rain falling, on the 

 average, vertically, becomes 



i cot u(h= — B sin a cot 3 «, (2) 



