Problems of Denudation. 181 



Thus on these hypotheses friction would considerably exceed 

 gravity, which contradicts the initial assumption. Hence 

 the hypothesis that friction is less than a fraction of gravity 

 which never approaches near to unity is untenable, and 

 therefore friction must be nearly equal to gravity, and the 

 accelerations can be neglected in the equations of motion. 



This holds for any depth of the liquid ; when the depth is 

 small enough to make viscous resistance (simply proportional 

 to the velocity) exceed the type here considered, the resistance 

 here assumed is still great in comparison with the accele- 

 rations, and a fortiori the viscous resistance is more im- 

 portant than the accelerations. 



The forces acting on the element of fluid therefore reduce 

 to two : gravity acting down the line of: greatest slope, and 

 friction acting opposite to the velocity, these two being prac- 

 tically equal and opposite. It follows at once first, that the 

 motion of the liquid is always down the line of greatest slope, 

 and second, that the velocity is given by 

 /V?=0£sina. 



For a different law of resistance the velocity will have a 

 different value. 



The equation of continuity has not yet been considered. 

 Consider the surface of the ground covered by two orthogonal 

 systems of curves, specified by \=constant and /z, = constant, 

 where \ and /j, are functions of the position. Let the elements 

 of length along these curves be ds± and ds 2 , where 



h x dsi = d\ and li 2 ds 2 = dfju i .... (1) 



hi and h 2 being in general functions of X and /j,. 



If the velocity at any point has components (w, v) in the 

 directions of dX and dfju respectively, the amount of liquid 

 crossing ds x in unit time Is v^ipand accordingly it is seen 

 that the equation of continuity is 



3A.UJ e/AV MA b*> * ■ ( ; 



where A is the rate of supply of water per unit area. When 

 the motion is steady 'dty'dt is zero. Now u and v are known in 

 terms of f, so that this becomes a partial differential equation 

 to find f. Again, f only enters through the combination fV. 

 If the law of resistance were different from that assumed 

 here, the equation would still hold, but V would be a different 

 function of f ; nevertheless the equation would be satisfied 

 by the same value of Vf, so that the solution for one law of 

 resistance can easily be deduced from that for any other law. 



