TRANSPORTATION AND DISTRIBUTION OF SEDIMENTS. 19 



carries stones many hundred times as large as grains of gravel. Let us 

 investigate the law. 



Law of Variation. — If the surface of the obstacle is constant, the 

 force of running water varies as the velocity squared : / a v 2 (1). This 

 may be easily proved. Suppose we have an obstacle like the pier of a 

 bridge, standing in water running with any given velocity. Now, 

 if from any cause the velocity of the current be doubled, since mo- 

 mentum or force is equal to quantity of matter multiplied by velocity 

 ( M = Q X V), the force of the current will be quadrupled, for there 

 will be double the quantity of water striking the pier in a given time 

 with double the velocity. If the velocity of the current be trebled, 

 there will be three times the quantity of matter striking with three 

 times the velocity, and the force will be increased nine times. If the 

 velocity be quadrupled, the force is increased sixteen times, and so on. 



Next, if the velocity of the current remains constant, w T hile the size 

 of the opposing obstacle varies, then evidently the force of the current 

 will vary as the opposing surface : if the opposing surface is doubled, 

 the force is doubled ; if trebled, the force is trebled, etc. But in similar 

 figures, surfaces vary as the square of the diameter. Therefore, in this 

 case, force varies as diameter squared : /* ex d 2 (2). Therefore, when 

 both the velocity of the current and the size of the stone or other 

 obstacle vary, then the force varies as the square of the velocity of 

 the current multiplied by the square of the diameter of the stone : 

 Fcxv 2 xd 2 (3). 



This last equation gives the law of variation of the moving force. 

 But the resistance to be overcome, or the weight of the stone, varies 

 as the cube of the diameters : W <x d s . We have, therefore, both the 



{Foe y 2 x d* 

 W 7* 



Now the case we wish to consider is that in which the current is just 



able to move the stone, or when F oc W. In this case d* oc v 2 X d*, 



or d oc v*. Substituting, in the third equation, for d its value, 



F oc v 3 X v* = v*. We place these equations together, so that they 



may be better understood : 



When surface = constant 

 When velocity = constant 

 When both vary . , 



But 



And when W oc F, then . 

 Dividing by d? 

 Substituting in 3 

 Or 



/OCfl'(l) 



P oc d* (2) 



F a v 2 x d 2 (3) 



TFa d* 



d* oc v* x d 2 



d a v* 



F a v 2 x v* 



F oc v 6 



That is, the transporting power of a current or the weight of the largest 

 fragment it can carry, varies as the sixth power of the velocity. This 



