162 



TRANSPORTATION OF DEBBIS BY RUNNING WATER. 



ing the mean velocity and increasing the depth. 

 If mean velocity and depth be restored by suit- 

 able changes in slope and discharge, a new and 

 different vertical velocity curve will be obtained 

 for each depth. While I am not able to de- 

 duce the exact nature of the changes, it seems 

 clear that those portions of the new curves 

 within the zone of saltation will be contrasted 

 in some such way as are the potential curves of 

 the drawing, and that for purposes of traction 

 the advantage will still be with the shallow 

 current. Provided the portions of the ve- 

 locity curves within the zone of saltation show 

 steeper gradient for the current of less depth, 

 tractional capacity will for that current bear 

 a higher ratio to mean velocity. 



RELATIVE SENSITIVENESS TO CONTROLS. 



The synthetic index of relative variation for 

 capacity in relation to mean velocity, when tho 

 limiting condition is constant discharge, namely, 

 Ira, has been estimated (p. 158) as 2.60 times 

 the corresponding index, I w for capacity in 

 relation to slope. The same index has been 

 estimated (Table 53) as 1.07 times I V d and 1.19 

 times Ira. Combination of ratios indicates 

 that Ivd is 2.43 times and Ira 2.18 times as 

 great as /. While these figures have an 

 appearance of exactitude, their order of pre- 

 cision is really low. They are built on aver- 

 ages of individual values of indexes, which 

 among themselves are highly diversified. At 

 best they represent the average of values 

 covered by the range of experiments in the 

 laboratory, but hi part they are based on 

 values covering much narrower ranges. More- 

 over, it was not possible, except in the case of 

 Ivq and /, to derive the compared series of 

 values of the index from data representing 

 exactly the same conditions. For these rea- 

 sons the numerical results should be accepted 

 only as indicating an order of sequence and an 

 order of magnitude. The quantitative re- 

 sponse of capacity to the change of mean 

 velocity is much larger than its response to 

 change of slope, probably more than twice as 

 large. Minor differences depend on the con- 

 ditions under which mean velocity varies. 

 The response is greatest when velocities are 

 subject to the condition of constant discharge, 

 less when the restrictive condition is constant 

 depth, and least when it is constant slope. 



COMPETENT VELOCITY. 



The demonstrations by Leslie, Hopkins, 

 Airy, and Law of the proposition that the 

 diameter of the largest particle a current can 

 move is proportional to the square of the 

 velocity involve the principle that the pressure 

 of a current is proportional to the square of its 

 velocity, and also the assumption that the 

 forward pressures on different parts of the 

 particle are the same, so that the total pressure 

 is proportional to the sectional area of the 

 particle. 1 Under that assumption the total 

 pressure may be conceived as applied to the 

 center of gravity of the particle, a considera- 

 tion of importance when the motion given to 

 the particle is of the nature of rolling or over- 

 turning. These assumptions are not strictly 

 true, because in the immediate vicinity of the 

 channel bed the velocity increases with dis- 

 tance from the bed. Moreover, as wo have 

 seen (p. 29), the rate of increase is a diminish- 

 ing rate. As a consequence of the inequality 

 of velocity and its mode of distribution (1) 

 the average pressure on the upstream face of a 

 large particle is greater than the corresponding 

 average pressure on a small particle, (2) the 

 point of application of the total pressure (the 

 point which determines the lever arm in over- 

 turning and rolling) is always above the center 

 of gravity, and (3) the point of application may 

 be differently related to the center of gravity 

 in particles of different sizes. The general 

 effect of these qualifying circumstances is to 

 reduce the difficulty of moving large particles, 

 and thus to make the rate at which competent 

 diametar of particle increases with velocity (at 

 any particular level) somewhat greater than 

 that of the square of the velocity. 



On the other hand, it is to be observed that 

 in stream traction the roughness of the channel 

 bed is denned by the coarseness of the load, 

 and the system of velocities near the bed is a 

 function of several things, one of which is the 

 roughness. It is by no means impossible that 

 the vertical velocity curve of a stream flowing 

 over a bed of coarse debris is an enlarged 

 replica of the curve of a shallower stream 

 flowing over a bed of finer debris, in which case 

 the law of Leslie might hold despite the quali- 

 fications mentioned above. The problem is too 



1 For references see footnote on page 16. 



