'120 



TRANSPORTATION OF DEBEIS BY RUNNING WATER. 



data for the same range of slope as that covered 

 by the value of I s . The table thus comprises 

 data for the coordinate values of the indexes 

 under conditions of constant form ratio and 

 constant width. 



The most important generalization from the 

 data in Table 23 is contained in the adjusted 

 means which appear at the bottom. In the 

 derivation of these means due account was 

 taken of the fact that the several individual 

 values from the same overlap are not inde- 

 pendent and also of the fact that the coordinate 

 data for constant width are incomplete. The 

 mean for constant form ratio, 1.88, is practi- 

 cally the same as that one for constant width 

 which is associated with the wider channels, 

 1 .89, and is notably greater than the mean as- 

 sociated with the narrower channels, 1 .72. On 

 the whole it is indicated that the sensitiveness 

 'of capacity to slope is the greater for traction 

 conditioned by constant form ratio, in the pro- 

 portion of 1.88 to 1.805, or as 1.04 to 1. 



When the values of I s and /, are arranged 

 according to the associated values of E, the fol- 

 lowing relations are brought out : 



SUMMARY. 



Only a small fraction of the observational 

 data are available for the discussion of the 

 capacity-slope relation under the condition of 

 constant form ratio, and the discussion is there- 

 fore limited to a comparison of its features un- 

 der that condition with corresponding features 

 under the condition of constant channel width. 

 The results of such comparison may be sum- 

 marized as follows: The sensitiveness of trac- 

 tional capacity to variation of slope is in gen- 

 eral greater under the condition of constant 

 form ratio, but the difference is of moderate 

 amount. The difference is somewhat less (at 

 least within the limits of available data) for 

 broad and shallow streams than for streams 

 that are narrow and deep. The range of sensi- 

 tiveness, or its variation with variation of slope, 

 appears to be somewhat less under the condi- 

 tion of constant form ratio. The generaliza- 



tions in regard to traction by currents of varia- 

 ble depth but invariable width may be ex- 

 tended, with only moderate qualification, to the 

 case of currents which retain geometric simi- 

 larity of section while slope is varied. 



REVIEW. 



With increase of the slope of descent goes in- 

 crease of a stream's energy (per unit time, per 

 unit distance). With the increase of energy 

 goes increase of capacity for the transportation 

 of ddbris along the channel bed. The increase 

 of energy is strictly proportional to the increase 

 of slope, but the increase of capacity follows a 

 different law. The law is not simple, but one 

 feature persists through all its manifestations: 

 The capacity for traction increases more rap- 

 idly than the slope. The difference in rapidity, 

 or the magnitude of the difference between the 

 rates of change for capacity and slope, is itself 

 a variable, depending on a variety of condi- 

 tions. The study of the relation of capacity to 

 slope is here treated as a study of the influence 

 of conditions on the magnitude of the differ- 

 ence between the two rates of change. 



The magnitude of that difference is indicated 

 by a quantity, of the nature of an exponent, 

 called the index of relative variation (of capac- 

 ity, as compared to slope), and designated by 

 i t . The index may be defined as the first 

 differential coefficient of log with respect to 

 log S. It is illustrated by saying that the 

 capacity varies, instantaneously, as the i 1 

 power of the slope. 



For the greater part of the field covered by 

 the experiments the index falls between 1.4 

 and 3.0, but under some conditions it is con- 

 siderably higher. It varies with slope, being 

 higher for low slopes and lower for high. It 

 varies with discharge, being relatively high for 

 small discharges. It varies with fineness, being 

 relatively high for coarse debris. Briefly, it 

 varies inversely with slope, discharge, and 

 fineness. It varies also with width of channel, 

 decreasingly for relatively narrow channels and 

 increasingly for relatively broad channels; so 

 that, for any particular combination of slope, 

 discharge, and fineness there is a width charac- 

 terized by a minimum value of the index. 



It is furthermore true that no one of these 

 variations is itself constant in rate, the rate of 

 each having its own law of variation. Thus 

 the complexity of the relation of capacity to 



