144 



TRANSPORTATION OF DEBRIS BY RUNNING WATER. 



TABLE 35. Values of the exponent i 3 arranged to show vari- 

 ation in relation to fineness of debris. 



In summary, the sensitiveness of capacity to 

 variation of discharge is greater as slope, dis- 

 charge, form ratio, and fineness are less. 



;,=/(,<>, 



(68) 



DUTY AND EFFICIENCY. 



The variation of capacity with discharge is 

 indicated in general terms by an equation of 

 the type of (33) : 



ff-v.e*-.- (69) 



Duty being the quotient of capacity by dis- 

 charge, this gives 



^=-. % L = v 3 Q i *- 1 -- - (70) 



and, as efficiency is the quotient of capacity by 

 discharge and slope, 



3i 



- _? O'*- 1 



S V 



(71) 



That is, the index of relative variation for both 

 duty and efficiency, in relation to discharge, is 

 less by unity than the corresponding index for 

 capacity. Therefore the values of the index 

 in Table 32 need only to be reduced by unity 

 to apply to duty and efficiency. 



Under ordinary conditions the index for duty 

 and efficiency falls between unity and zero ; or, 

 in other words, duty and efficiency increase 

 with increase of discharge, but their increase is 

 less rapid than that of discharge. Exception- 

 ally the increase is much more rapid, the excep- 

 tions being associated with discharges little 

 above the limit of competence. On the other 

 hand, there appear to be conditions under which 



the index falls below zero, so that duty and 

 efficiency diminish with increase of discharge. 

 The diminution indicated by the figures in the 

 column (of Table 32) for grade (D) and width 

 0.66 foot is only of the order of magnitude of 

 the probable error; but a pronounced diminu- 

 tion would be inferred from the values of the 

 index for grade (B) and width 0.23 foot. As 

 the results from the last-mentioned group of 

 observations stand by themselves in various 

 respects, some reservation is felt in regard to 

 them, and there is at least room for doubt 

 whether the diminution is actually demon- 

 strated. 



With respect to all conditions the variations 

 of the index for duty and efficiency follow the 

 same laws as the index for capacity; but, as a 

 consequence of the uniform reduction by unity, 

 the rates of variation are higher. If, for ex- 

 ample, in passing from a smaller to a larger dis- 

 charge, the index for capacity falls from 1.40 to 

 1.20, a reduction of one-seventh, the index for 

 efficiency falls from 0.40 to 0.20, a reduction 

 of one-half. 



Lines of reasoning strictly parallel to those 

 employed in the last section of Chapter III 

 yield the following conclusions: 



The synthetic index of relative variation for 

 the duty of water in relation to discharge is 

 less by unity than the corresponding synthetic 

 index for capacity in relation to discharge. 



The synthetic index of relative variation for 

 efficiency in relation to discharge is less by 

 unity than the corresponding synthetic index 

 for capacity in relation to discharge. 



If the duty of water, or if efficiency, be as- 

 sumed to vary as some power of Q K, the ex- 

 ponent of that power (expressing the instanta- 



K 



neous rate of variation) equals o ~ 



charge increases from K toward infinity, the 

 exponent diminishes from o toward o 1 . 



If the relation of efficiency to discharge ( and 

 similarly for the relation of duty to discharge) 

 be expressed by 



E^BAQ-Kj'n ---------- (71a) 



As dis- 



the value of o n is always less than the corre- 

 sponding value of o, the difference approaching 

 but not exceeding unity. The value of /q is 



