122 



TRANSPORTATION OF DEBRIS BY RUNNING WATER. 



of slope, efficiency varies as the i, 1 power. 

 The index of relative variation is 1 less for 

 efficiency than for capacity. Table 15 could 

 therefore be adapted to efficiency by diminish- 

 ing all its values by unity. As the lowest 

 known values of \ are greater than unity the 

 lowest in Table 15 is 1.31 it follows that effi- 

 ciency is an increasing function of slope under 

 all tested conditions. 



The combination of (44) with 



vields 



-(40) 

 -(46) 



showing that in passing from the field of ca- 

 pacity to that of efficiency the exponent asso- 

 ciated with a constant coefficient also is reduced 

 by unity. 



Following the form of equation (35), we have 



log c"- log a , 35 , 



7 ' = lo g S"-logS'--- 



where C' and C" are specific values of capacity 

 corresponding severally to the slopes S' and S". 

 Designating by I e the synthetic index of effi- 

 ciency in relation to slope, and by E' and 

 E" the efficiencies corresponding to C' and 

 C" , we have 



"" 



log ff"-l 



E' 



log S' '-log S' 



.(35b) 



As C= ESQ, log C' = log E' + log S' + log Q, and 

 log C"' = log E" +log S" +log Q. Substituting 

 these values in (35a) and reducing, we have 



Subtracting the members of this expression 

 from those of (35b) and transposing, we have 



/.-/,-!- (47) 



That is, the synthetic index of relative varia- 

 tion of efficiency with reference to slope is less 

 by unity than the corresponding index for 

 capacity. 



It is evident that at competent slope, when 

 capacity is zero, efficiency also is zero. Like 

 capacity, it increases with increase of slope. 

 Under the assumption that its law of increase 

 is of the same type, its value varying with a 



power of (S a), two expressions have been 

 derived, but neither has been found reducible 

 to simple form. The algebraic work being 

 omitted, they are 



n , ?W 



S-ir 



logS-2 



log (S-)-2 



In the first of these expressions the coeffi- 

 cient is variable, being a function of S; so that 

 the exponent may be regarded as the index of 



relative variation for efficiency in relation to 



r* 



(S a). As s falls to zero when S falls to 



the limiting value a, and as it approximates 

 unity when S is indefinitely large, the values 

 of the exponent He between n and n 1 for all 

 practical cases. In the second expression the 

 coefficient is constant, with respect to slope, 

 but the exponent is transcendental and intract- 

 able. 



Thus it appears that the derived expression 

 for efficiency as a function of S a is not 

 simply related to the coordinate expression 

 for capacity and is not available for practical 

 purposes; but it does not necessarily follow 

 that the actual relation of efficiency to slope 

 can not be formulated for practical purposes by 

 an equation of the sigma type. All that is 

 really shown is that if capacity and efficiency 

 are both formulated in that way, the results 

 are not consistent. Formula ( 10) was adopted 

 for the capacity-slope relation, not because it 

 expresses a demonstrated law of relation, but 

 because it so far simulates the real law of rela- 

 tion as to be available for the marshaling of the 

 observational data. It seems quite possible 

 that had the data been first translated from 

 terms of capacity into terms of efficiency, the 

 type of formula would have been found equally 

 available. 



By way of testing the matter a few com- 

 parative computations were made, observa- 

 tional series being selected for the purpose 

 from those which in the adjustment gave small 

 probable errors. From the original data in 

 Table 4 values of efficiency were computed, and 

 these were plotted on logarithmic paper in rela- 

 tion to S a, the values of a being those em- 

 ployed in the adjusting equations. In four of 

 the nine cases treated the locus indicated was a 



