RELATION OF CAPACITY TO SLOPE. 



121 



slope is such as to be characterized by varia- 

 tions of the rates of variation of rates of varia- 

 tion; and there are even vistas of higher orders 

 of variability. The law connecting capacity 

 with slope may be susceptible of much more 

 compact expression, but such formulation must 

 probably await the development of a mechan- 

 ical theory of stream traction. 



Formulation founded on the index of rela- 

 tive variation, while bringing out clearly cer- 

 tain general features of the law, is not able to 

 afford a complete quantitative statement. It 

 may be likened to a map in definite hachures 

 as contrasted with one in definite contours. 

 As the hachure tells the direction and rate of 

 slope but omits the absolute altitude, so the 

 index tells the relative change under given con- 

 ditions but omits the absolute capacity. To 

 remedy this defect, the experiment was tried 

 of substituting for the equation C=v l S l> , in 

 which v l is variable, the equation C=c l S il , in 

 which Cj is constant for each series of observa- 

 tions. Formulation by means of c t and j l is 

 more nearly analogous to the contour map, 

 but the variability of the exponent j t is no less 

 formidable than that of \ while the definition 

 and derivation of y\ are less simple and its 

 significance is less clear. 



Further utilizing the analogy of the map, we 

 may think of the capacity-slope relation as an 

 undulating topography, in which the vertical 



f(C) 

 element is -^ or s,c,\ and the horizontal ele- 



ments are qualifying conditions. Formulation 

 is a mode of representing this topography, the 

 hills and valleys of which do not depend on the 

 mode but are real. Two modes have been 

 tried, each with limitations, but the ideal mode 

 is not known. The contour map or the relief 

 model would serve admirably if the qualifying 

 conditions were two only, but as they number 

 at least four, a graphic or plastic expression is 

 possible only in space of n dimensions. 



By reason of the complexity of the relation 

 of capacity to slope and because of the lack of 

 a mechanical theory of flow and traction, the 

 laboratory data do not warrant inferences as to 

 the quantitative relations of capacity to slope 

 for rivers. 



Of various attempts to evade the complexity, 

 two are thought worthy of record. In each 

 series of experiments the mode of traction 

 changes with increase of slope, first from dune 



to smooth, then from smooth to antidune, but 

 the critical slopes are not the same for different 

 discharges or widths or degrees of fineness. It 

 appeared possible to gain in simplicity by 

 treating separately the data associated with a 

 single mode of traction, and data for the 

 smooth mode were accordingly segregated and 

 discussed. Greater simplicity was not found, 

 but the range of variation is somewhat smaller 

 for the single mode of traction. 



The second attempt was connected with the 

 form of cross section of the current. Within a 

 single series of experiments the width was con- 

 stant and the depth varied, so that the capacity 

 was conditioned not only by slope but by form 



ratio, R = ~. By comparing one observational 



series with another it is possible to obtain data 

 conditioned by difference in slope, but without 

 difference in form ratio. The discussion of 

 such data developed only moderate modifica- 

 tion of the results previously obtained and no 

 reduction in complexity. 



DUTY AND EFFICIENCY. 



For the purposes of this paper duty has been 



defined as the ratio of capacity to discharge: 



n 

 U= -Q. Combining this with C=f(S) , the most 



general expression for the relation of capacity 



to slope, we have U= 



Under no form dis- 



covered for/() is this expression reducible to 

 simpler terms. For each value of discharge, 

 duty is simply proportional to capacity; and 

 the entire discussion of this chapter applies to 

 duty as well as capacity. The parameters, n, 

 \, ji, !,, and a may be transferred, without 

 modification, to formulas for duty. 



Efficiency has been defined as the ratio of 

 capacity to the product of discharge by slope: 



-(44) 



The combination of this with 

 C=v 1 S<i. 



yields 



--.(36) 



= v = v -iS'>- 1 .(45) 



The transformation of the exponent is im- 

 portant. While capacity varies as the i t power 



