130 



TRANSPORTATION OF DEBRIS BY RUNNING WATER. 



Substituting from (57) into (54), we have 



Equations (54) and (58) are alternative ex- 

 pressions of the same relation, the one involving 

 a without p, the other p without a. Each has 

 its field of superior convenience, alike for com- 

 putations and discussion. 



To obtain the maximum capacity, p is sub- 

 stituted for R in (58) . The equation reduces to 



DISCUSSION OF EXPERIMENTAL DATA. 



SCOPE AND METHOD OF DISCUSSION. 



The equations adopted for the formulation of 

 C=f(R), namely, equations (54) and (58), in- 

 volve four constants. & 2 is a quantity of the 

 unit of capacity; p is a ratio, the form ratio cor- 

 responding to maximum capacity; a is a ratio 

 connected with side-wall resistance; and m is 

 an exponent. There is a mutual dependence 

 between a and p; but 1 2 , a, and m, grouped 

 together in (54), are independent; and so are 

 5 2 , p, and m, grouped in ( 58) . In the following 

 discussion of the relation of capacity to form 

 ratio, equations of the form of ( 54) and ( 58) are 

 derived from groups of experimental data; and 

 these are compared in such way as to show the 

 control of their constants by the conditions of 

 slope, discharge, and fineness. 



A " group " of experimental data, for this pur- 

 pose, includes values of capacity and form ratio 

 from at least three observational series, all per- 

 taining to the same fineness, slope, and dis- 

 charge, but to different widths. Three pairs of 

 observational values suffice to determine the 

 three independent parameters; with a greater 

 number the problem is usually one of adjust- 

 ment, to determine the most probable values of 

 the parameters. 



The computation of the constants bv alge- 

 braic methods is tedious, and a graphic method 

 was substituted. When an equation of the 

 form of (58) is plotted on logarithmic section 

 paper the shape and size of the curve are deter- 

 mined wholly by the exponent m: its position 

 measured in the direction of the axis of log R 

 is determined by the value of p; and its position 



with respect to the axis of log C by the value 

 of & 2 . A graphic process based on these prop- 

 erties gave solutions of sufficient approxima- 

 tion for the purposes of the discussion. 



SENSITIVENESS AND THE INDEX OF RELATIVE 

 VARIATION. 



The sensitiveness of capacity to variation of 

 form ratio is indicated graphically by the incli- 

 nation of the logarithmic locus C=f(R). As 

 that locus is a curve, the sensitiveness varies 

 with R. The form of the curve, as mentioned 

 in the last paragraph, is determined by the ex- 

 ponent m, and the steepness (see fig. 42) of its 

 legs varies directly with m. The exponent is 

 thus a general index of sensitiveness. 



FIGURE 42. Logarithmic plots of C-& 2 (l Z-\ ^, corresponding 



to the same value of 6 2 and />, but different values of m. 



The logarithmic equivalent of (54) is 



log (7= log & 2 + log (1 - aR) +mlogR 

 Differentiating, we have 



d log C=*d log (l-aR)+md log R 

 Dividing by d log R, we have 



dlog C d log (1 - aR) 

 d log R ~ d log R 



Making substitutions from 

 dlog C 



d log (1 - aR) 



-adR 

 " 1 - aR 



and 



and reducing, we have 



dR 



aR 



(60) 



