72 



TRANSPORTATION OF DEBRIS BY RUNNING WATER. 



with a power of discharge, approximately the 

 0.5 power. It varies inversely with a some- 

 what smaller power of the bulk fineness of 

 de'bris. It varies inversely with the range of 

 fineness of the grades of debris, this variation 

 serving to qualify the preceding. It varies 

 with width of channel, the variation including 

 a minimum. Without attempting to give 

 definite symbolic expression to these laws of 

 variation, they were applied to the practical 

 problem, and by a series of adjustments the 

 skeleton of values of a in Table 8 was developed 

 into a system covering the whole range of 

 experimental conditions. That system is pre- 

 sented in Table 1 1 . 



TABLE 11. Values of a, in per cent of slope, as adjustedfor 

 use in interpolation equations of the form C=bi (S a) n . 



INTERPOLATION. 



The values of a having been assigned, the 

 data of Table 4 were once more plotted on 

 logarithmic paper, the ordinates again repre- 

 senting load or capacity, and the abscissas rep- 

 resenting S a, or observed slope less the con- 

 stant slope a. As in the preliminary plotting 

 for inspection, the primary data were (1) the 

 estimates of load from the quantity of de'bris 

 delivered at the end of the trough and (2) the 

 associated slopes of the bed of debris, and 

 accessory data were added with distinctive 



notation. The secondary data were used 

 chiefly to indicate the relative precision of the 

 primary data, the primary having greater 

 weight when agreeing more closely with the 

 secondary. The illustrative plot, figure 26, 

 shows only the primary data. 



The next step was to draw through and 

 among the observational points the best rep- 

 resentative straight line. It is a property of 

 the logarithmic plot that its distances repre- 

 sent ratios, and the scale of ratios is every- 

 where the same. Similar errors of observa- 

 tional positions are shown by similar distances 

 in all parts of the plot, provided the errors are 

 considered as fractional parts of the plotted 

 quantities. Each observational point should 

 be given the same influence in determining the 



o * 

 & 



FIGURE 26. Illustration of the method used to adjust values of capacity, 

 in relation to slope, by means of a logarithmic plot of observed values 

 of capacity in relation to slope minus a. 



representative line, provided the (fractional) 

 probable errors of the observations are the 

 same. In this case, however, the fractional 

 probable errors for low capacities and slopes 

 are much greater than for high capacities, and 

 the best representative lines can not be drawn 

 without consideration of weights. Adjust- 

 ment by the least-squares method was con- 

 sidered and experiments were tried, but the 

 labor entailed by the necessity of using weights 

 was not thought to be warranted by the 

 quality of the data. The following simpler 

 and less rigorous method was employed : 



A group of observational points correspond- 

 ing to the highest slopes was selected by in- 

 spection of the plot, and its center of gravity 

 was computed by a graphic method. In the 



