74 



TRANSPORTATION OF DEBRIS BY RUNNING WATER. 



Table 12 contains the computed probable 

 errors for all those series, 82 in number, in 

 which the number of observations is not less 

 than 4. For a smaller number of observations 

 it was possible so to frame adjusting equations 

 as to leave no residuals; and although this was 

 not done, the propriety of applying the com- 

 putation to such cases was not evident. 



The arithmetical mean of the 82 values of 

 probable error for adjusted capacities is 2.50 

 per cent. The corresponding mean for observed 

 capacities is 8. 80 per cent. Another measure 

 of the precision of the observations, collectively, 

 is the arithmetical mean, irrespective of sign, 

 of all the residuals (966 in number) involved 

 in the 82 series. That mean is 11.5 per cent. 



While these estimates of precision are com- 

 puted specifically for capacities or loads, the 

 sources of the errors are not restricted to the 

 observations of loads but include also the obser- 

 vations of slope and discharge. With use of 

 the same diagrams it would be possible to con- 

 sider slope as a function of capacity and com- 

 pute the probable errors of slope determina- 

 tions. The relations are such that, within any 

 series, the probable error of slope, considered 

 as a percentage, is less than the probable error 

 of capacity. 



The residuals within a series are, as a rule, 

 relatively great for the lower slopes; but this 

 is true only when the residuals are considered 

 as fractional parts of the capacity values to 

 which they pertain. If the residuals be 

 measured in the unit of capacity, then they are 

 relatively great for the higher slopes. The 

 gradation of precision in relation to slope was 

 discussed with some care, and an elaborate 

 system of weights was prepared for the ad- 

 justed values of capacity. While these weights 

 were of service in connection with various 

 combinations afterward made, it has not 

 seemed necessary to include them in the 

 printed tables. It is to be understood, how- 

 ever, that the probable errors associated with 

 each series in Table ]2 apply to the values of 

 capacity as a group, the probable errors (in 

 per cent) of the smaller values being relatively 

 great and those of the larger values relatively 

 small. 



The residuals represent chiefly the errors of 

 observation, but they include also errors intro- 

 duced in the process of adjustment. To what- 



ever extent the formula of adjustment mis- 

 represents the actual relation between the 

 variables, to whatever extent the assigned 

 values of a are inaccurate, and to whatever 

 extent the representative lines of the plot are 

 misplaced, factors of error are introduced which 

 tend to increase the residuals. The estimates 

 of probable error of observations are therefore 

 larger than they would be if the methods of 

 adjustment were perfect. 



The observations giving large percentage 

 residuals were, as a class, treated in the graphic 

 adjustments as of relatively low weight; but in 

 the computations of probable error they were 

 treated as of equal weight. The estimates of 

 probable error, and their average values, are 

 larger than they would be if computed with 

 regard to weights. The influence of this factor 

 can not be definitely evaluated, but ratios 

 brought out in the discussion of residuals for 

 the assignment of weights indicate that it has 

 some importance. It is thought that the 

 average probable error of the adjusted values 

 may be as low as 2.0 per cent, and that of the 

 observations as low as 7.0 per cent. 



On the other hand, the computations take 

 account only of the discrepancies revealed by 

 comparing observations of the same series and 

 do not cover such discrepancies as exist be- 

 tween one series and another. The estimates 

 of probable error are smaller than they would 

 be if both classes of discrepancies were included. 

 This matter receives further consideration in 

 Chapters V and VI. 



DUTY. 



The duty of water traction, as defined by the 

 units adopted for this paper, is the capacity in 

 grams per second for each cubic foot per second 



Q 



of discharge, and its formula is U=-^.. The 



V 



duty corresponding to each adjusted capacity 

 has been computed, and the values appear in 

 Table 12. 



It is some tunes desirable to treat duty as the 

 ratio which the mass of the load, or capacity, 

 bears to the mass of the carrier. To obtain the 

 value of duty as a ratio of masses the corre- 

 sponding value of U should be divided by 

 28,350, the number of grams in a cubic foot of 

 water. 



