94 



TRANSPORTATION OF DEBRIS BY RUNNING WATER. 



PRECISION. 



Probable errors were computed in the manner 

 described on page 73. The largest error 

 found for a series of adjusted values of depth 

 is 5.5 per cent; and the arithmetical mean 

 of the 66 determinations is 1.70 per cent. 

 The mean of 66 determinations of the probable 

 error of an observation of depth is 5.47 per 

 cent; and the arithmetical mean of the (710) 

 residuals, or differences (irrespective of sign) 

 between observed and adjusted depths, is 

 6.65 per cent. 



The residuals of any series, computed as 

 fractional parts of the quantity measured, are 

 greater for small depths than for large. Com- 

 puted in fractions of a foot, they do not vary 

 notably with depth. 



The average value of the depths to which 

 these measures of precision pertain is 0.176 

 foot. Using this factor to convert the pre- 

 ceding average percentage errors into (ap- 

 proximately) equivalent linear errors, we have, 

 for the average of probable errors of adjusted 

 depths 0.003 foot; for the average of probable 

 errors of observed depths 0.010 foot, and 

 for the average residual 0.011 foot. 



The errors thus estimated include (1) the 

 strictly accidental errors of observation, (2) the 

 more or less systematic influences exerted on 

 the measurements by the diverse modes of 

 traction, and (3) errors occasioned by the as- 

 sumptions underlying the method of reduc- 

 tion; but they do not include such systematic 

 errors as are shown by comparing gage meas- 

 urements with profile measurements. (See p. 

 26.) In connection with the descriptions of 

 methods of measurement it is stated that the 

 probable error of a single measurement by gage 

 was computed from the comparison of values 

 by two methods (in all cases where both are 

 used), and on the assumption that the measures 

 by the profile method are relatively accurate. 

 This computation gave a value smaller than 

 0.010 foot, namely, 0.007 foot. The groups 

 of measurements to which the two estimates of 

 precision apply are not the same, although they 

 overlap. One group includes all gage measure- 

 ments of 66 series, their number being 710; the 

 other group includes all those gage measure- 

 ments of the 92 series which were checked by 

 profile measurements, their number being 118. 



By comparing these results with those re- 

 ported on page 74, it is seen that the computed 

 probable errors for depth are smaller than those 

 for capacity. It is nevertheless believed that 

 the measurements of load were more precise 

 than those of depth. The discrepancy is ac- 

 counted for by considering, first, that the esti- 

 mates of precision, instead of applying simply to 

 the measurements of load and depth, apply to 

 capacity for load as a function of slope and to 

 depth as a function of slope; and, second, that 

 the relation of load to slope 'is subject to con- 

 tinual rhythmic variation, while the relation of 

 depth to slope is little influenced by that varia- 

 tion. 



MEAN VELOCITY. 



As the discharge, Q, equals the product of the 

 sectional area of the current, wd, by the mean 

 velocity, V m , we have 



F ra = 4- -(23) 



For each observational series, Q and w are con- 

 stant and 



T7 .1 



.(24) 



Substituting for d its value in the interpolation 

 equation (21), and remembering that 6' is a 

 constant for each series, we obtain 



.(25) 



By means of (23) a value of mean velocity was 

 computed for each adjusted value of d, and 

 these values are given in Table 14. They in- 

 volve all the assumptions of the formula for the 

 reduction of the depth observations and have 

 the same fractional measures of precision. 



FORM RATIO. 



The adjusted values of d were used also for 

 the computation of the form ratio, R, which is 

 the quotient of the depth of the current by its 

 width; and a value of R is tabulated with each 

 value of d. Within each observational series 

 the form ratios are proportional to the depths, 

 and they have the same measures of precision. 



GRAPHIC COMPUTATION. 



All these computations were made by graphic 

 methods. For each observational series a plot 



