166 



TRANSPORTATION OF DEBRIS BY RUNNING WATER. 



least two interpretations may be considered 

 (1) that the actual variation is usually inverse 

 but under some conditions is direct; (2) that 

 the law of inverse variation is general and that 

 the apparent exceptions are due entirely to the 

 imperfection of the data. The second inter- 

 pretation accords the better with the analogies 

 afforded by other branches of the subject and 

 is provisionally accepted. 



The interpretation directs attention to the 

 general discordance of the computed values of 

 Idr, and a suggestion may be made as to the 

 cause of that discordance. First, the measure- 

 ment of depth was the most difficult of the 

 direct measurements performed in the labora- 

 tory, and it was peculiarly subject to possi- 

 bilities of systematic error. Second, the range 

 of slopes through which depth could be ob- 

 served was much less than in the case of 

 capacity, and this made the work of adjust- 

 ment less satisfactory. Third, in the compu- 

 tation of this particular index the depth data 

 enter twice, and they enter in such way that 

 their errors cumulate instead of canceling. 



In Table 58 the relations of I& to various 

 conditions are shown by the comparison of 

 corresponding means. 1& is seen to vary 

 directly with mean velocity, and it may fairly 

 be inferred to vary inversely with fineness, but 

 the data as to width of channel are contradic- 

 tory and inconclusive. 



The control of the index by mean velocity 

 affords information as to its relation to slope 

 and discharge. Mean velocity varies directly 

 with both slope and discharge, and it does not 

 change without corresponding change in at 

 least one of these factors. It follows that the 

 index, which varies directly with mean velocity, 

 also varies directly with at least one of the 

 factors slope and discharge. In the groups of 

 data from which values of the index were com- 

 puted, increase of velocity was associated 

 either with increase of slope or with increase of 

 both slope and discharge but in no case with 

 decrease of slope or discharge. It seems, there- 

 fore, proper to infer that I& varies in magni- 

 tude directly with both slope and discharge. 

 In this respect it stands as an exception among 

 the indexes of relative variation connected 

 with traction. All other species of 7 and also i 

 vary inversely with slope and discharge. As a 

 check on this exceptional result, certain prob- 



able errors have been computed. In the upper 

 division of Tablo- 58 ' -0.19, the mean of 12 

 values of 7^ with *a velocity of 2 ft./sec., is 

 compared with 0.41, the corresponding mean 

 for a velocity of 3 ft./sec. The difference 

 between these means, 0.22, has a probable 

 error of 0.13. A similar difference, appear- 

 ing in the comparison of indexes for velocities 

 of 3 and 4 ft./sec., is -0.24 0.18. As the 

 two differences give testimony of the same 

 tenor, their joint evidence is stronger than 

 that of cither separately, so that the discussion 

 of the residuals leaves the presumption in favor 

 of the conclusion that this particular index 

 constitutes a real exception in its relation to 

 slope and discharge. 



TABLE 58. Partial means based on Table 57, illustrating the 

 control of I^v, by mean velocity, fineness and width. 



THE THREE CONDITIONS COMPARED. 



When depth is increased without change of 

 slope (or width or grade of d6bris), its increase 

 is effected by increase of discharge, with the 

 result that capacity is increased, so that ca- 

 pacity is an increasing function of depth. 

 When depth is increased without change of 

 discharge, its increase is effected by reducing 

 slope, with the result that capacity is reduced, 

 so that capacity is a decreasing function of 

 depth. When depth is increased without 

 change of velocity, its increase requires increase 



