60 



TRANSPORTATION OF DEBRIS BY RUNNING WATER. 



distinguished experiments made with use of 

 the contractor from those with free outfall. 



A critical study of these plots led to several 

 conclusions. (1) Judged by the internal evi- 

 dence of regularity and irregularity, the ob- 

 servations of de'bris slope and water slope 

 have about the same quality. (2) Similarly, 

 the observations of load fed and load deliv- 

 ered have about the same quality. (3) As 

 already stated, the law of sequence is not 

 discovered to change in passing from one 

 process of traction to another; the assump- 

 tion of a continuous law is the best practic- 

 able. (4) Except for very low slopes, the 

 results obtained with the use of the con- 

 tractor do not differ widely from those with- 

 out it. (5) There are differences between 

 bodies of experimental data obtained at dif- 

 ferent stages of the work differences of un- 

 certain source but presumably connected with 

 modes of manipulation which make it desir- 

 able to treat such bodies separately whenever 

 practicable. (6) The best representative line 

 is not straight but curved, and its curvature is 

 always in one direction. 



If the representative line were straight, func- 

 tion (3) would have the form 



log C=log A+nlog S. 



-(4) 



in which A is a constant capacity and n a ratio ; 

 and function (2) would have the form 



C^AS". 



-(5) 



On some of the plots the observational points 

 are too irregular to afford trustworthy evidence 

 of curvature. On most of them the indicated 

 curvature is slight. From inspection of the 

 data during the progress of the experiments it 

 was thought that the true representative line 

 would prove to be straight, in which case the 

 accurate determination of two points on the 

 line would suffice; and some of the experimental 

 work was adjusted to that theory. The series 

 giving data for but two points on the logarith- 

 mic plot of course furnished no evidence as to 

 curvature of the representative line. 



For all those cases in which the position of 

 the best representative line could be inferred, 

 with close approximation, from the arrange- 

 ment of the observational points, a satisfactory 

 adjustment could be made by simply drawing 



the line and then converting its series of posi- 

 tions into figures; but this procedure would 

 afford no control for the curvature of repre- 

 sentative lines in cases where the observa- 

 tional points were few or inharmonious. In 

 order to make the stronger series support the 

 weaker, the plan was adopted of (1) connecting 

 the lines with a formula of interpolation, (2) 

 correlating the constants of the formula with 

 conditions of experimentation, and thus (3) 

 giving deductive control to the lines of the 

 weaker series. 



SELECTION OF AN INTERPOLATION FORMULA. 



The best interpolation formula is one which 

 embodies the true theory of the relation to 

 which the observations pertain. In the present 

 case the true theory is not known, but there 

 are certain conditions which a theory must 

 satisfy, and these may be used as criteria in 

 the selection of a form for empiric formulas 

 of interpolation. Subject to these criteria, 

 the form selected should serve to minimize the 

 discrepancies between observed and adjusted 

 values. 



For the study of the character of the curve 

 to represent best the logarithmic plot of ob- 

 servations, the data for de'bris of grade (G) 

 were selected. The experiments furnishing 

 those data were all performed by one method, 

 the method using moderate contraction of the 

 trough at outfall; and for that grade the ap- 

 parent curvature of the logarithmic graph is 

 greater than for most others. The data were 

 first plotted (on logarithmic paper) in groups, 

 each group containing the data for three graphs 

 which pertain to the same width of channel 

 but to different discharges. It was assumed 

 that the three graphs, if correctly drawn, would 

 constitute a system, the one for the medium 

 discharge being intermediate in form and posi- 

 tion between the other two; and in drawing 

 them on this assumption the forms were mu- 

 tually adjusted. Then a rearrangement was 

 made which grouped together data agreeing as 

 to discharge but differing as to width of chan- 

 nel, and further adjustment was made. Selec- 

 tion was finally made of the graph for w = 0.66 

 foot and Q = 0.734 ft. 3 /sec., and this was drawn, 

 through the selected positions, by the aid of a 

 flexible ruler. Thus the curve in figure 19 is 

 a graphic generalization not only from its 



