88 



TRANSPORTATION OF DEBRIS BY RUNNING WATER. 



ditions, it is evident that as the slope is flat- 

 tened, the current is slowed and the depth in- 

 creased, and that zero slope gives infinite depth. 

 The theoretic curve, therefore, has an asymp- 

 tote in the vertical line corresponding to zero 

 slope. Similarly the depth is reduced by in- 

 crease in slope but remains finite for very high 

 slopes. The theoretic curve has as asymptote 

 a horizontal line corresponding, exactly or 

 appioximately, with the (horizontal) line of 

 zero depth. These asymptotes relate the curve 

 d=f(S) to the hyperbola. In the same figure, 

 but with use of a different scale, are a series of 

 crosses which show the same observations as 

 they appear when plotted on logarithmic sec- 

 tion paper. They are the plot of observations 

 ou log d =/, (log 8) ; and their arrangement sug- 

 gests that the representative line may be 

 straight. 



Plots were made to show the relations of 

 depth observations to associated capacity, and 

 these also suggest the hyperbola and the 

 straight line. If, however, the locus of d =/ u ( f) 

 is a hyperbola it differs materially from that of 

 d=f(S), for as depth increases and current 

 slackens, capacity becomes zero when current 

 reaches the value of competence, and depth 

 is not then infinite. So the line of zero capac- 

 ity is not an asymptote to the curve. 



It is to be observed also that the represen- 

 tative lines for log d =/, (log S) and log d =/ 

 Gog (7) can not both be straight, for if they were 

 there could be derived from them a straight 

 line representing log C=f v (log S), and it has 

 already been found that that line is curved. It 

 is, indeed, probable that neither of the loga- 

 rithmic plots involving depth is straight; yet 

 there are cogent practical reasons for assuming 

 one or the other to be so. One reason is the 

 very great convenience of the straight-line func- 

 tion, and another that the relatively small range 

 of the depth data renders impracticable such a 

 discussion of the curvature of logarithmic loci 

 as was made in the case of capacity versus 

 slope. Accordingly, the most orderly plots of 

 log d =f L (log S) and log d =/ re (log C) were com- 

 pared with special reference to curvature. For 

 the function of log C the plots were found to 

 indicate curvature in one direction only, while 

 for the function of log S they indicated slight 

 curvatures in both directions, with the straight 

 line as an approximate mean. The function 



log d =/! (log S) was accordingly selected for the 

 adjustment of the depth observations; and the 

 representative line on the logarithmic plot was 

 assumed to be straight. In accordance with 

 that assumption, the adopted formula of inter- 

 polation was 



*--- - <21) 



with its logarithmic equivalent, 



log d = log b' n t log S. 



.(22) 



The coefficient 6' is a depth, the depth cor- 

 responding to a slope of 1 per cent. 



The data were all plotted on logarithmic 

 section paper. The notation was made to 

 distinguish depth measurements made at a 

 single point by means of the gage (see p. 25) 

 from those based on full profiles of water sur- 

 face and bed of debris. The former were used 

 exclusively in the drawing of the representative 

 lines, but not because they were regarded as 

 of higher authority. It was thought best not 

 to combine data which in certain cases were 

 known to be incongruous ; and the gage observa- 

 tions covered the whole range of the work, 

 while the profiles did not. The measurements 

 by profile were used in criticising the measure- 

 ments by gage, and they determined the accept- 

 ance or rejection of certain gage measurements. 

 It was noted that in some series of observa- 

 tions the depth measurements by the two 

 methods were in close accord, while in others 

 there was a large systematic difference; and 

 certain series were rejected because of such 

 large differences. 



The plots were made to distinguish also the 

 observations associated with different modes 

 of traction the dune, smooth, and antidune 

 modes. The observations with the smooth 

 mode were assumed to be best, as a class; and 

 these, together with the observations connected 

 with the transitional phases of traction, were 

 used to fix an initial point of each representa- 

 tive line. The direction of the line was then 

 adjusted by eye estimate to make it repre- 

 sentative of all the points of the particular 

 series. In this adjustment consideration was 

 given to the conditions affecting the measure- 

 ments of both depths and slopes. 



The lines were first drawn for those obser- 

 vational series which appeared from the plots 

 to be most harmonious, and for these there was 



