128 



TRANSPORTATION OF DEBRIS BY RUNNING WATER. 



ously varied in such way as to maintain the 

 slope unchanged, and the changes in depth are 

 to be compared with the changes in load or 

 capacity. To make the results strictly coordi- 

 nate with those for the control of capacity by 

 width, we should deal only with channels so 

 broad and shallow that the lateral portions 

 (fig. 39) are small in comparison with the medial 

 portions, but this is not practicable. It is pos- 

 sible, however, to minimize the influence of 

 lateral retardation by selecting groups of data 

 in which the form ratio is small. 



Table 25 contains data pertaining to debris 

 of grade (C), a channel width of 0.66 foot, and 

 a slope of 1 per cent. The values of capacity 

 are taken from Table 12 and the values of 

 depth from Table 14. It appears by inspection 

 that the capacity increases with the depth. 

 On plotting the pairs of values on logarithmic 

 section paper, it is found that they may be 

 represented approximately by a straight line. 

 The examination of many such plots showed 

 that the most accurate representative line has 

 a gentle curvature, but for the present purposes 

 it suffices to assume that the line is straight. 

 That is to say, it is found to be approximately 

 true, and the assumption is made, that C varies 

 with some power of d, or 



Coed" 



(50) 



In the particular case m 1 = 1.90, and other 

 values of m 1 are shown in Table 26. 



TABLE 25. Values of capacity and depth for currents trans- 

 porting debris of grade (C), when the width is 0.66 foot and 

 the slope 1 .0 per cent. 



TABLE 26. Valuesofm l in Coc rf" 1 ', when slope is constant. 



Comparison of the tabulated values of m l 

 shows that they have considerable range and 



that their variations in magnitude are defi- 

 nitely related to several conditions. The sensi- 

 tiveness of capacity to variation of depth (when 

 width and slope are constant) is greater as the 

 slope is less, is greater as the fineness is less, 

 and, with one exception, is greater as the width 

 is greater: 



m, =/(, / (51) 



As w is by postulate constant, and asdwR, 

 it follows that dxR. R may therefore be sub- 

 stituted for d in proportion (50), giving 



CccR" 



.(52) 



This is an expression for the relation of 

 capacity to form ratio, so far as that relation 

 depends on variation of depth. It is the com- 

 plement of proportion (49) on page 126. 



CAPACITY AND FORM RATIO. 



We now return to the assumption of con- 

 stant discharge. Having obtained an approxi- 

 mate expression for the law of capacity's varia- 

 tion in response to change of width and a 

 coordinate expression for variation on response 

 to change of depth, we next inquire how these 

 may be combined into an expression for the 

 response of capacity to simultaneous changes 

 of width and depth, when those changes are of 

 such character as to leave discharge constant. 



If the functions (49) and (52) be assumed 

 to be independent, their simple and direct 

 combination gives " 



C oc 



R 



This formula, as rnay readily be shown, satis- 

 fies an important condition by providing for a 

 maximum value of C, but the assumption in- 

 volved in its construction is not strictly war- 



is based 



ranted. The law embodied in 



H 

 o 



on the constancy of d and may not be valid 

 when d is made variable, while the law em- 

 bodied in R 1 may not be valid when w is made 

 variable. 



Viewing the matter from another side, we 

 may say that the difficulty would not exist if 

 the values of oc and m l depended only on fine- 

 ness, slope, and discharge and were independent 

 of R; but they are in fact functions of R. (We 

 have already seen that m, is an increasing 

 function of w, and the variation of with d 



