BELATION OF CAPACITY TO DISCHARGE. 141 



TABLE 34. -Adjustment of system of values of K for a channel width of 1 foot and slopes of 1.0 and 1.2 per cent. 



In making these adjustments due account 

 was taken of the fact that not all the data used 

 pertain to the same slope. Allowance for slope 

 difference was made in plotting the values of 

 K to adjust for fineness, and again in applying 

 the results of that adjustment. The adjust- 

 ments for slope are based on the following 

 consideration. 



By use of the data in Table 12, values of K 

 pertaining to the same grade and width but 

 differing as to slope were compared. In one 

 case K appeared to increase with increase of 

 slope, but in most cases it appeared to decrease. 



FIGPBE 49. Logarithmic plot otn-f(D'). 



Little weight, however, was given to such 

 determinations, because it was evident that 

 they were controlled by the values of a, which 

 had been somewhat arbitrarily assigned in the 

 first adjustment of the observations. A more 

 satisfactory indication was obtained by reason- 

 ing based on the relative sensitiveness of 

 capacity to the several controls of slope and 

 discharge. The index of relative variation of 

 capacity with respect to slope being i 1; we may 

 designate the corresponding index with respect 

 to discharge by i s . As capacity varies simul- 

 taneously with S' 1 and Q' 3 it is evident that for 

 any particular value of capacity 5" 1 and Q'' 



vary inversely, 

 yields 



The proportion 



S 7 ' 



oc 



From a discussion of the relations of the two 

 indexes, to be found on a later page of the 

 present chapter, it appears that the average 

 ratio of i, to i 3 is 1.36 but that the ratio varies 

 somewhat with conditions. For the condition 



1 



=0 it is assumed to be 1.3; or K oc 



S 1 - 3 



While considerable uncertainty attaches to 

 the numerical relations, there can be no ques- 

 tion that K varies directly with width and 

 inversely with fineness and slope. As any 

 change in width affects the form ratio in the 

 opposite sense, we may write 



,fy ........ -.-(66) 



A system of values of having been thus 

 arranged, logarithmic plots were made of the 

 capacities of Table 32, in relation to Q K, and 

 straight lines were drawn representing equa- 

 tions of the type of (64). In cases where the 

 plotted points did not fall well in line, recourse 

 was had to weights based on the probable 

 errors recorded in Table 12. The lines thus 

 drawn gave, by their inclinations and intersec- 

 tions, the values of the other parameters, o and 

 6 3 , as well as the adjusted values of capacity, 

 C T , all of which are presented in Table 32. The 

 plots for data of grades (E) and (F) and for 

 two of the three groups under grade (G) 

 exhibited so great irregularities that it was 

 decided not to use them. 



The similarity of equation (64) to equation 

 ( 10) makes it unnecessary to repeat the reason- 

 ing by which (38) was educed, and the equation 

 for the index of relative variation for capacity 

 in relation to discharge may be written directly: 



I** 



3 ~ Q i 



(67) 



By its use the values of i 3 in Table 32 were 

 computed. 



