RELATION OF CAPACITY TO VELOCITY. 



157 



A preliminary remark applies to all. For 

 each4grade of d6bris and width of channel, 

 and for each specific assumption of a constant 

 discharge, slope, or depth, there is necessarily 

 a competent mean velocity, below which no 

 traction takes place. The conception of such 

 a competent velocity has underlain all the 

 discussions of competent slope, competent dis- 

 charge, competent fineness, and competent 

 form ratio. A broad analogy therefore points 

 to the propriety of formulating the capacity- 

 velocity relation as other relations of capacity 

 have been formulated. And the inference 

 from analogy finds support in logarithmic 

 plots of C=f( V m ) under each of the three above- 

 mentioned conditions. It may fairly be as- 

 sumed, therefore, that the index of relative 

 variation for capacity and velocity itself varies 

 with velocity, being relatively small for high 

 velocities, being relatively large for low veloci- 

 ties, and becoming indefinitely large as com- 

 petent velocity is approached. 



For the purposes of this chapter, however, 

 it has seemed best to employ a simpler method, 

 using only the synthetic index of relative 

 variation characterized by the symbol I. 

 Calling the synthetic index for the variation 

 of capacity with respect to mean velocity I v , 

 we may conveniently distinguish by I r<t , I ra 

 and Ivd the values associated severally with 

 the special cases of constant discharge, con- 

 stant slope, and constant depth. 



The computations of the index are made 

 chiefly \>j the formula 



..(80) 



log <7- log (7" 

 logF m '-logTV 



in which C' and C" are specific capacities, and 

 V m ' and V m " are the corresponding mean 

 velocities. Graphically, I v is the inclination of 

 a line connecting two points of which the coor- 

 dinates are, for the first, log C' and log V m ', and 

 for the second, log C" and log V m ". Where 

 the available data serve to place more than two 

 points on the logarithmic plot of C=f( V m ), defi- 

 nite suggestion may thereby be made that the 

 line connecting the extreme points does not 

 constitute the most probable location of the 

 chord theoretically corresponding to I Y ', and in 

 such cases a line is drawn with regard to all 

 the data, and its inclination is measured on the 

 plot. 



The subject of competent velocity, which is 

 of interest independently of the formulation of 

 capacity and velocity, will be considered at the 

 end of the chapter. 



THE SYNTHETIC INDEX WHEN DISCHARGE IS 

 CONSTANT. 



In Table 14 are 73 series of values of V m , 

 each value corresponding to a stated value of S. 

 The coordinate series in Table 12 contain values 

 of C corresponding to the same values of S. 

 From each pair of series were taken the highest 

 and lowest values of V m and the corresponding 

 values of C, and from these four quantities was 

 computed a value of I v<t . The 73 values of the 

 index are shown in Table 46, where the arrange- 

 ment is such as to exhibit the variation of the 

 values with respect to discharge. 



TABLE 46. Values of lyq, the synthetic index of relative 

 variation for capacity in relation to mean velocity, when 

 discharge is constant. 



The values which lie in any horizontal lino 

 agree as to all conditions except discharge. On 

 comparing the columns for discharges 0.093 and 

 0.182 ft. 3 /sec., it is seen there are six lines 

 carrying values in both columns. The means 

 of these values are 4.80 and 4.21, the greater 

 mean belonging with the smaller discharge. In 

 the column for discharge 0.182 are nine values 

 coordinate with values hi the column for dis- 

 charge 0.363, and the means for the two groups 



