RELATION OF CAPACITY TO DEPTH. 



167 



of discharge accompanied by diminution of 

 slope; and as these changes have opposite in- 

 fluences on capacity, it is not evident a priori 

 whether capacity will be enlarged or reduced. 

 The experimental data show that it is slightly 

 reduced, so that capacity is a decreasing func- 

 tion of depth. 



When depth is reduced without change of 

 slope, and the reduction is continued progres- 

 sively, a stage is eventually reached in which the 

 velocity is no longer competent for traction. 

 It is probable, therefore, that, under this con- 

 dition, an approximate formula for C=f(d) 

 might involve a depth constant and be similar 

 to the formula (64) used for 0=f(Q). 



Reduction of depth without change of dis- 

 charge involves increase of velocity, and it is 

 evident that competence does not lie in that 

 direction. But increase of depth involves re- 

 duction of velocity and leads eventually to a 

 competent velocity. The limiting depth cor- 

 responding to competence is therefore a great 

 depth instead of a small one. As mean ve- 

 locity now varies inversely with depth, a coor- 

 dinate formula might take the form 



c=. 



B being used as a general constant and d as a 

 constant depth. 



When depth is reduced without change of 

 mean velocity, the efficiency of the mean 

 velocity is enhanced and competence is not 

 approached. When depth is increased, the 

 efficiency of the unchanged mean velocity is 

 diminished and a (large) competent depth 

 may, under some conditions, be realized. 



To show the numerical relations of the in- 

 dexes by which these three capacity-depth func- 

 tions are severally characterized, certain means 

 are assembled in Table 59. It was not found 

 possible to procure values of the different in- 

 dexes representing closely the same conditions, 

 and what was done was to derive for each index 

 the mean of all determinations made for each 

 particular grade of debris. 



The arrangement by grades points again to 

 the fact that all the indexes vary inversely with 

 fineness of debris, but the rate of variation is in 

 fact' somewhat greater than these series of 

 values suggest. The observations on the 

 coarser grades were made with steeper average 



slopes and larger average discharges than those 

 on the finer grades, and the effect of high slopes 

 and large discharges (except in case of !&} is 

 to reduce the indexes of relative variation. 



TABLE 59. Comparison of synthetic, indexes of relative vari- 

 ation for capacity and depth, under the several conditions of 

 constant discharge, constant slope, and constant mean 

 velocity. 



From the general means at the bottom of the 

 table it appears that, for the range of laboratory 

 conditions, capacity is 2.34 times as sensitive 

 to the control of depth when the limiting con- 

 dition is constant discharge as when the limit- 

 ing condition is constant slope, and about nine 

 times as sensitive as when the limiting condi- 

 tion is constant mean velocity. 



One of the results of the discussion is to 

 emphasize the importance, when considering 

 the relation of tractional work to depth, of 

 sharply discriminating the conditions under 

 which depth is regarded as a variable. 



So far as the variations of the capacity-depth 

 indexes admit of formulation in the symbols 

 used for other indexes, 



= ft 



P) 



-.(84) 

 (84a) 

 (84b) 



COMPARISON OF CONTROLS BY SLOPE, DIS- 

 CHARGE, MEAN VELOCITY, AND DEPTH. 



In Chapter V the general sensitiveness of 

 capacity to slope is compared with that of 

 capacity to discharge by means of coordinate 

 values of the exponent i. In Chapter VII the 

 sensitiveness to slope is compared with sensi- 

 tiveness to mean velocity (with discharge con- 

 stant) by means of coordinate values of /. 

 The two methods of comparison are so far 



