192 



TRANSPORTATION OF DEBRIS BY RUNNING WATER. 



fore had to theoretic considerations alone, and 

 the conclusions reached can claim no direct 

 support from the tabulated values of exponents 

 and competence constants. 



The conclusion (105) that the competence 

 constants collectively vary inversely with ca- 

 pacity is based on the fact that individually 

 they vary inversely with things which promote 

 capacity, namely, slope, discharge, and fine- 

 ness. Let us now assume that those three con- 

 ditions remain constant and consider the effect 

 of varying form ratio. Initially let the form 

 ratio be so small that the bed velocity is com- 

 petent; a, K, and <f> are severally equal to the 

 competent values of 8, Q, and F. Now change 

 width and depth so as to increase the form 

 ratio and capacity becomes finite. That ca- 

 pacity may be finite, a, K, and cj> must be less 

 than S, Q, and F, and as the latter have not 

 changed the competence constants have been 

 reduced by the increase of form ratio. By 

 parity of reasoning it can be shown that if the 

 initial form ratio be so large as to make the 

 bed velocity competent a reduction of form 

 ratio will cause a reduction of a, K, and <. 

 Somewhere between the two form ratios of 

 competence lies p, the form ratio of maximum 

 capacity, and between the same limits lie mini- 

 mum values of the competence constants. 



The greater the capacity induced by adjust- 

 ment of form ratio, the greater the reduction 

 of slope, for example, necessary to reduce 

 capacity to zero, and as this reduction varies 

 directly with the depression of a below the 

 initial value of S, it follows that the minimum 

 value of a (and similarly of and </>) coincides 

 with the maximum of capacity. 



The conclusion that the competence con- 

 stants vary inversely with capacity is therefore 

 true for the case in which changes in capacity 

 are caused by changes in form ratio. It can 

 be shown also that the exponents, n, o, and p, 

 follow the same law. 



The extension of this principle to the domain 

 of form ratio gives assurance that the conclu- 

 sions embodied in equations and propositions 

 (99) to (108), conclusions which were reached 

 from phenomena of slope, discharge, and fine- 

 ness, are not vitiated by the traversing phe- 

 nomena of form ratio. 



The function in the form-ratio factor of (109) 

 being characterized by a maximum, the varia- 

 tions of parameters with respect to form ratio 



are characterized by a minimum. This laia 

 may be so combined with those of (93), (94) 

 (61), and (62) as to yield the following sys 

 tern of equations for the trends of changes in 

 parameters consequent on changes in the foia 

 independent variables of equation (109): 



P = 



n =f, 

 o =/ 



P =/7 



=/. 



P, R) 

 , P, R) 



..(110) 



In the development of the form-ratio factor 

 of equations (58) and (109), detailed in Chap- 

 ter IV, the factor first appeared as (1 - nE)R m , 

 the quantity a being a numerical coefficient in- 

 troduced to represent the resistance to the cur- 

 rent occasioned by the sides of the channel. It 



was afterward shown that<r = -^ - - and that 



m+ 1 p 



form of coefficient was substituted. These re- 

 lations show that either a or m varies in value 

 with the character and amount of the resist- 

 ance by the channel sides; and, in point of fact, 

 both do. Nor is that control restricted to the 

 parameters of form ratio. Lateral resistance 

 affects also, and in comparable degree, the para- 

 meters of slope, discharge, fineness, and ca- 

 pacity. The sides of the laboratory channels 

 were vertical and were of wood, planed and 

 painted. Had they been smoother or rougher, 

 or had they been inclined, the whole system of 

 values given by the experiments would have 

 been different. There is no reason, however 

 to question that they would have yielded the 

 same qualitative results. 



DUTY AND EFFICIENCY. 



The discussions of duty and efficiency in 

 Chapters III and V give reason for the belief 

 that the variations of either quantity in rela- 

 tion to controlling conditions may advanta- 

 geously be expressed by an equation identical 

 in form with (109). Such an equation would 

 not be interconvertible with (109), nor would 

 an equation for duty be the exact equivalent 

 of one for efficiency. By the aid of reasonable 

 assumptions the parameters of either equation 

 might be derived from the parameters of an- 

 other, but the results of computations by the 

 several equations would not be strictly com- 

 patible. These discordances may be demon- 

 strated as properties of the algebraic forms. 



