REVIEW OF CONTROLS OF CAPACITY. 



187 



ranges for o and p correspond approximately 

 with the middle part of the range for n. 



It will be recalled that while the forms of the 

 equations involving a,K, and <j> were based on the 

 conception of competence, it was not found 

 possible to correlate those parameters strictly 

 with competent slope, discharge, and fineness. 

 The correlations were obstructed by phenomena 

 of dune rhythms and of diversified fineness and 

 could not be completed, but the forms of equa- 

 tion were found to be well adapted to the com- 

 bined expression of observational data above 

 the region of competence. Their relation to 

 competence is not absolute but intimate, and 

 it is so intimate that certain properties of the 

 parameters may properly be inferred from the 

 physical theory of competence. 



When the swiftest velocity on the bed is 

 barely able to move d6bris, there is a threefold 

 condition of competence. For the particular 

 discharge and fineness, the slope is competent; 

 for the particular slope and fineness, the dis- 

 charge is competent; for the particular slope 

 and discharge, the fineness is competent. The 

 conditions of competence for the three factors 

 controlling capacity are thus not only similar 

 but simultaneous and coincident. Neither 

 factor can sink alone to the limiting level of 

 competence, but the three arrive together. 

 This is an important principle and lies at the 

 foundation of the systematic interdependence 

 of parameters and variables. 



.(92) 



In equation (91) the quantities 

 (S-a), (Q-K), and (F-& be- 

 come zero simultaneously. When 

 S = a, then also Q = K, and F= <f> ; 

 and vice versa. 



As capacity varies directly with (S a), 

 (Q K~), and (F<f>), it is also true that each of 

 these varies directly with capacity. Any 

 change of condition which affects capacity 

 affects those three quantities in the same sense. 

 For example, suppose discharge to be increased. 



This not only increases (Q K) and thereby 

 increases capacity, but it also increases (S a) 

 and (F<f>). One mode of expressing this fact 

 is to say that capacity measures the remoteness 

 of each controlling factor from the initial status 

 of competence, and all recede or approach 

 together. 



Let us now make a more definite assumption, 

 that discharge is increased while slope and 

 fineness remain the same. The resulting in- 

 crease of (S a), as S is unchanged, implies a 

 diminution of a; and the increase of (Fcf>) 

 implies a diminution of <f>. That is, a and <f> 

 vary inversely with discharge. Parallel reason- 

 ing shows that a and K vary inversely with 

 fineness, and that K and <j> vary inversely with 

 slope. 



These relations are here developed deduc- 

 tively from the theory of competence. They 

 have been developed inductively from the ob- 

 servational data, for equations (26), (66), and 

 (77) include 



No way has been found in which to study the 

 exponents deductively. The only evidences of 

 order discovered by comparison of observa- 

 tional data pertain to n, which has been found 

 (equation 27) to vary inversely with discharge 

 and fineness. The question whether o and p 

 follow similar trends could not be answered by 

 the adjusted data because of the cumulative 

 effect of accidental errors. There is, however, 

 considerable force in analogic reasoning, based 

 not only on equations (93), but on other ele- 

 ments of symmetry in the relations of capacity 

 to the several factors elements to be noted 

 later. The state of the evidence may be ex- 

 pressed by 



.(94) 



n =/,(& 

 [o =/($, 

 [P =fy (S, 



It is convenient to have a name for the group 

 of constants designated by Greek letters, and 

 as they define the conditions of competence, 

 they may be called competence constants. 



The exponent n and the associated compe- 

 tence constant a, as they vary with Q and F 



