190 



TRANSPORTATION OF DEBRIS BY RUNNING WATER. 



and are thought worthy of examination, al- 

 though the data at hand do not suffice for their 

 testing. 



.... (106) 



(91) 



The equation under discussion, 



is an expression of relation between capacity 

 and three of its controls, namely, slope (S), 

 discharge (Q), and fineness (F). It involves 

 seven parameters, of which six are functions 

 of the independent variables, S, Q, and F. It 

 is thus a bare framework, and the completion 

 of the structure calls for the replacement of the 

 six parameters by their values in terms of the 

 variables. The laws contained in the equa- 

 tions and propositions numbered (92) to (98), 

 with their corollaries, (99) to (105), are con- 

 tributions toward the completion of the struc- 

 ture, but they are largely of the nature of re- 

 strictions. They impose conditions to be sat- 

 isfied by the perfected equation. 



Some of the conditions are already embodied 

 in the form of (91), and with reference to such 

 conditions it is important that the origin of that 

 form be not overlooked. The form assumes 

 that the three competence constants are the 

 values of the corresponding variables when ca- 

 pacity is zero, whereas their identification with 

 those values is by no means complete. The 

 definition and recognition of the status of com- 

 petence are so obstructed by the complicating 

 conditions of nonhomogeneous de'bris and dune 

 rhythm that no more can be asserted than an 

 indefinitely representative relation. For most 

 purposes, however, we are little concerned with 

 conditions in the immediate neighborhood of the 

 competence limit, so that this qualification is of 

 small practical moment. Outside of the neigh- 

 borhood of competence the support of the form 

 is empiric ; it has served well as a scheme for the 

 marshaling of the observations. The support 

 is qualified, in turn, by the fact that the obser- 

 vations are not of such harmony and precision 

 as to discriminate nicely among formulas of 

 adjustment. In view of these qualifications, 

 the possibility has been recognized that some 

 of the laws above enumerated might emanate 

 from the form of the equation and have no 

 other basis; and in view of this possibility the 



foundations of each conclusion have been scru- 

 tinized. I believe that all the inferred laws, 

 from (92) to (105), are essentially inductive. 



It is easy to understand that any construc- 

 tive effort which should hang all supplementary 

 conditions on the framework of (91) would re- 

 sult in a formula so unwieldly as to be useless. 

 It is a matter of faith with me that if our data 

 were so precise as to substitute definite quanti- 

 tative relations for the fascicle of trends and 

 indefinite parallelisms they have actually fur- 

 nished, some way would be found leading from 

 complexity to simplicity. I am not without 

 hope that the presentation here made may sug- 

 gest to the mechanist, familiar with the aspects 

 of solved problems of similar difficulty, a ra- 

 tional theory under which the data may advan- 

 tageously be recombined. 



In an effort to discover unities among the 

 complexities of the capacity relations, equation 

 (91) was given the following form: 



The three factors making the second division 

 of the second member, being independent of the 

 units of measurement, seemed well adapted to 

 the expression of comprehensive harmonies, if 

 such exist. 



The following negations were demonstrated: 



SO F 



The quantities -, -, and -7 are not equal, nor 



are the ratios between them constant. 



The quantities - - 1, ^ - 1, and -, - 1 are not 



(f A. G) 



equal, nor are the ratios between them con- 

 stant. 



The quantities f--l), (~-l), and 



/F V 



( -7 1 ) are not equal, nor are the ratios be- 



tween them constant. 



It was also found that the symmetric factors 

 in equation (91), namely, (S e)", (Q-K), and 

 (F<f>)P, are not equal, nor are the ratios be- 

 tween them constant. . 



THE FORM-RATIO FACTOR. 



In its relation to form ratio capacity has 

 two zeros, one corresponding to a high ratio, 

 the other to a low. Each of these corresponds 

 also to a competent bed velocity, so that into 



