188 



TRANSPORTATION OF DEBRIS BY RUNNING WATER. 



and do not vary with S, are controlled by Q 

 and F. As they both vary inversely with Q 

 and with F, it follows that they vary directly 

 one with the other. It is evidently true in 

 general that each exponent varies directly with 

 the associated competence constant. 



--(95) 



?-/l(0J 



With a constant, the variation of S a is 

 determined by variation of S. Considered as 

 additive, their variations are identical; but if 

 we regard the changes as ratios, the changes in 



(S a) are proportional to o_ and those in S to 



. The ratio between these fractions, which is 



o 



t; , is a measure of the sensitiveness of (S a) 



o a 



to changes in S. It is evident that as S in- 

 creases the sensitiveness diminishes. As ca- 

 pacity varies with a power of (S a), the sensi- 

 tiveness of capacity to slope becomes less as the 

 slope increases. 



Any change in S causes, according to (92) 

 and (93), a change of opposite character in K 

 and o. When S is increased, K and o are re- 

 duced. As the sensitiveness of (Q ) to 



change in Q is measured by Q_ K , it is evident 



that the reduction of K lessens the sensitiveness. 

 This has the effect also of lessening the sensi- 

 tiveness of capacity to discharge ; and that sen- 

 sitiveness is further lessened by the reduction 

 of o. Parity of reasoning shows that increase 

 of slope lessens the sensitiveness of capacity to 

 fineness, so that the effect of increasing slope 

 is to reduce the sensitiveness of capacity to 

 all three of its controlling factors. It is evi- 

 dent also that a similar result would be reached 

 if the analysis began by assuming an increase 

 of discharge or fineness. 



It is a general principle that any change in 

 one of the control factors, slope, discharge, and 

 fineness, causing capacity to increase, has the 

 effect also of making capacity less sensitive to 

 changes in each and all of the control factors; 

 and the inverse proposition is of course equally 

 true. The statement being phrased to include 

 both, the sensitiveness of capacity to the three 

 controlling conditions varies inversely with 

 capacity. 



The term "sensitiveness," as used in the pre- 

 ceding paragraphs, is equivalent to the more 

 specific "index to relative variation," for 

 which the symbol i has been used; and by 

 reference to various studies of the control of 

 the index by conditions it may be seen that 

 the entire scope of the general principle just 

 stated has been covered by essentially in- 

 ductive generalizations. From equations (39), 

 (68), and (79), 



.(96) 



*<-/($,& F) 



This checking of deductive by inductive results 

 helps to establish the second and third equa- 

 tions of (94), which were inferred from anal- 

 ogies. 



Very little is known of the nature of the 

 functions in (93) to (96), beyond the fact 

 that those of (95) are increasing and the 

 others decreasing. Deductive reasoning has 

 not been successfully applied, and induction 

 has escaped the entanglement of accidental 

 errors in only a single instance and to a limited 

 extent. The exceptional instance is that 

 represented by the first equation of group (96). 

 The symbols being translated into words, that 

 equation reads: The index of relative varia- 

 tion for capacity in relation to slope varies 

 inversely with slope, with discharge, and with 

 fineness. There are in fact three distinct 

 propositions, and each of these might be 

 expressed by a separate equation. As to the 

 first proposition, that the index varies in- 

 versely with slope, it was found, inductively, 

 that the rate at which it varies with slope is 

 itself a decreasing function of slope and also 

 of discharge and fineness; and knowledge of 

 similar character was gained as to the second 

 and third propositions -(pp. 104-108). Repre- 

 senting by diis, di ig , and di iF the rates of varia- 

 tion of the index in relation to slope, discharge, 

 and fineness, severally, we have 



(97) 



di 1F =f 3 (F) 



These fragmentary determinations are all of 

 one tenor, and in view of the remarkable 

 symmetries already discovered among the ele- 



