CHAPTER III. RELATION OF CAPACITY TO SLOPE. 



INTRODUCTION. 



A series of chapters, beginning with this one, 

 are given to the discussion of the observational 

 data. The discussions make use of the ad- 

 justed values of capacity for stream traction, 

 slope of stream bed, and depth of current, 

 with their derivatives, contained in Tables 12 

 and 14. Associated with those adjusted values 

 are certain grades of transported material, or 

 degrees of fineness, certain widths of channel, 

 and certain discharges of the transporting 

 stream. The leading subjects of discussion are 

 the relations of capacity to slope, discharge, 

 fineness, and form ratio, but consideration is 

 also given to the relations of capacity to depth 

 and velocity, and to the relations which duty, 

 efficiency, and depth bear to various condi- 

 tions. The discussion is essentially empiric, its 

 course being guided in small degree only by 

 theoretic considerations. 



The treatment of the relation of capacity to 

 slope first views it as conditioned by channels 

 of fixed width, and then as subject to the rela- 

 tively ideal condition of fixed form ratio. 



IN CHANNELS OF FIXED WIDTH. 



THE CONDITIONS. 



In each observational series the width of 

 channel was constant, and so also were the 

 discharge and the grade of debris constituting 

 the load. As the load was changed, the slope 

 responded; velocity responded to change of 

 slope; and with variation of velocity went varia- 

 tion of depth. The ratio of depth to width, or 

 the form ratio, was therefore a variable; so 

 that the stream which dragged a large load 

 down a steep slope differed in form, and to that 

 extent in type, from the stream which moved 

 a small load along a gentle slope. In a few 

 cases it is possible so to combine data from 

 different series as to discover the relation of 

 capacity to slope for streams which have simi- 

 lar cross sections; and these will be examined 

 in another place; but the principal discussion 

 relates to streams with constant width and 

 variable form ratio. 



96 



THE SIGMA FORMULA. 1 



Those properties of the formula 



(10) 



which determined its adoption for the reduction 

 of the observational data to a more orderly 

 system led also to the consideration of its 

 availability as an empiric formula for the gen- 

 eral relation of the stream's capacity for trac- 

 tion to the slope of its bed. With a view to 

 this second use, the specific values of & 1; <j, and 

 n derived for the purpose of the reduction 

 values which are recorded in Table 15 were 

 arranged and combined in various ways, in 

 order to discover, if possible, definite relations 

 to the several conditions in accordance with 

 which the experiments were varied. It has 

 already been noted (p. 71) that the critical 

 slope, a, varies inversely with fineness of 

 debris, with discharge, and (probably) with 

 range of fineness within a grade, and that it 

 varies inversely with width of trough when 

 that width is relatively small, but directly 

 with width when width is relatively large. 



(26) 



written to express these relations in symbols, 

 introduces H to designate range of fineness and 

 distinguishes trend of function by means of 

 accents. As the notation by accents will be 

 frequently used, its definition may be made 

 explicit. Where the function is direct, or 

 increasing, its value increasing with the in- 

 crease of the independent variable, the symbol 

 of the variable is given the acute accent ('). 

 Where the function is inverse, or decreasing, its 

 value decreasing with increase of the variable, 

 the grave accent ( v ) is used. For a maximum 

 function, first increasing and then decreasing 

 ( A ) is used, and for a minimum function ("). 

 The discussion of the values of b t showed (1 ) 

 that they vary directly and in marked degree, 

 but irregularly, with F 2 , (2) that they vary 



i Since these lines were penned 1 have discovered that this title 

 duplicates one in the field of higher mathematics. Nevertheless I 

 retain it because of its mnemonic convenience. The two fields of 

 application are so distinct that serious confusion will not be occasioned. 



